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HARVARD COLLEGE

SCIENCE CENTER
LIBRARY

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M¥ ELEMENTS

GEOMETRY.

BY SEBA SMITH.

NEW YORK: •

GEORGE P. PUTNAM, 155 BROADWAY.
LONDON : RICHARD BENTLEY.

1850.

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s \c>r,5o

Entered according to act of Congreu, in the year 1650, by

SEBA SMITH,

in the Clerk's office of the Distiict Coart of the United Statei, for the Southern Diitrict of

New York.

S. W. BENEDICT,

ITEacOTTtEa AZn> PEIIfTBB, 16 SPEUCB AT.

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THREE PARTS.

PART FIRST.
THE PHILOSOPHY OF GEOMETRY.

, PART SECOND.
DEMONSTRATIONS IN GEOMETRY.

PART THIRD.

HARMONIES OF GEOMETRY.

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PREFACE.

A FEW remarks of a personal and prefatory character, it may be pro-
per in this place to address to the reader. Some thirty years ago,
while in college, I had paid some little attention to Geometry, having
gone with my class through three or four of the fifteen books of Euclid's
Elements. But the knowledge obtained, even of the few books read,
was somewhat. superficial ; and pursuits in after-life not requiring exer-
cise in the science, thirty years disuse had suffered every demonstration
and almost every principle derived from Euclid to fad^ from the mind.
About two years and a half ago, John A. Parker, Esq., of New York,
a gentleman whose life had mainly been passed in mercantile and com-
mercial pursuits, applied to me, as an old acquaintance and friend, to
examine some original papers, in which he claimed to have solved the
most celebrated problem in mathematics, the quadrature of the circle.
He had several years before discovered what he believed to be the true
ratio of the circumference of a circle to its diameter, and had, during
the pterval, made repeated endeavors to have his papers examiaed, and
his positions acknowledged by mathematicians. But he had found very
few to give them even a slight examination, and none to concede the
truth of his conclusions.

I took up his papers and read them with great care. I was at once
much impressed witii the boldness, strength, and originality of his rea-
soiting, and finally convinced of the truth of his solution of that remarka-*
ble problem j which had long since been pronounced by mathematicians •
and learned societies as an impossibility. I became strongly interested

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PREFACE.

in the whole subject of Geometry. I took down mj M. Euclid, and
brushed off the dust of thirty years ; I went to the bookstalls and book-
stores and searched for different works on Geometry, till I had picked
up fifteen or twenty, which I examined, some partially, some thoroughly,
but all with a zest. Mr. Parker's reasonings and demonstrations led
to the conclusion that the circumference of a circle was not a line coin-
ciding with the perimeter of the circle, as geometers had hitherto con-
sidered it, but a line wholly and perfectly outside of the circle, and
consequently that it must be a magnitude entirely distinct from the
circle, and must have breadth. In addition to these reasonings of Mr.
Parker, which were entirely original with him, I found upon research
that the learned and acute mathematician. Dr. Barrow, had come to
the decided conclusion that mathematical number always expressed
magnitude. And I also found some remarks of Aristotle, which seemed
to lead to the same conclusion.

Here a great question intensely pressed upon my mind, — ^if mathema-
tical number always represents magnitude, mathematical Unes represent-
ed by numbers are magnitvdes^ and must have breadth ; and if they
have breadth, is it not possible by some geometrical demonstration to
prove what that breadth is 7 The thought pursued me day and night,
for it would not leave me even in my sleeping hours. I set myself down
steadily to the task for a year and a half, and the present volume is the
result. The breadth of mathematical Unes i^ not only perfecdy estab-
lished, but the whole subject of Geometry is simplified, cleared from
obscurities and difficulties, and placed, as it were, on a new foundation.
But let it not be supposed that the new laws and principles of Geome-
try, developed and demonstrated in this work, have been derived from
hypothesis and theoretical reasoning. They rest not upon so unsafe a
basis. They were reached by the pure methods of the mductivo philo-
sophy of Bacon. I went to work upon original diagrams with the Greek
rule and compasses in my hand, and spent a long and laborious year in
diggmg out mj facts, I examined such varieties of geometrical forms

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PREFACE. vii

as the imagination conld Buggest, and as patient thonght and labor were
able to inrest^te. I measured, computed and compared diameters,
areas, and circumferences of plane figures, and diameters, solidities, and
surfi&ces of solid figures, at the same time examining^ and comparing the
roots of all these various quantities ; and from ilie facts tiius gradually
collected and arranged, tiie general laws and principles of the science
presented themselves clearly to view, and demanded the acknowledg-
ment of their high prerogatives.

It is proper here also to remark, that a work made up almost entirely
of new geometrical principles, and embracing a great variety of original
arithmetical calculations, all prepared by one individual, without being
revised by others, cannot reasonably be expected to be entirely free
from errors. Some slip of tiie pen, some oversight of the eye, some
figure missed, or some typographical error unperceived or uncorrected,
may very probably be found to mar, in some degree, the work. But if
nothing shall be discovered to invalidate the principles laid down, as
they are intended to be explained, the Author trusts that minor errors,
should such appear, will be charitably and cheerfdlly overlooked by the
reader.

The work of Mr. Parker on the Quadrature of the Circle is in pre-
paration for the press, and is expected soon to be published. It is
therefore unnecessary, and would be hardly appropriate here for me to
enter into any elaborate consideration of it. I have already expressed
my conviction of the truth of his ratio of the circumference of a circle to
its diameter. That ratio is 20612 for circumference, and 6561 for di-
ameter, which is the smallest expreseion of the perfect ratio that can be
^ven in whole numbers. This ratio, in a circle whose diameter is 1,
gives for circumference 3. 141594-)-. The approximate ratio obtained
by geometers, and generally received as correct, is 3. 141592+ • Mr.
Parker, it is seen, differs from this in the sixth decimal figure. And
he shows conclusively that the method of geometers in obtaining this
approximate ratio, which is by means of inscribed and circumscribed

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PREFACE.

polygons, necessarily leads to an error in the sixth decimal place. To
test the truth of his perfect ratio, Mr. Parker has, with a bold concep-
tion and singular originality, applied it to some of the astronomical cir
cles, and obtained remarkable and startling results, indicating that in
the motions and periods of the heavenly bodies there are perfect mathe-
matical relations much more wonderful and extensive than have yet
been understood.

Hippocrates squared a portion of a circle more than two thousand
years ago, in the figure called ^^ the lune of Hippocrates ;" and I have
myself squared other portions of a circle by similar methods. And I
think when the reader has seen, in the demonstrations and principles
exhibited in the following pages, what perfect harmony prevails between
the circle and all rectilineal figures, and how the circle controls all rec-
tineal figures by one simple and uniform law, he will have no doubt
that the whole circle may be perfectly squared.

SEBA SMITH.

New York, My 4, 1860.

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PART FIRST.

THE PHILOSOPHY OF GEOMETRY.

SECTION I.

IMPORTANCE OF THE SCIENCE, AND ITS DIFFICULTIES.

" The invention of forms," says Lord Bacon, the
great founder of inductive philosophy, " is of all other
parts of knowledge the worthiest to be sought, if it
be possible to be found." And in the same connec-
tion he adds: "As for the possibility, they are ill-
discoverers that think there is no land when they can
see nothing but sea."

Plato also regBrdedforms as the true object of know-
ledge; but in the judgment of Bacon he "lost the
real fruit of his opinion, by considering of forms as
absolutely abstracted from matter ;^^ by which means
he was led into theological speculations, " wherewith
all his natural philosophy is infected."

In the opinion of Pythagoras, the study of the ma-
thematics, including geometry, was "the first step
toward wisdom." The pupils in his school first be-
came mathematicians ; and after they had made suffi-
cient progress in geometrical science, they were con-
ducted to the study of nature, the investigation of

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1<^ PART FIRST.

primary principles, and the consideration of the. at-
tributes of Deity.

Plato arrived at such a reverence for geometry, that
he had inscribed over the door of his academy where
he taught philosophy, " Let no one who is ignorant of
geometry enter here/* And when his opinion was
asked concerning the probable employment of Deity,
he is said to have replied, " He geometrizes contin-
ually;** by which he undoubtedly meant that the
great Author of nature established and governs the
universe by geometrical laws.

Also the learned and pious Dr. Barrow held geome-
try in such estimation, that he considered the contem-
plation of it as not unworthy of the Deity ; and in
publishing an edition of the works of ApoUonius, he
inscribed it with the wqrds, " God himself geome-
trizes. O Lord, how great a geometer art thou !"

And in testimony of the truth and immutability of
the principles of geometry, Aristotle, the great mas-
ter of ancient philosophies, declared that " the poles
(Of the world will be sooner removed out of their
places, and the fabric of nature destroyed, than the
foundations of geometry fail, or its conclusions be
convinced of falsity."

And yet this first and most important of the scien-
ces — most, important, because lying at the foundation
of all other sciences — so clear in its principles, so
certain in its conclusions, so venerable for its anti-
quity, hoary with the lapse of thousands of years, and
honored in every age by the earnest investigations of
the master-minds of men — this grand fabric of geo-
metry, so beautiful in its proportions, and so magnifi-

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THE PHILOSOPHY OF GEOMETRY. 11

cent in extent, rests in part on a false foundation.
One of the corner-stones upon which it was first
erected was given with false dimensions, and must be
removed, and its place supplied by the true comer-
stone, before the structure can be made perfect
throughout, and present an unbroken harmony in all
its relations.

To remove that false corner-stone and supply the
true, is the object intended by the present treatise.
Should I be met at the threshold by the incredulous
world, and reproachfully or satirically asked, in the
words of Paul, " Who is sufficient for these things ?"
I shall reply only in the humble spirit of the same
apostle, when he declared that " God hath chosen the
foolish things of the world to confound the wise ; and
God hath chosen the weak things of the world to con-
found the things which are mighty.''

The error, which I allege to exist in the fundamen-
tal principles of geometry, is embodied in the first
definitions of the science, as given by Euclid, and as
adopted and followed in the many hundreds of works
written on the subject for the last two thousand
years. Euclid, and I believe all other geometers
who have written hitherto, take their stand upon
these definitions, viz. :

" A line is length without breadth ;" and " a sur-
face is length and breadth without thickness."

I meet these definitions at once, and declare that
every mathematical line has a breadth, as definite, as
measurable, and as clearly demonstrable, as its length;
and that every mathematical surface has a thickness as

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12 PART FIRST.

definite, as measurable, and as clearly demonstrable,
as its length or breadth.

Will it be answered here, that the demonstrations
given by geometers are clearly and unquestionably
true, and therefore, if there be an error in one or two
of their definitions or assumed principles it affects not
their conclusions, and must be a matter of but little
consequence ? Though such an answer could scarce-
ly be expected to come from a mind imbued with
sound principles of philosophy, it may still be worth
while to dwell upon it for a moment. Astronomers
got along very well before Galileo's time, upon the
hypothesis that the earth was the center of the solar
system. They calculated eclipses on that hypothesis,
and the eclipses came out right, and verified their
calculations. Was it, therefore, a matter of but little
consequence, that their whole system was based on
a false foundation ? The conclusions which they
could reach from that foundation were clearly and
unquestionably true; but there are many vast and
important truths in the science of astronomy, which
they never could have reached till their fundamental
error was discovered, and the earth allowed to re-
volve- about the sun.

So there are many important truths in geometry
demonstrated in this work, and many more doubtless
yet to be demonstrated, which never could have
been reached till the true nature of linfes and surfaces
vras discovered and their proper quantities demon-
strated.

It is certainly unphilosophical to admit that a truth,
lying at the foundation of any science, can be unim-

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THE PHILOSOPHY OF GEOMETRY. 13

portant. Professor Playfair has well and forcibly said :
" The, truths of geometry are all necessarily connected
with one another, and the system of such truths can
never be rightly explained, unless that connection be
accurately traced wherever it exists. It is upon this,
that the beauty and peculiar excellence of the mathe-
matical sciences depend. It is this, which, by pre-
venting any one truth from lieing single and insulated,
connects the different parts so firmly, that they must
all stand, or fall together."

If it is a truth, therefore, that mathematical lines
have a definite and measurable breadth, and that ma-
thematical surfaces have a definite and measurable
thickness, that truth must unquestionably be of great
importance to the science of geometry and to all ma-
thematics. And the want of a knowledge of this
truth, I think, has hitherto prevented the true relation
between numbers, magnitudes, and forms, from being
clearly and properly understood. It is not strange
therefore, that while a fundamental principle in ma-
thematics remained shrouded in darkness, the profes-
sors of that science should have been led into a thou-
sand laborious and useless speculations, upon questions
in which that unknown prinaiple was necessarily in-
volved. Indeed from these causes, the mathematical
sciences, like a very luxuriant vine left without prun-
ing, have run out into immense quantities of foliage,
bearing comparatively but little fruit. This state of
things has become a reproach to mathematics. The
writer of an able article in the Edinburgh Encyclopedia
remarks, *' The luxuriance of modern analytical spe-
culation is arrived at such a point as to startle the most

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14 PART FIRST.

industrious, and to render an equally perfect know-
ledge of all its parts, no longer attainable by one indivi-
dual'^ And a writer of equal ability in the London
Encyclopedia pursues the subject in a vigorous and
satirical vein as follows. " Let the mathematics be
encouraged and patronized. Let t)iem be cultivated
to the fullest extent, even with considerable waste of
mental power and loss of money, to discover the
north-west passage in the polar regions of fluxionary
creation ; to find out some new calculus, whether dif-
ferential or integral. Were there no probable, or
even possible results, as to such a mixed, impure,
vulgar entity, as utility, in contending with practica-
bility, and penetrating to a high mathematical latitude
of discovery ; were it merely for the sake of the con-
tention of mind, or to have it proved how far the alge-
braic analyst can go up out of sight in some new-
invented infinitesimal balloon ; in short if the mathe-
matical progression were a thousand miles ahead of
any practical purpose or advantage whatever, we
would not be for terminating its career. We have
spare hands enough, and spare heads too ; and as all
cannot find useful employment, it is better perhaps
that they should be out of the way of idleness and
mischief, by digging mathematical holes and filling
them up again, or in perpetual motion to discover new
methods of contention of mind, new calculuses, new
analyses, new fluxions, new infinitesimals, to rival and
supersede the old ones, ad infinitum. Only let us
^ know, if possible, what the mathematics are about,
and wherein their infinite quantity of excellence con-
sists."

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THE PHILOSOPHY OF GEOMETRY. 15

On this point I may remark further, that quite re-
cently in some of the most valuable of the English
scientific magazines, I have observed articles from
able professors and distinguished mathematicians,
gravely discussing the question of the relative value
of three times nothings and twice nothing; 0x3, and
>c 2. I have not the magazines before me, but I
recollect that certain quantities had been carried
through an algebraical process, duly invested with
the signs of plus and minus, in which it was argued
that the zeros used in the operation became invested
also with a positive and negative character, and that
the two zeros in the equation, which resulted from
the operation, had acquired some sort of an infinites-
imal value, and reaJiy had a ratio to each other as three
to two.

When learned professors find themselves driven to
such conclusions by their received principles of a sci-
ence, it would seem to be high time for them to go
back to first principles, and see whether there is not
something wrong in the very foundations of that
science.

But thus it must ever be, while men attempt to
reason about nothing instead o{ something; whether it
be by an algebraical process to establish the value of
a cipher, which is absolutely without value, or by
geometrical demonstrations to fix the value of lines
which are assumed to be entirely and absolutely
without breadth or thickness. The value and the re-
sults of such labors are well portrayed by Lord Bacon,
in reference to the "Schoolmen," before his time,
who, he says, " shut up in the cells of monasteries and

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16 PART FIRST.

colleges, and knowing little history, either of nature
or time, did out of no great quantity of matter, and
infinite agitation of wit, spin out unto us those labo-
rious webs of learning which are extant in their
books. For the wit and mind of man, if it work
upon matter, which is the contemplation of the crea-
tures of God, worketh according to the sttcff^y and is
limited thereby. But if it work upon itself, as the
spider worketh his web, then it is endless, and brings
forth indeed cobwebs of learning, admirable for the
fineness of thread and work, but of no substance or
profit."

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THE PHILOSOPHY OF GEOMETRY. IT

SECTION II.

THE COMMON VIEW OF GEOMETRY.

Geometers have always felt embarrassed by their
definitions of lines and surfaces. The explanations
and illustrations of these definitions by Robert Simson,
the distinguished Professor of Mathematics in the Uni-
versity of Glasgow, are the most elaborate, and gene-
rally deemed the clearest and mofst satisfactory of any
that have been given. They aore copied and adopted
by Professor Play&ir, and many other geometers.
And yet these very illustrations of Professor Simson,
in which he endeavors to prove that a line has no
breadth, and that a surface has no thickness, embody
a fallacy which entirely destroys the validity of hi&
conclusions. I am ]K)t willing to make this remark,,
however, without adding, that I think the very high
reputation acquired by Professor Simeon was most
eminently deserved, and that science is greatly in-
debted to him for the zeal with which he devoted a
good portion of his life to recover and restore to purity
some of the lost and mutilated works of the Greek
geometers.

I will endeavor to explain the reasoning of Profes^
sor Simson, upon these definitions as briefly as possi-
ble. He remarks, " It is necessary to consider a solid,
that is, a magnitude which has length, breadth and

thickness, in order to understand aright the definitions
2

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18 PART FIRST.

of a point, line, and superficies; for these ail arise
from a solid, and exist in it. The boundary, or boun-
daries, which contain a solid, are called superficies ;
or the boundary which is cmnmon to two solids which
are contiguous, or which divides one solid into two
contiguous parts, is called a superficies/' To illus-
trate the argument more briefly, if not more clearly,
than is done by Professor Simson's diagram, we will
suppose two perfect cubes, or dice, with feces geome-
trically contiguous, or in perfect contact. We will
call one cube A, and the other B. The boundary,
where these cubes meet, is common to them both,
and, says the Professor, ** is therefore in the one, as
well as in the other solid, called a superficies, [or sur-
face,] and has no thickness." For, proceeds the ar-
gument, if the surface, which is thus common to the
two solids A and B, have any thickness, it must be a
part of the thickness of A, or a part of B, or a part of
each of them. But it cannot be a part of A; because
if A be removed, the surface of B still remains as it
was. Nor can this common surface be a part of B ;
because if B be removed, the surface of A remains as it
was. Therefore he comes to the conclusion, that the
two solids being geometrically in contact, and the
common surface between them being no part of either
solid, it can have no thickness.

By precisely the same course of reasoning, he ar-
gues that a line has no breadth. If we suppose a sur-
face or plane to be divided by a geometrical line, and
call the two parts of the plane C and D, he considers
the geometrical line the common boundary alike of
each part, and afiirms that this line can have no

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THE PHILOSOPHY OF GEOMETRY. W

breadth. For, says he, if the line have any breadth^
it must be a part of the plane C, or of the plane D, or
a part of each of them. But it cannot be a part of C,
because if C be removed the line still remains as the
boundary of D. Nor can it be a part of D, because if
D be removed, the line still remains as the boundary
of G. Therefore, considering the edges of the two
planes as in geometrical contact, and the boundary
line between them being no part of either plane, he
comes to the conclusion that the line is absolutely
without breadth.

To see the fallacy of this conclusion, or rather of the
premises on which it rests, it seems only to be neces-
sary to look at both solids at the same time, the one
which is removed as well as the one which remains
unmoved. The argument is, if A be removed from
B, the common surface between them still remains as
it was, the surface of B. But what becomes of poor
A in this predicament? Is it sent off into the world
without any surface to its back ? Have we not
as good reason to say that the common surface
goes with A, as to say it remains with B ? It cannot
of course be pretended that A, after being removed,
has no surface at all upon that side where it had been
in contact with B. And if the two solids while in
contact, have but one common surface between them,
how shall it be decided which of the two retains the
surface when they are separated ? Has B, in conse-
quence of remaining unmoved, a stronger attraction
for the common surface, than is possessed by A, which
is removed ? To prevent the possibility of any ima-
ginary advantage being possessed by B in this respect,

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PART FIRST.

let the two solids be separated by removing both
equally, the one to the right and the other to the left.
Which of the two solids, in that cdse, tshM retain the
surface that was inherent in them both, and common
to them both while in contact ? Perhaps it may then
be said that the surface is divided between them,
each taking its part. If so, and it be still contended
that their common surface, when in contact, had no
thickness^ it would seem to follow that the surface of
each, when separated, must be half the thickness of no-
thing.

The truth is, the pi-emises in the case are all wrong,
and the conclusions, to use again the words of Bacon,
can only lead to those " cobwebs of learning, admira-
ble for the fineness of thread and work, bat of no sub-
stance or profit."

Now, I entirely deny that the two solids, when in
contact, have but one surface which is common to
them both. On the contrary, I affinn, that each has
its own surface entirely distinct and separate from the
other ; and not only distinct and separate, but that the
surface of each occupies a position and place entirely
different from the other; and also that each surface
has a thickness as definite and as measumble as the
solid itself. But before proceeding to describe these
surfaces and define their position and value, it may be
profitable to pass to a consideration of other topics,
which are requisite to afford a clear and distinct view
of the whole matten

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THE PHILOSOPHY OF GEOMETRY. 21

SECTION III.

THE TRUE VIEW OF GEOMETRY.

I w0iri«D call Geometry, the science of magnitude,
which measures and compares extension and forms.
And in order to start aright, and scatter light and not
darkness in our path, it is important for the mind to
obtain a clear view of the nature of the extensions and
magnitudes which we measure and compare.

The great philosopher of antiquity, whom Dr. Bar*
row calls " the most subtle and very learned Aris-^
totle/' makes this very important remark, viz. —
•* Mathematicians do neither want nor use infinite
magnitude, but take as much as they please, when they
are minded to terminate it.*' And this is really the
true foundation of all geometry. Our measures of ex-
tension or magnitude are not, and cannot be, drawn
from infinity. They have no proportion or relation to
space infinitely extended or infinitely diminished.
Our standards of measure are made by ourselves, from
material substances which can be reached and com-
prehended by our senses, and are consequently ^mV^.
And by these measures, we never measure positive
magnitude, or positive space, as a definite portion of
ali space, but only measure and compare magnitudes
or spaces, that are relative and proportional to the
standards which we have ourselves made and adopted.

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82 PART FIRST.

All that is measured in space, may be called quantity
or magnitude, and is always and only relative to the
standard which we have assumed for the measurement.
Our measures are noted and designated by numbers ;
and the standard of every kind of measure must of
necessity be unity, or one.

In making or determining any standard of measure,
in the words of Aristotle, " we take as much magni-
tude from the infinite space as we please, and then
terminate it/* For instance, we take the common
breadth of a man's thumb, and agree to call it one inch.
And twelve of these being about equal to the length
of a man's foot, we agree to call twelve inches one foot.
And in this, and similar ways, all measures are estab-
lished. When a standard of measure is established,
we can apply it to any quantity or magnitude which
we can reach or comprehend, and tell whether such
magnitude is less or greater than our measure, or how
many times our measure must be repeated to equal
the magnitude. But it must not be forgotten that
this gives us no knowledge of the nature or value of
positive quantity in infinite space. The human intel-
lect is not capable of fathoming or comprehending
positive or absolute quantity, ^o finite quantity is of
itself great or small, but only so by comparison, or re-
lative to some other quantity. The mind cannot con-
ceive a magnitude so small, but there may be another
still smaller, nor yet one so large, but that a larger
one may still lie beyond it. Absolute quantity or
magnitude can- be comprehended by Him alone,
whose instrument it is;

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THE PHILOSOPHY OF GEOMETRY

" To Him, no high, no low, no great, no small,
He fills, he bounds, connects, and equals all.*'

We cannot obtain the idea of extension or form,
except from material substances. Could our minds
exist in entirely free space, void of all matter, they
could know nothing of extension or form. Infinite
space would be one uniform thing; an unbroken, in-
variable unit. It is therefore impossible that such a
thing as a line or a surface can have existence in nature,
unless it is formed of some material substance, which
occupies a portion of space. And if a line and a sur-
face in nature are of necessity formed of material sub-
stances, and of necessity occupy a portion of space,
they must with equal necessity possess breadth and
thickness ; and they must with equal necessity possess
extension in every direction from their center. Now,
the measurement of extension is precisely the object
of geometry ; and lines without breadth and surfaces
without thickness are imaginary things, of which this
perfect and exact science can take no cognizance.
How vain, therefore, are all those speculations, where
these airy nothings are attempted to be forced upon
geometry and mingled with its pure demonstrations.
And with how much force does the language of Bacon
apply here — " For the wit and mind of man, if it work
upon matter, which is the contemplation of the crea-
tures of God, worketh according to the stuf, and is
limited thereby. But if it work upon itself, as the
spider worketh his web, then it is endless.^'

I think it clear that every thing which can come
within the reach of geometry, must have extension;

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94 PART FIRST.

must have magnitude ; must occupy a portion of space ;
must have extension in every direction from its center.

Now what is the instrument, and the name of the
instrument, with which geometry always works ? The
instrument, by which it measures all magnitudes, is a
limited magnitude of a definite form ; and the nanle
of that magnitude, that instrument, is unit, or one.
One square, one cube, one circle, one sphere, one
triangle, one tetrahedron, &c., are in their nature units
of magnitude, having definite but different forms. The
square form, or rather the cubic form, is the one which
has been universally adopted, and is undoubtedly the
most convenient in practice, for the measurement of
magnitudes of all forms. And when other forms have
to be measured, they require to be geometrically de-
composed, so to speak, and reduced to the cubic form,
in order to express their value in magnitude ; that is,
to express the number of cuUc units which they con-
tain. It is seen, then, that the cubic unit is the proper
instrument of geometry, wherewith it accomplishes all
its wonderful work. The cubic unit is its starting
point, its first stepping-stone. It has already been
seen, when we put this instrument into the hands of
geometry to work with, how we fix or determine its
size or quantity. " We take as much magnitude as
we please,'* and call it one; and geometry does all the
rest. It takes the instrument given it, and applies it
in a thousand ways, to all definite magnitudes, and
among all definite forms, and returns to us an exact
account of its labors, with every thing perfectly mea-
sured, nothing remaining over and nothing falling short.
If we follow the footsteps of geometry by the light of

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THE PHILOSOPHY OF GEOMETRY. 25

these principles, it will not lead us into a dark laby-
rintli from which we cannot escape ; and while it un-
folds to us the beautiful relations and harmonies of all
forms, we shall not, like Plato, " lose the real fruits of
our labors by considering of forms as absolutely ab-
stracted from matter.''

Geometers say, there are three kinds of quantity in
geometry — lines, surfaces, and solids ; and that these
quantities are not homogeneous ; that they are differ-
ent in their nature, each having its own peculiar unit ;
and that, as quantities or magnitudes, they cannot be
measures of each other.

I say there is but one kind of quantity in geometry,
and that lines, surfaces, and solids are all of the same
nature, having identically the same unit, and of course
are always perfect measures of each other, both arith-
metically and geometrically. They are measures ol
each other in numbers, and they are measures of each
other in quantities or magnitudes.
, The nature of appoint in geometry is rightly giveft
in the books. It has position, but not magnitude. It
is not a thing which geometry recognizes as a measure
of any magnitude, or a constituent part of any magni-
tude. It forms no part of a line, or a surface, or a solid;
but is simply an index of place, or position of lines, sur-
faces, and solids. If we bisect a line, which is a posi-
tive magnitude, the place of bisection, where the two
halves of the line meet each other and are in contact,
we designate by calling it a point; th^is really accom^
plishing the poetic paradox of giving " to airy nothing
a local habitation and a nafne.^'

But when we come to lines and surfaces, geometry

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36 PART FIRST.

may be said to have already stepped upon solid ground
for it is then really on the ground of solids. A mathe-
matical line is not a filmy, airy thread, reasoned down
to infinity — to an imaginary nothing; but it is a real
magnitude, a positive quantity, used to measure and
compare positive quantities. It has already been in-
dicated that a unit is the name or representative of
any assumed magnitude to which it is applied. The
definite size of the unit may be infinitely varied, as
the magnitudes or quantities in nature or space are
infinitely various. " We take as much as we please,"'
and call it one. The unit not only represents a mag-
nitude, but it represents a magnitude of a definite
form. And, as already stated, the form universally
adopted in the world as the standard of measure is
the cubic form. Since the unit always represents a
definite magnitude — and a magnitude from its very
nature has an extension in evert/ direction from its center
— the unit is necessarily the representative of some-
thing that is extended in every direction from its cen-
ter. Therefore the unit means not only one in length,
but one also in breadth, and one in thickness. One
inch, for instance, in pure geometry or mathematics is
always one cubic inch ; but when the object in any
process is only to measure a line, or extension in one
direction, it is necessary to use in the measurement
only one dimension of the unit — that is, the linear
edge of the cube ; and this we apply along the line,
repeating it, tiU the measure is completed. And in
this operation, having no use whatever for the breadth
or thickness of the unit, geometers have &llen into the
error of regarding a line as length without breadth.

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THE PHILOSOPHY OF GEOMETRY. 27

In like manner, when the object is to measure a
surfi^^e, or extension in two directions only, length
and breadth, it is necessary to use in the measure-
ment only two dimensions of the unit — Yiz., its length
and breadth. That is, we use one face of the cube,
which is a simple square, and this we apply to the
quantity or area to be measured, a sufficient number
of times to complete the measurement. And in this
operation, having no use whatever for the thickness
of the unit, geometers have fallen into another error
— ^that of regarding a surface as length and breadth
without thickness.

But in both cases, the unit we have been using is
the representative of a magnitude, and a magnitude of
a definite form and value; and the unit never changes
its form or value, because in measuring a line we dis-
regard its breadth, nor because in measuring a surface
we disregard its thickness. In every unit of the line,
and in every unit of the surface, the perfect cube has
an implied existence; — ^that is, the unit, wherever
and however employed, possesses in its own right the
value of the perfect cube, and is capable of vindicating
its claim to that value, in every diagram and geome-
trical demonstration that can be presented.

It follows that a mathematical line is made up of a
succession of single and equal units ; and therefore a
mathematical line has always a breadth of one. Also,
that a mathematical surface is made up of a succession
of single lines, and therefore a mathematical surface
has always a thickness of one.

The^ couclusions are stated with boldness and
without hesitation, because they are abundantly ca-

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28 PART FIRST.

pable of being proved in all geometrical demonstra-
tions^ and are proved in a great variety of demonstra-
tions in the following pages. The very simplicity of
these conclusions — that all lines have a breadth of
one, and all surfaces have a thickness of one, may
strike some as an argument against their validity.
But I regard it as an argument, if any argument at all
were needed, decidedly in favor of their truth ; for
all the operations and reasons of nature, when we
once get at them, are found to be very simple. Sir
Isaac Newton, who probably looked more widely and
deeply into the works of nature than any other philo-
sopher has hitherto done, remarks that "Nature is
pleased with simplicity, and affects not the pomp of
superfluous causes."

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THE PHILOSOPHY OF GEOMETRY.

SECTION IV.

COMMENSURABLE AND INCOMMENSURABLE QUANTITIES.

Strictly speaking, there are no quantities or mag-
nitudes in nature that are incommensurable. We
can bind and tie up portiqps of magnitude or quantity
by the units we ourselves make and limit, so that
while bound by these fetters they cannot measure
each other. But all magnitudes and quantities, whe-
ther of matter or space, are in their own absolute
natures commensurable. Whether matter is infinitely
divisible, or whether possible division at last termin-
ates in ultimate particles or atoms, is a question on
which the most acute thinkers have not been agreed.
It is probably a question beyond the reach of the hu-
man intellect. Sir Isaac Newton, in speaking of the
properties of bodies or solids, remarks as follows :

" The extension, hardness, impenetrability, mobil-
ity, and vis inertUB of the whole, result from the ex-
tension, hardness, impenetrability, mobility, and vires
inerticB of the parts; and thence we conclude the least
particles of all bodies to be also all extended, and
hard, and impenetrable, and movable, and endowed
with their proper vires inertice ; and this is the foun-
dation of all philosophy. Moreover, that the divided
but contiguous particles of bodies may be separated
from one another, is matter of observation ; and, in

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PART FIRST.

the particles that remain undivided, our minds are
able to distinguish yet lesser parts, as is mathemati-
cally demonstrated. But whether the parts so distin-
guished, and not yet divided, may by the powers oj
nature be actually divided and separated from one
another, we cannot certainly determine. Yet, had we
the proof of but one experiment that any undivided
particle, in breaking a hard and solid body, suffered
a division, we might, by virtue of this rule, conclude
that the undivided as well as the divided particles
may be divided and actually separated to infinity.**

But whether matter iis infinitely divisible or not, in
either case the common measure of two quantities of
matter will be reached if the division of them be con-
tinued far enough. For if matter be infinitely divisi-
ble, there is an infinite number of divisions in which
to seek the .common measure. Or if the possible
division of matter terminates in ultimate particks,
which are indivisible, the common measure of two
quantities made up of those particles, if not reached
before, will certainly be found at last in the ultimate
particle.

Whatever may be the fact in nature with regard to
the infinite divisibility of matter, there seems to be
as much reason to believe space infinitely divisible, as
there is to believe it infinitely extended. And the
laws of geometry apply to all space as well as to all
matter. Space is both divisible and extended, be-
yond the reach (^ the human intellect. The mind
can reach no terminus in either direction. To us,
therefore, space is both infinitely divisible and infin-
itely extended. The profound and subtle Aristotle

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THE PHILOSOPHY OF GEOMETRY.

affirmed, that ''whatsoever is continued is divisible into
parts, again divisible/' By '' continued'' is here
meant, whatever has extension, as extension df mat-
ter or space, or extension of time or duration. He
also tells us that ''Plato does therefore make two
kinds of infinites, because he thinks there is an infin-
ite procedure both in augmentation and diminution."
With regard to incommensurable quantities, there-
fore, which have always been so troublesome and
perplexing to mathematicians from Euclid's time
down to the present day, it may be one step toward
getting OTer these difficulties to know that no such
quantities really exist in nature. They are the crea-
tiims of the mathematicians themselves in tying up
quantities into indivisible units, and then attempting
to measure the constituent parts of a unit by the
whcde unit, while thus bound up and indivisible ; or
attempting to m^isure a unit of one size by a unit
of another size, while both are bound hand and foot,
limited and indivisible. And here is seen the reason,
for instance, why the diagonal of any square is incom-
mensurable with the side of the square ; and why the
diagonal of a cube is incommensurable with the linear
edge of the cube. They are all relative and consti-
tuent parts of the unit, and while thus bound, they
form an indivisible whole ; — that whole is a perfect
cube, its linear edge is 1, its face is 1, and its solid-
ity or whole body is 1. And whether the positive
size or breadth of that cube equal the thousandth part
of a hair, or a thousand times the breadth of the earth,
it is still, as a unit, one indivisible thing. The dia-
gonal of one of its feces is the square root of 2, and is

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aa PART FIRST.

expressed in numbers by 1.4142+ j and the diagonal
of the cube, or of the whole unit, is the square root
of 3, which is 1.732+ ; and neither of these expres-
sions can be measured by 1, because the quantities
they represent, while bound in their present form,
are nothing but the constituent and indivisible parts
of the cube or unit; and the cube cannot be applied
to its own parts and measure itself. It cannot mea-
sure its own diagonal, or the diagonals of its faces.
Therefore these diagonals are not commensurable
with the side of the square or face, which is 1, or with
the whole cube, which is 1. But though, while inhe-
rent in the cube, and parts of it, they cannot be com-
mensurable with the cube, yet the very limitation
imposed upon them, while thus bound, makes them
the roots of other quantities, which are commensur-
able with the cube. The diagonal of the cube is the
square root, or root of a square, whose quantity is just
three times as large as the cube ; and the diagonal of
the face is the square root, ot root of a square, whose
quantity is jiist twice as large as the cube.

Here, too, is seen the reason why the solution of
the celebrated problem, styled "the duplication of
the cube,*' is in its very nature absolutely impossible.
This famous problem has been a puzzle to mathema-
ticians, equal to that of the quadrature of the circle,
for two thousand years ; and is said to have been first
proposed by the Oracle of Apollo at Delphos. If so,
it proves the Oracle to have been an expert geometer
in the received principles of the science, or that he
had one of the most expert geometers of antiquity to
whisper in his ear. The story is this : While a plague

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THE PHILOSOPHY OF GEOMETRY. 33

was raging j^t Athens, the Oracle of Apollo at Delphos
was consulted to know when and how the progress of
tfie pestilence could be stayed. The Oracle gave for
answer, that the plague should cease when Apollo's
altar, which was in the form of a cube, should be
doubled. This answer was probably a safer one for
the Oracle than even he himself was aware j for as
the Omniscient ruler of the -world alone could tell
when the plague should cease, so none but the
Omnipotent geometer of the universe had the power
to solve the problem proposed. To man, who has no
means of solving it, but by the use of numbers, the
principles already laid down show the solution to be
an utter impossibility.

The meaning of the question proposed was this :
Apollo's altar being supposed to be a perfect cube,
and its dimensions exactly given — to find the exact
dimensions of another perfect cube contmning just
double the solidity or bulk. of Apollo's altar. Two
such cubes may unquestionably exist in nature ; but
we have no power to measure one by the other, so as
to compare exactly their contents; which I think
must be evident, if we recur again to the origin of the
unity its nature §nd value, and its uses in geometry.
It has already been shown that we can only measure
magnitudes by a limited magnitude of some definite
form ; and that the form universally adopted in the
world, as the standard of measure, is the cube ; that
being the most convenient in practice. Now, let us
suppose, at a point in space, a quantity of matter or
space exists, infinitely small, or as small as the ima-
gination can reach, but in the form of a perfect cube.
3

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34 PART FIRST.

And if we suppose this cube to increase in magnitude,
to grow, as it were, with a uniform increase of exten-
sion in every direction from its center, till it becomes
a cube infinitely extended, or as large as the imagina-
tion can reach; it is manifest that it will have been a
perfect cube at every point of progress during the
augmentation, and that in this way the whole may be
said to contain an infinite series of perfect cubes, from
magnitude infinitely diminished to magnitude infin-
itely extended. Here, then, is the infinite material,
out of which we are to make our units ; and, to recur
again to the idea of Aristotle, ^' we take as much as
we please," and call it one. Suppose we arrest the
growing cube at that point where it has exactly
reached one vcubic inch, and call it one. We have
theffi fixed and determined a standard of measure ;
we have made our unit ; we have tied up our quan-
tity or magnitude into one indivisible thing, which,
even though it may be nothing but empty space, is
still, in the hands of geometry, harder than adamant,
forever invariable in quantity and forever invariable
in form. To this quantity or unit we give the name
of one cubic inch. Now, suppose the great Geometer
of the universe to arrest the growii||; cube at that
point where its quantity or bulk is just double the
cubic inch, .and to require of us its measurement.
We apply our standard of measure, the unit we have
made, to this new cube, and endeavor to get a com-
parison ; but we find no agreement. The linear edge
of our unit is too short, and its face is both too short
and too narrow. If we apply the linear edge of our
ainit twice, we find no agreement there, for its double

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THE PHILOSOPHY OF GEOMETRY. 35

is too long for the linear edge of the new cube. And
since our unit is one indivisible thing, of a fixed
form, it is manifest that it can never reveal to us
exactly the quantity or bulk of a magnitude, to which
it can show no equality or agreement when applied
to it. No cube, therefore, can be an exact measure
of another cube of a different size, till it reaches
one exactly eight times its own bulk. For if we
take our cubic inch and place it on a table, and place
another cubic inch by its side, face meeting face,
we then have a line of two inches. Our unit can
measure that line, because it agrees with its fellow-
unit as well as with itself. If we place two more
units by the side of this line, we have a square of four
inches, and our unit can measure this square, because
it agrees with each of the units in it. If we place
four more units upon the top of this square, we then
arrive at another perfect cube containing eight of
the original units. Now, if we apply our unit to
the new cube, we find that its linear edge, applied
twice, exactly measures the linear edge of the new
cube, and the face applied four times, exactly mea-
sures the face of the new cube. And we are de-
lighted with the certainty of the truth revealed, that
one perfect cube is just eight times as large as the
other. Our unit can measure this new cube, be-
cause it agrees with each of the eight units of which
it is composed. But as no cube can possibly be com-
posed of units of the same size, and containing any
nuniber of units between one and eight, no cube can
be a measure of another cube that is double its quan-
tity, or triple its quantity, or any larger quantity, till

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36 PART FIRST.

it reaches the cube which is just eight times as large
as itself. And hence the reason why the double of
Apollo's altar could not be perfectly given.

The next perfect cube in size, which can possibly
be composed of units, must of necessity contain twenty-
seven units. For if we increase the linear edge be-
yond two, the next perfect measure must be three ;
and still retaining the cubic inch as the unit, we shall
have a line of three inches. Placing three such lines
side by side we obtain a square, containing nine inches.
And three such squares, placed in succession one
above another, will form a perfect cube, containing
twenty-seven inches or units. This cube can be per-
fectly measured by the unit, because the unit agrees
with every single unit of which it is composed.

In like manner, if we seek for the next perfect
cube which can possibly be composed of units, we
And it to contain sixty-four units. Four units in a line
make the linear edge. Four such lines side by side
make a square of sixteen units ; and four such squares,
placed in succession one upon another, make a perfect
cube of sixty-four units. In like manner, if we take
a linear edge of five units, we shall find that the next
perfect cube which can possibly be composed of units,
contains a hundred and twenty-five units. Again, if
w^e take the next perfect linear edge, which must be
six, we shall find that the next perfect cube which
can possibly be composed of units, contains two hun-
dred and sixteen units. And by the same process,
taking seven units for a linear edge, we shall find that
the next perfect cube which can possibly be composed
of units, contains three hundred and forty-three units.

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THE PHILOSOPHY OF GEOMETRY. 37

And taking eight units for a linear edge, we find the -
next possible perfect cube composed of units, contains
five hundred and twelve units. And with nine units
for a linear edge, we find that the next possible per-
fect cube composed of units, contains seven hundred
and twenty-nine units. And in like manner the next
possible perfect cube must contain a thousand units,
having ten for its linear edge.

Though mathematicians talk about " the three roots
of a cube, one real and two imaginary,'* I cannot pos-
sibly conceive of a cube having more than one root,
and that is its linear edge. The "two imaginary
roots" I presume must be the result of some very
acute algebraical process, so searching in its operation
as to discover the difference in the value of " three
times nothing, and twice nothing^* And perhaps the
imaginary roots may, like the zeros^ have a ratio to
each other as 3 to 2.

But it is the " real root'* of the cube that we arc
now dealing with. And as it is not possible to have
more than nine perfect cubes composed of units, till
we arrive at one thousand, so there are but nine num-
bers from 1 up to 1000, of which a perfect cube root
can be found. These nine numbers are — 1 ; 8 ; 27 ;
64; 125; 216; 343; 512; and 729.

The same reason, which limits the number of per-
fect cubes and perfect cube roots, also limits the num-
ber o{ perfect squares to thirty-one, which can possibly
be composed of units under one thousand. So there
are but thirty-one numbers under a thousand, which
can possibly have a perfect square root. The smallest
of these numbers is 1, and the largest is 961. And

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38 . PART FIRST.

there are but two numbers under one thousand which
can have both a perfect cube root and a pegrfect square
root ; and these numbers are 64, and 729.

From these considerations it must be obvious that
all surd quantities in mathematics are the necessary
results of the fixed and invariable ybrm of the assumed
unit, and do not arise from any incommensurable quan-
tities actually existing in nature. But though we are
not able to obtain the perfect root of these quantities,
it is well known that by fractions, or assuming smaller
units which shall have an exact proportion to the
given unit, we may obtain an approximate root of all
numbers whatsoever, and carry the approximation as
near to the true root as we please, or as far as we can
handle numbers and comprehend their value ; but, go
as far as we may, we can never reach the perfect root,
and therefore cannot in this operation entirely satisfy
the requirements of geometry, which is content with
nothing short of perfect agreement.

It is interesting to observe that all the perfect
squares which can possibly be formed of units, are
formed by the successive additions of all the odd num-
bers from one^ upward. Thus the first perfect square
is 1, arid its root is 1. Add to 1 the. next odd num-
ber, 3, and it makes 4, which is the next perfect
square, whose root is 2. Add the next odd number,
5, to 4, and it makes 9, which is the next perfect
square, whose root is 3. Add to 9 the next odd num-
ber, 7, and it makes 16, which is the next perfect
square, whose root is 4. And so on, as in the follow-
ing table, embracing all the perfect squares which can
possibly be formed of units under one thousand.

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THE PHILOSOPHY OF GEOMETRY. 39

Successive Sums

Perfect

of the odd Numbers,

Squares,

Roots.

1+ 8

1 + 3+6

1 J-3 + 6+ 7
1 + 8 + 6 + 7+9

1 + 8 + 6 + 7 + 9 + 11

36 + 13

49 + 16

64 + 17

81+19

100

100 + 21

121

121 + 23

144

144 + 25

169

169 + 27

196

196 + 29

225

225 + 31

256

256 + 33

289

289 + 35

324

324 + 37

361

861 + 39

400

400 + 41

441

441 + 43

484

484 + 45

629

529 + 47

676

576 + 49

625

625 + 51

676

676 + 53

729

729 + 65

784

784 + 57

841

841 + 69

900

900 + 61

961

The necessity of the results exhibited in the pre-
ceding table will more clearly appear from an exami-
nation of the annexed diagram.

We will take the small square in the corner of the
diagram, marked 1, for a unit. It is then a fixed
quantity both in magnitude and form. It is 1 in
lengthy and 1 in breadth, and is really also 1 in thick-
ness. But we now entirely disregard the thickness,
because we are considering it as a plane figure, in

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PART FIRST.

which extension is measured or considered only in

length and breadth. This unit,
which we have assumed, is an
indivisible thing, invariable in
form. It is 1 square. This in-
strument is just what we have
made it ; it can never be any-
thing else ; and we must work
with it as it is, or with other
units of precisely the same size and form. Now, from
inspection or trial, it is obvious that the next square
which can possibly be formed of such units, or mea-
sured by such units, must be made by adding three
more units to the first square, as is done in the dia-
gram by adding the three units or squares marked 3.
It is then seen to be true on the diagram, as it was in
numbers in the preceding table, that the two first odd
numbers, 1 and 3, added together, make a perfect
square, containing four units. On further inspection,
it will appear that the next square, which can possi-
bly be formed of such units, must be made by adding
five more units, as those marked 5 in the diagram.
We then get another perfect square, containing nine
units. This square is made up, both in magnitude and
numbers, by adding together the three first odd num-
bers, 1 unit, and 3 units, and 5 units. In like manner,
the next square, which can possibly be formed of such
units, must be made by adding seven more units, as
those marked 7 in the diagram. This gives a per-
fect square, containing sixteen units, made up of the
first four odd numbers, or of the squares which the
numbers represent, 1 unit, 3 units, 5 units, and 7

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THE PHILOSOPHY OF GEOMETRY. 4J

units. And by adding nine more units, the next odd
number, as is done by adding those marked 9 in the
diagram, we have another perfect square, containing
twenty-five units. This square has a perfect root,
which is 5, for the length of its side is 5 units. In
like manner perfect squares and perfect roots may be
shown by diagram to agree with numbers, as in the
table, as far as we choose to carry them.

From these considerations and many others which
will subsequently appear, it will be manifest, that in
handling abstract numbers we are in reality handling
positive magnitudes; and that numbers in their own
essence are nothing but sig7is of those magnitudes,
while absolute geometrical quantities having exten-
sion are truly the things signified.

It is worthy of remark here also, that the difference
of the squares of any two consecutive numbers, from
one upward, is an odd number, and this odd number
equals the sum of the two consecutive numbers which
are the roots of the squares. Thus, take the two con-
secutive numbers 2 and 3. The square of 2 is 4, and
the square of 3 is 9 ; and the difference between 4
and 9 is 5, and that equals the sum of 2 and 3.

Take the consecutive numbers 3 and 4. The
square of 3 is 9, and the square of 4 is 16 ; and the
difference between 9 and 16 is 7, and that is the sum
of 3 and 4.

Take the consecutive numbers 4 and 5. The square
of 4 is 16, and the square of 5 is 25 ; and the differ*
ence between 16 and 25 is 9, and that is the sum of
4 and 5. And so on, for all consecutive numbers.

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43 PART FIRST.

SECTION V.

CIRCUMFERENCE, DIAMETER, AND AREA OF PLANE FIGURES.

Plane figures are those in which extension is mea-
sured or considered in two directions only, length and
breadth, disregarding entirely their thickness. The
area, or quantity of extension of a plane figure, is de-
termined and measured by a diameter, a line passing
through the center of the figure, and its circumference,
a line outside of the figure and inclosing it. And
these lines, as well as all other lines, have always a
breadth of one.

By a necessary law of numbers and quantities, twice
the square root of any given quantity or number, less
than 4, is greater than the given quantity. And twice
the square root of any given quantity or number,
greater than 4, is less than the given quantity. But
if 4 is the given quantity, twice the square root is
equal to the given quantity. And here we arrive at
a general, simple, and beautiful law, by which diame-
ter controls the relation between area and circumfer-
ence, and by which circumference controls the rela-
tion between area and diameter. This law, in general
terms, is as follows : — ^If diameter be 1, the area equals
one-fourth the circumference. If diameter be 2, the
area equals two-fourths, or one-half the circumference.
If diameter be 3, the area equals three-fourths the

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THE PHILOSOPHY OF GEOMETRY.

circumference. If diameter be 4, the area equals
the circumference. And if diameter be greater than
4^ circumference is less than area. This general law
applies to all regular plane figures^ the circle, equila-
teral triangle, square, pentagon, hexagon, and all re-
gular polygons of any number of sides.

Buty in all these cases, the law requires that the
diameter, used in the measurement, shall be the dia-
meter of the inscribed drck. In the equilateral triangle,
if the inscribed circle (touching each side of the tri-
angle) has a diameter of 1, then the area of the triangle
equals one-fourth of its circumference. If the diame-
ter of the inscribed circle be 2, the area of the triangle
will equal one-half its circumference. If the diameter
be 3, the area will equal three-fourths of the circum-
ference. And if the diameter of the inscribed circle
be 4, the area of the triangle will just equal its cir-
cumference. It will just equal the circumference if
computed in numbers, and just equal it if measured in
space or geometrical quantity. For the line of cir-
cumference having a breadth of 1, it occupies and fills
a portion of space as truly as the area of the triangle.
And when the diameter of the inscribed circle is 4,
the quantity of space occupied by the line of circum-
ference of the triangle, is just equal to the quantity of
space or area of the triangle itself.

So also in the square, the pentagon, hexagon, or any
regular polygon, if the inscribed circle (touching each
side of the figure,) has a diameter of 1, the area of the
figure equals one-fourth of its circumference. If the
inscribed circle has a diameter of 2, the area of the
figure equals one-half its circumference. If diameter

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44 PART FIRST.

is 3, the area equals three-fourths of the circumference.
If the diameter of the inscribed circlets 4, the area of
the figure equals its circumference. And if diameter
is greater than 4, circumference is less than area.

By the same general law> and in the same manner,
circumference, when given or assumed, controls the
relation between area and diameter. In the circle,
triangle, square, pentagon, hexagon, and all regular
polygons of any number of sid^s, if circumference be
1, area equals one-fourth of the diameter. If circum-
ference be 2, area equals one half thfe diameter. If
circumference be 3, area equals three-fourths of the
diameter. If circumference be 4, area and diameter
are equal. And if circumference be greater than 4,
diameter is less than area.

But here also the law requires, that in all these
cases, the diameter, used in the computation, shall be
the diameter of the inscribed circle. Thus it seems
that circumference and diameter, in a certain sense,
cross each other at the point of 4. It is' a universal
law of all plane figures, if circumference is less than 4,
diameter is greater than area. If circumference is
greater than 4, diameter is less than area. And on
the other hand, if diameter is less than 4, circumfer-
ence is greater than area; and if diameter is greater
than 4, circumference is less than area.

After tracing these general laws of circumference,
diameter, and area thus far, I perceived that they
have a still more general and wider application than
has been already laid down; for they apply, not only
to all regular plane figures, but also to all irregular
rectilineal figures, provided their lines of circumfer

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THE PHILOSOPHY OF GEOMETRY. 46

ence are so drawn that an inscribed circle shall touch
every side. Thus, we may have plane figures of three,
or four, or a dozen, or twenty sides, and all the sides
irregular or unequal in length, and yet if these lines
are so arranged that an inscribed circle shall touch
every side of the figure, the magic power of the circle
holds them all to the same general law. If the dia-
meter of the circle is 1, the area jof the figure is one-
fourth its circumference. If diameter is 2, area is one-
half the circumference. If diameter is 3, area is three-
fourths the circumference ; and if diameter is 4, area
and circumference are equal.

If the circumference of the figure is 1, its area is
one-fourth the diameter of the inscribed circle ; if cir-
cumference is 2, area is one-half the diameter; if
circumference is 3, area is three-fourths the diameter ;
and if circumference is 4, the area of the figure is
equal to the diameter of the inscribed circle.

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49 PART FIRST.

SECTION VI.

DIAMETER, SOLIDITY, AND SURFACE OF SOLID FIGURES.

Solid figures are those in which extension is
measured or considered in three directions — length,
breadth, and height, or thickness.

Plane figures really have extension from their cen-
ters to every point of their circumferences; but as
they are all measured by being brought into squares,
and squares are considered as having extension in
length and breadth only, therefore all plane figures are
said to be those in which extension is measured in
two directions, length and breadth.

So all solid figures really have extension in every
direction from their centers to their surfaces; but •as
they are all mensured by being brought into the form
of a cube, and a cube is considered as having exten-
sion in length, breadth, and thickness only, therefore
all solid figures are said to be those which have three
dimensions, or in which extension is measured in
three directions, length, breadth and thickness.

The term solidity ^ in geometry, simply means bulky
or the amount or quantity of extension which is to be
measured. So that, in geometry, a cubic inch of air,
or even a cubic inch of empty space, has exactly the
same solidity as a cubic inch of gold.

The terms ^urfubce and mperficieSy which are used

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THE PHILOSOPHY OF GEOMETRY. 47

indiscriminately in geometry as synonymous, require
some particular consideration. As we have given
thickness to surface, and made it a solid body, or body
having extension, it becomes more important to find
a place for it and define its locality, than it would be,
if it were the airy nothing which it has heretofore
been considered. In the view taken of surface by
Professor Simson and other geometers, it is considered
as inherent in the solid, but existing at the extreme
limits of the solid. In the cube, for instance, they
make the surface to be identically the same thing as
the six faces of the cube. But face and surface are
different words, and literally and truly have different
meanings. Face is a very proper word to represent
the extreme limit of the cube ; but swr-face, or super'-
faciei, literally and properly means that whibh is upon
the face. And that is precisely what is wanted in a
term to express the true meaning and locality of sur-
face. So that A^e find the right and proper term al-
ready in use, though always used heretofore without
its true and legitimate meaning being attached to it.
The surface of a cube, therefore, is a quantity or mag-
nitude lying upon and exactly covering each face of
the cube, and always having a thickness of one. And
the same definition of surface applies to all soJids, of
whatever form. The surface covers all the faces witli
a thickness of one.

Now, as in plane figures the diameter of an inscribed
circle holds all rectilineal figures to one general and
simple law, so in solid figures the diameter of an in<-
scribed sphere holds all solids with plane surfaces to
a similar law, equally general and equally simple.

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48 PART FIRST.

The only difference in the two cases is this ; the point
of equality in plane figures is 4, while the correspond-
ing point of equality in solid figures is 6, And the
reason of this will he manifest^ if we look again for a
moment at the cubic unit, by which all magnitudes,
both plane and solid, are really measured. Let us
take a cubic inch and place it upon a table, and call
it a unit, or one inch. First, if we consider it as a
plane figure, its extension is measured in two direc-
tions, length and breadth, and we call it one square
inch. It has four sides, or faces turned outward hori-
zontally, and its circumference must cover these four
faces. Its circumference therefore is 4 ; which will
more distinctly appear when we come to the demon-
strations in Part Second. If we regard the cubic inch
as a solid figure instead of a plane, we must then con-
sider its extension in three directions, length, breadth
and thickness ; and instead of circumference it must
be provided with a surface. Then, ta the four faces,
which are turned outward horizontally we must add
the face at the top and the face at the bottom, mak-
ing six faces, which must be covered by the surface.
The surface therefore is 6. The same identical unit,
which, as a plane figure, has a circumference of 4, as
a solid figure, has a surface of 6. And as circumfer-
ence and diameter, in governing the area of plane
figures, make a point of equality at 4, so surface and
diameter, in governing the solidity of solid figures,
make a point of equality at 6. Accordingly we find,
if the diameter of a sphere is I, its solidity is equal to
one-sixth of its surface. If its diameter is 2, its soli-
dity equals two-sixths or one-third its surface. If its

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THE PHILOSOPHY OT GEOMETRY. 49

diameter is 3^ its solidity equals three-sixths or one«
half its surface. If diameter is 4, solidity equals four«
sixths or two-thirds of the surface. If diameter is 5,
solidity equals five-sixths of the surface. If diameter
is 6^ solidity equals the sur&ce.

On the other hand, if the sur&ce of a sphere is 1,
solidity equals cme-sixth of the diameter. If the sur-
face is 2, solidity equals two-sixths or one-third of the
diameter. If the surface is 3, solidity equals one-half
the diameter. If the surface is 4, solidity equals two-
thirds of the diameter. If the surface is 5, solidity
equals five-sixths of the diameter. If the surface is 6,
solidity and diameter are equal.

This same general law which thus governs the soli-
dity of the sphere, applies also to all solids with plane
surfaces. But the law requires that the diameter used
shall be the diameter of an inscribed sphere ; that is,
a sphere which shall touch every plane constituting
the surface of the solid. Thus in the tetrahedron, the
solid bounded by four plane equilateral triangles, if
the diameter of the inscribed sphere is 1, the contents
or solidity of the tetrahedron equals one-sixth of its
surface. If the. diameter of the inscribed sphere is 2,
the solidity of the tetrahedron equals one-third its sur-
face. If the diameter is 3, solidity equals one-half the
surface. If the diameter is 4, solidity equals two-
thirds of the surface. If the diameter is 5, solidity
equals five-sixths of the surface. If the diameter is 6,
the solidity of the tetrahedron and the surface of the
tetrahedron are equal. On the other hand, if the sur-
fiice of the tetrahedron is 1, its solidity equals one-
sixth of the diameter of its inscribed sphere. If the

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10 PART FIRST.

surface of the tetrahedron is 2, its solidity is one-third
the diameter of the inscribed sphere. If the surface
is 3, solidity is one-half the diameter. If surface is 4,
solidity is two-thirds the diameter. If surface is 5,
solidity is five-sixths of the. diameter. If surface is 6,
solidity and diameter are equal.

And so the hexaedron or cube, the octaedron, the
dodecaedron, and icosaedron, are all perfectly and
rigorously bound by their inscribed sphere to tlie same
general law.

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THE PHILOSOPHY OF GEOMETRY.

SECTION vn.

NUMBERS ARE NOTHING BUT SIGNS OP qUANTTTIES OR
MAGNITUDES.

All geometrical quantities or magnitudes may be
said to have a possible existence everywhere and al-
ways^ throughout all space ; that is^ in any part of in-
finite space any possible magnitude may be assumed
to exist, and the quantity of space thus assumed has
a permanent existence ; it is the same before we con-
sider it, and the same afterward. It is not so with
mathematical numbers. They are not things existing
of themselves. They have no place or being in na-
ture till we apply them to quantities as signs or names,
by which we may express our knowledge or ideas of
those quantities. Numbers therefore always being
nothing but signs, and quantities or magnitudes al-
ways being the things signified, it follows of necessity
that abstract numbers can do nothing whatever of
themselves ; but all they seem to do is really done by
the quantities they represent. It is not the sign, but
the thing signified, which truly performs the operation
in every mathematical process. The number does
not make or limit the value of the quantity, but the
quantity stamps its own value, as it were, upon the
number, making it the representative of the quantity.
If we assume any abstract number first, say the num-
ber 1, it can give us no knowledge or idea whatever,

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§2 PART FIRST.

for it has no meaning. We know not whether it may
mean 1 egg, or 1 apple, or 1 mountain, or 1 anything
else. But if we first assume some definite quantity or
magnitude, say one cubic inch, the quantity is a real
thing, making itself known to our senses, and giving
us an idea of its amount of extension, without any
sign or name being attached to it. If we then apply
the number 1 to it, as the sign or name to represent
it, the quantity stamps its value upon the number 1,
so that it cannot mean 1 egg, or 1 apple, or 1 moun-
tain, but must of necessity mean 1 cubic inch, and
nothing else.

Mathematical numbers therefore must sdways re-
present magnitudes. Dr. Barrow, the distinguished
and learned predecessor of Sir Isaac Newton in the
mathematical chair at Cambridge,^ also took this view
of numbers. But I have found no other writer who
has considered them in the same light. ^^ I am con-
vinced," says Dr. Barrow, " that number really differs
nothing from what is called continued quantity ; but
is only formed to express and declare it. And conse-
quently, that arithmetic and geometry are not conver-
sant about different matters, but do both equally de-
monstrate properties common to one and the same
subject. And very many and very great improve-
ments will appear to be derived from hence upon the
republic of mathematics.'^

Again, he says, "There is neither any general
axiom nor particular conclusion agreeing with geome-
try, but what by the same reason also agrees with
arithmetic. And on the other hand, nothing can be
affirmed^ concluded, or demonstrated, concerning

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THE PHILOSOPHY OF GEOMETRY.

numbers, which may not in like manner be accommo-
dated to magnitudes. Whence accrues a remark-
able light and vast improvement to both sciences."

And yet again, says this very acute and philoso*
phical thinker, " No geometrical argument is of force,
which agrees not exactly with an arithmetical calcu-
lus. Also all true conclusions and lawful demonstra-
tions in geometry may be illustrated and confirmed
by help of an arithmetical calculus."

And again he adds, '* I say that a mathematical
number has no existence proper to itself, and really
distinct from the magnitude it denominates."

Thus it is evident that Dr. Barrow arrived at the
clear conviction, that mathematical number always
represents magnitude; and to me it seems wonderful
that he did not go a step further, and discover how
much magnitude, that is, hxm much relative magnitude,
every number truly reptesents. But the progress of
every mind, clothed in the flesh, has its limits, as if
infinite Wisdom had said to it, thus far shalt thou go,
but no fiirther. •

As all geometrical quantities or magnitudes may be
said to have a possible existence everywhere and al-
ways, throughout all space, so all geometrical forms
may be said to have a possible existence everywhere
and always, throughout all matter and all space. That
is, all geometrical forms are truly inherent and have a
possible existence in every assumed quantity of mat-
ter, and in every assumed portion of space. This idea
is also clearly and elegantly expressed by Dr. Barrow,
as follows : *^ All imaginable geometrical figures are
really inherent in every particle of matter; I say

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04 PART FIRST.

really Inherent in fact, and to the utmost perfection,
though not apparent to the sense ; just as the effigies
of Cesar lies hid under the unhewn marble, and is no
new thing made by the statuary, but only is discov-
ered and brought to sight by his workmanship, by re-
moving the parts gf matter which involve and over-
shadow it. Which made Michael Angelus, the most
famous carver, say, that sculpture was nothing else
but a purgation from things superfluous ; for take all
that is superfluous from the wood or stone, and the
rest will be the figure you intend."

In the fact, that the intimate and inseparable rela-
tion existing between numbers, magnitudes, and forms,
has not been clearly understood, may be found the
principal cause of the difficulties and mysteries which
have always attended the investigation of many ma-
thematical subjects. From a want of this knowledge.
Lord Bacon, in classing the sciences, seemed to be
puzzled and in doubt, whether to place mathematics
with the physical or metaphysical sciences. He, hoTv^-
ever, came to the conclusion that it w^as *^ more agree-
able to the nature of things, and to the light of order,
to place it as a branch of metaphysic ; for the subject
of it being quantity, not quantity indefinite, which is
but relative, but quantity determined, or proportion-
able, it appeareth to be one of the essential forms of
things, causative in nature of a number of effects.'*

The want of this knowledge, also, was undoubtedly
the principal cause of the divisions and disputes be-
tween some of the celebrated schools of antiquity.
Lord Bacon says, " In the schools of Democritus and
Pythagoras, the one did ascribe .^gwre [form] to the

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THE PHILOSOPHY OF GEOMETRY. Sff

first seeds of things^ and the other did suppose numbers
to be the principles and originals of things." These
schools might undoubtedly have shaken hands and
united, had they but known, that numbers are nothing
but signs of magnitudes and forms, and forever follow
magnitudes and forms, as the shadow follows the sub-
stance, and can no more exist without magnitudes
and forms than the shadow can exist without the sub-
stance which it represents.

The want of this knowledge also caused a strange
confusion of terms among the ancients, which could
not fail to throw a mist over their subjects of discus-
sion, and lead to endless disputes. " For,*^ says Dr,
Barrow, "Aristotle improperly makes the Greek to
poson to be the common genus of both miUtiticde, as it
is numerable, and magnitude^ as it is measurable;
which word properly signifies quotityy and only re-:
spects number. And, on the contrary, quantitas, by
which word the Latins used to express the to poson of
Aristotle, denotes magnitude only, and cannot properly
be referred to number; which defect of a common
name to magnitude and multitude is perhaps an argu-
ment that the ancients had no common perception of
them.'*

Numbers are entirely blind with regard to absolute
or positive magnitudes, for they know no difference
between great and smaH, and can only recognize rela-
tive quantities of definite forms.

If we divide the circumference of any square, how-
ever great or however small, by the circumference of
its inscribed circle, it will always produce the same
quotient— viz., L273+, and this, too, whether the

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PART FIRST.

square contain an area of one inch or ten thousand
inches ; — ^and the square root of this quotient, thus ob-
tained from all squares, is the diameter of a circle
whose area equals 1 square.

So also, if we divide the area of any square what-
ever by the area of its inscribed cirele, we always
obtain the same quotient as when we divide the cir-
cumference of the square by the circumference of the
circle — ^viz., 1.273+.

If we divide the diagonal of any square, however
great or however small, by the side of the same square,
we always obtain the same quotient, which is the
square root of 2 — ^viz., 1.4142+.

If we divide the surface of any cube, however great
or however small, by the surface of its inscribed
sphere, we always obtain one and the same quotient,
viz., 1.9098+; and the c/ube root of this quotient is the
diameter of a sphere whose soUdity equals 1 cube.
So also, if we di vidfc the solidity of any cube by the
solidity of its inscribed sphere, we obtain always the
same quotient — viz., 1.9098+.

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THE PHILOSOPHY OF GEOMETRY.

' SECTION VIII.

IDENTITY Ot' PURE AND MIXED MATHEMATICS.

" The speculation was excellent," says Lord Bacon,
^^ in Farmenides and Plato, although but a speculation
in them, that all things by scale did ascend to unity.
So then always that knowledge is worthiest, which is
charged with least multiplicity/'

In nearly all our works on mathematical subjects,
we are told that mathematics are of two kinds, pure
and mixed. If there be in reality such a distinction,
founded in the nature and principles of the science, it
should undoubtedly be recognized, and the bounda-
ries of the two branches of mathematics clearly de-
fined, so that the principles which belong exclusively
to the one branch may not be confounded with those
which apply to the other. But if it be merely a dis-
tinction without a difference ; if all the principles of the
one are identically the same with the principles of the
other, is it not high time that so useless an absurdity
were expunged from our books of science and instruc-
tion, and our mathematics clothed with such habili-
ments of simplicity as their principles will allow ?

Pure mathematics, they tell us, considers and mea-
sures abstract quantities. And mixed mathematics
considers and measures quantities as they exist in ma-
terial bodies. Thus, if a mathematician considers and
calculates the surface of a cube whose linear edge is

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SB PART FIRST.

2, he finds it ta be 24 ; and this is called a process of
pure mathematics. If a carpenter makes a box, whose
linear edge is 2 feet, the length, breadth, and height,
all being the same, and measures and, calculates the
number of square feet of boards used in the construc-
tion, he finds it to be 24 ; and this is called a process
of mixed mathematics. But the laws of computation
are precisely the same in the two cases, nor is it pos-
sible to conceive any principle involved in the one,
which does not apply equally to the other. In the
cube, the linear edge is 2, which being squared makes
one face of the cube to be 4, and there being 6 faces,
4 repeated six times, or multiplied by 6, makes alJ
the faces or the whole surface to be 24. So in the
box, the linear edge is 2 feet, which squared, or mul-
tiplied into itself, makes one side or one face of the
box to be 4 feet, and this multiplied by 6, the number
of sides or faces of the box, makes the whole surface
24 feet.

The truth is, the 2 used by the mathematician, in
the abstract operation, as truly represents a qtiantity,
or magnitude, as the 2 used by the carpenter. The
only difference is, the quantity or size of the unit used
by the carpenter is defined, and we know what it is,
while the unit used by the mathematician is not de-
fined, but left unlimited, and may mean a cube of any
magnitude whatsoever, from magnitude infinitely
diminished, to magnitude infinitely extended. And
all the laws, principles, and results of abstract num-
bers in any process carried on by the mathema-
tician, agree precisely with the laws, principles and
results of the numbers used by the mechanic or

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THE PHILOSOPHY OF GEOMETRY. 99

the natural philosopher in their dealings with ma-
terial bodies. One set of these numbers is cer*-
tainly no more pure than the other, nor is one
set any mofe mixed than the other. Why then
should they be distinguished by different names, as
though they possessed different natures, or different
powers, or could work with different degrees ol
purity ?

The absurdity of this division of mathematics has
been observed and condemned by several writers of
high authority, though they had not discovered the
true nature and value of the unit. The author of the
able article on mathematics in the London Encyclope-
dia, remarks on this subject as follows :

^'Mathematics are commonly distinguished into
pure and mixed. Pure mathematics, it is said, con-
siders quantity abstractedly ; and mixed mathematics
treats of magnitudes as subsisting in material bodies,
and consequently are interwoven everywhere with
physical considerations. This is one of the objection-
able distinctions, positions, and definitions, too fre-
quently to be met with in connection with a science
which boasts of accuracy and certainty. The notion
of quantity abstractedly, or separately from material
bodies and physical considerations, is manifestly ab-
surd ; for where or how can quantity exist, or be con-
ceived of as existing, but in some material body ?
We might philosophize about color, form, or shape,
solidity, fluidity, elasticity, gravity, &c., &c., abstract-
edly from material bodies, and physical considerations,
[as we long indeed attempted,] and call this pure
philosophy ; but it would be a pure fiction of the brain

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PART FIBST.

mere absurdity. If pure mathematies really con-
sisted in such abstractions^ they might be defined the
science of non-existents. But even in the most ima-
ginary quantity of the most absolute abstmction, the
imagination [to say nothing of the understanding] of
the purest or most speculative mathematician must
have something of the nature of materia firma to rest
upon."

To show that the process of reasoning and handling
numbers is precisely the same in mixed mathematies
as in the pure« a writer in the Edinburgh Encyclope-
dia gives the following illustration.

" The geometer, who reasons on the comparative
weights of a globe of brass and a cube of water, is
perfectly indifferent whether, in any particular globe
presented to his view, the microscope may not disco-
ver superficial irregularities, or whether the instru-
ments used for taking specific gravities be susceptible
of mathematical precision. He reasons on them as
creatures of his own imagination, agreeing in their
forms and qualities with certain arbitrary definitions,
and is fully aware, that in so far as bodies are to be
found in nature which conform to those definitions, in
so far only will his conclusions apply to them.*'

The truly philosophical and always correct Dr. Bar-
row remarks as follows: "In reality, those which are
called mixed, or concrete mathematical sciences, are
rather so many examples only of geometry, than so
many distinct sciences separate from it; for when
once they are disrobed of particular circumstances,
and their own fundamental and principal hypotheses
come to be admitted, they become purely geometri*

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THE PHILOSOPHY OF GEOMETRY.

cal.'^ And again the same learned author remarks ;
" There is no reason why the doctrine of generals
should be separated from the consideration of particu-
lars^ since the former entirely includes and primarily
respects the latter. Whence it is altogether amiss to
disjoin geodesy from geometry ; for the multitude of
sensible things, to which number and magnitude may
be applied, is too diffusive to be circumscribed within
these limits ; and consequently if this supposition be
admitted, the field of mathematics will become far too
wide and extensive/'

Sir Isaac Newton, whose ideas were not only al-
ways clear, but also clearly expressed, throws a dis-
tinct light on this subject as follows : ^' The ancients
considered mechanics in a two-fold respect: as ra-
tional, which proceeds accurately by demonstration ;
and practical. To practical mechanics all the manual
arts belong, firom which mechanics took its name.
But as artificers do not work with perfect accuracy, it
comes to pass that mechanics is so distinguished from
geometry, that what is perfectly accurate is called
geometrical; what is less so, is called mechanical.
But the error is not in the art, but in the artificers. He
that works with less accuracy is an imperfect mecha-
nic ; and if any could work with perfect accuracy, he
would be the most perfect mechanic of all ; for the
description [construction] of right lines and circles,
upon which geometry is founded, belongs to mecha-
nics. Geometry does not teach us to draw these lines,
but requires them to be drawn. * * * * There-
fore geometry is founded in mechanical practice, and is
nothing but that part of universal mechanics which

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92 PART FIRST.

accurately proposes and demonstrates the art of mea-
suring."

Pythagoras and his followers divided mathematics
into four kinds ; two of which they called pure and
primary — ^viz., arithmetic and geometry; and the other
two, mixed and secondary — ^viz., music and astron-
omy. " Thus did the Pythagoreans of old," says Dr.^
Barrow, " divide mathematics — I suppose because
they had not yet applied themselves to the other
parts — such as optics, mechanics, &c."

On the whple, there does not seem to be any good
reason why mathematics may not and ought not to
be further simplified by banishing from our books
the Imaginary distinction between the pure and the
mixed ; — for as geometry really has but one kind of
quantity, and does not furnish one kind of quantity
for a line, and another kind for a surface, and still
another kind for a solid — so mathematics truly deals
in but one kind of numbers, which are immutable in
their nature and invariable in their laws, and are
therefore always precisely the same things, whether
in the hands of the mathematician, or the mechanic,
or the natural philosopher.

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THE PHILOSOPHY OF GEOMETRY.

SECTION IX.

THE REASON WHY THE SCIENCE OF GEOMETRY HAS NOT

HITHERTO MADE ANY ESSENTIAL PROGRESS IN

MODERN TIMES.

It is a remarkable fact, that, while the world has
been making wonderful progress in arts and sciences,
and all departments of knowledge, during two or
three centuries past — new arts and new sciences
being continually discovered and carried to great per-
fection, and old ones improved wherever improve-
ment seemed possible — it surely is a remarkable fact,
and worthy of special attention, that geometry, the
most important of the sciences, though resting on an
essential fundamental error, has been left till the pre-
sent day, standing almost precisely where Euclid and
Archimedes left it two thousand years ago.

There is some reason to suppose that the ancient
Egyptians attained to a high knowledge of geometry,
though we have no positive evidence of the fact.
Thales is said to have brought the science from Egypt
into Greece ; and Grale, as quoted in Rees' Encyclo-
pedia, says, " The Egyptians used geometrical figures,
not only to express the generations, mutations, and
destructions of bodies, but the manner, attributes, &c.,
of the Spirit of the Universe, who, diffusing himself
frpm the center of his unity, through infinite concen.
trie circles, pervades all bodies, and fills all space.

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64 PART FIRST.

But of all other figures, they most affected the cirde
and the triangle'' This last remark is strong pre-
sumptive evidence that the Egyptians had a higb
knowledge of geometrical figures ; for the circle and
equilateral triangle sustain a very remarkable relation
to each other, being the two extreme limits of what
may be regarded an infinite series. That is, in all
the infinite variety of Tegular plane figures having the
same length of circumference, the equilateral triangle
contains the least possible area of the whole series, and
the circle contains the greatest possible area of the whole
series. There would seem to be a reason, therefore,
why the Egyjptians should " most affect the circle and
triangle," if they had a deep knowledge of geometri-
cal figures.

Pythagoras is said to have first discovered, that of
all regular plane figures, having the same length of
circumference, the circle contains the largest amount
of area ; and of all regular solids, having the same
extent of sur&ce, the sphere contains the greatest
bulk or amount of solidity. Mr. Parker shows in his
reasonings and demonstrations, that if a circle and
equilateral triangle have an equal amount of area, the
diameter of the circle and the perpendicular of the
triangle are in opposite duplicate ratio to each other.

But when we pass from the Egyptians to the Greeks,
among whom Dr. Barrow enumerates "the wonderfiil
Pythagoras, the sagacious Democritus, the divine
Plato, the most subtle and very learned Aristotle,
men whom every age has hitherto acknowledged and
deservedly honored as the greatest philosophers, the
ringleaders of the arts," we are not left in the dark

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THE PHILOSOPHY OF GEOMETRY. 65

as to the progress of geometrical science, for we find
it cultivated to a degree that sheds a luster over the
Grecian name, unequaled in that department of
knovt^ledge among any other people in any age of the
world. And yet the Greeks possessed but a very im-
perfect arithmetic and inconvenient and defective
methods of notation, while the moderns possess an
arithmetic of great perfection, and a method of deci-
mal notation, of wonderful practical facility and un-
failing accuracy.

It becomes then a question of great interest, and
still greater importance, to know why geometry made
such progress and rose to such high perfection among
the Greeks, with all their disadvantages, and why
among the moderns, with vastly superior facilities, it
has hitherto remained essentially upon the same level
where the Greeks left it.

The Greek method of investigating geometrical
subjects was with rule and compasses, and by dia-
grams of lines and circles. The modern method, since
the time of Descartes, has been principally by the al-
gebraical process. And to this different method of
handling geometrical subjects I cannot but think
must be attributed the striking difference in the re-
sults, so injurious to modern science. As I have but
little practical acquaintance with algebra, and judge
of it mostly from general principles, I should express
this opinion with great diflSdence, or perhaps not at
all, if I did not find it supported by some of the highest
and best authorities. It seems due to the importance
of the subject, that a few of the opinions alluded to,
should be presented here for consideration.

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66 PART FIRST.

Algebra is a very condensed, short-hand method of
handling mathematical subjects, and is undoubtedly
very convenient and very useful in many things ; but
it was probably a great error to apply it to geometry,
whereby that pure and clear-sighted science, walking
the earth with free and unerring footsteps, became, as
it were, blindfolded, embarrassed, and stayed in her
progress.

The Edinburgh Encyclopedia says: "The essen-
tial character of algebra consists in this — ^that when
all the quantities concerned in any inquiry to which
it is applied, are denoted by general symbols, the re-
sults of operations do not, like those of arithmetic' and
geometry, give the individual values of the quantities
sought, but only show what are the arithmetical or
geometrical operations, which ought to he performed on
the original given quantities in order to determine
their values."

Professor Simson, of Glasgow, became so thorough-
ly impressed with the superior accuracy and utility
of the Greek geometrical methods over the modern
algebraic, that he devoted almost his whole life in
labors to revive the methods of the ancients, and
to restore and repair some of their lost and mutilated
works. And we are told by the London Encyclo-
pedia, that Professor Simson " came at last to con-
sider algebraic analysis as little better than a kind of
mechanical knack, in which we proceed without ideas
of any kind, and obtain a result without meaning, and
therefore without any conviction pf its truth."

Bishop Berkley, one of the most acute and subtle
reasoners of modem times, "maintains, in the Analyst,

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THE PHILOSOPHY OF GEOMETRY. 67

that the differential calculus, new analysis, or doctrine
of fluxions, is inaccurate in its principles, and that if
it ever lead to true conclusions, it is from an acciden-
tal compensation of errors, that cannot be supposed
always to take place." The writer in the London
Encyclopedia, from whom this paragraph is quoted,
adds : " No one who knows Berkley's ability requires
to be told that the Analyst is a masterly production ;
nor can some of his arguments and charges be fairly
met."

Laplace, the incomparable Laplace, compares the
Greek method with the modern as follows: "The
geometrical synthesis has the advantage of never los-
ing sight of its object, and of illuminating the whole
path which- leads from the first axioms to their last
consequences; whereas the algebraic analysis soon
causes us to forget the principal object in order to
occupy us with abstract combinations."

Sir Isaac Newton himself, who invented the method
of fluxions, and carried the algebraic analysis to the
highest pitch of power and refinement, left his testi-
mony and the weight of his great example in favor
of the Greek geometry. When but a youth, and just
entering upon his collegiate studies, a biographer says
that, " Regarding the propositions contained in Euclid
as self-evident truths, he passed rapidly over this an-
cient system — a step which he afterward much regretted
— and mastered, without further preparatory study,
the Analytical Geometry of Descartes." In after life,
he "censured the handling of geometrical subjects
by algebraical calculations, and the maturest opinions
he expressed were additionally in favor of the geo-

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PART FIRST.

metrical method." So decided was his opinion on
this point, that it is said of him, " he thought that a
mathematical proposition ought not to be made pub-
lic, or was not fit to be seen, till invested in a synthe-
tic dress." And we know that he demonstrated and
published to the world his own sublime discoveries
by the methods of the Greek geometers.

Upon such testimony, is it not fair to conclude that
he, who is pursuing geometrical subjects by the me-
thods of algebra, is like one groping blindfolded to
hunt for gems among pebbles ? — for as the one may
pass over a thousand gems without seeing them, and
if he chance to get one in his hand, cannot under-
stand its value till he takes it to the light; so the
other is forever reaching in the dark; and, if by
chance he grasps a truth, is unable to tell what it is,
till he borrows the light of arithmetic or geometry to
reveal it. But let him descend into the deep caverns
of geometry with the Greek rule and compass in his
hand, guided by the perfect modern numerical nota-
tion, which the Greeks did not possess, and he carries
a torch before him, which lights up his entire path-
way, and the rich and bright gems of truth on all sides
come flashing upon his gladdened sight from every
crag and corner.

Let the teachers of the world give this important
question a fair hearing ; and if this shall prove to be
the right view of it, let them wrest geometry out of
the hands of algebra, strip the bandage from her eyes,
and let her walk forth again upon the earth with un-
clouded vision. Then shall she brush away the cob-
webs and dust of modern abstractions, and, clothed

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THE PHILOSOPHY OF GEOMETRY.

in a garment of new light and beauty, shall stand be-
fore the world more perfect and more comely than in
the days of her Grecian youth. Then shall she carry
forward her high mission to elevate the condition of
man, by teaching him God^s everlasting truths. Then
shall her dignity and divine importance be vindicated,
perhaps even to justify the assertion of Plato concern-
ing the probable employment of Deity, that " He geo-
metrizes continually.''

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10 PART FIRST.

SECnON X.

CONCLUDING REFLECTION.

It must be manifest from what has been already
presented, that geometry and mathematics are not
such hard and incomprehensible matters as the world
has generally regarded them ; and that their perplex-
ing difficulties and forbidding and endless labors are
not so mu(ih to be found in the nature of the sciences
themselves, as in the errors which have hitherto been
lying at their very foundation. The mathematics
have justly been regarded by the wise in all ages as
the best of all disciplines. They were considered by
Pythagoras, as " the first step toward wisdom." Dr.
Barrow in describing the importance of the science,
says, "If the fancy be unstable and fluctuating, it is
as it were poised by this ballast, and steadied by this
anchor ; if the wit be blunt it is sharpened upon this
whetstone ; if luxuriant, it is pared by this knife ; if
headstrong, it is restrained by this bridle ; and if dull,
it is rousAl by this spur."

Says Lord Bacon, " Men do not sufficiently under-
stand the excellent use of the pure mathematics, in
that they do remedy and cure many defects in the
wit and faculties intellectual. For if the wit be too
dull, they sharpen it ; if too wandering they fix it ; if
too inherent in the sense, they abstract it* So thai

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THE PHILOSOPHY OF GEOMETRY. 71

as tennis is a game of no use in itself^ but of great use
in respect it maketh a quick eye, and a body ready to
put itselC into all postures ; so in the mathematics,
that use which is collateral and intervenient, is no
less worthy than that which is principal and in-
tended/'

I most earnestly desire, therefore, to do something
to simplify the study of geometry, the real foundation
of all mathematics ; something to make it not only a
delight to' the student of the University, but a wel-
come guest in every common school, and a cherished
visitor at every fiunily fire-side ; something by which
the benefits of its admirable discipline may become
more widely diffused, and its beauties and harmonies
more generally enjoyed. The world should no
longer be afraid to come in contact with the works of
geometers and mathematicians, ^' the sight of whose
writings,'* says Dr, Barrow, "everywhere shining
with the rays of geometrical diagrams, the unskillful
in these things are afiraid oV* The true principles of
geometry are so simple, that he that runs may read,
and a child can understand them. I desire, therefore,
that none may feel deterred from reading these pages,
or examining the succeeding demonstrations, firom an
apprehension that their knowledge of such subjects is
too limited to enable them to understand them. Let
them bring to the labor a little patient and persever-
ing thought, and examine each step with some vigor
of attention, and they will be surprised at the light
resting on every diagram, and charmed by the beauty,
simplicity, and harmony, with which an endless vari-

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72 PART FIRST.

ety of forms blend and yield obedience to a few gene-
ral and simple laws.

Geometry should always precede arithmetic, or ra-
ther go hand in hand with it, in a system of education.
As soon as a child had learned to count his ten fingers
I would begin to teach him geometry ; for, as it is the
most simple and perfect of all sciences, so it is the
most readily comprehended if properly taught.
Through geometry he should learn all his arithmetic.
Then would he find the dark and puzzling labyrinths
of numbers to lighten up at every step of his progress.
Then would the toilsome and blind path of arithmetic
become a bright and pleasant road, and her mystic
and vague expressions would open to him full of clear
and beautiful meaning. Then would he see and com-
prehend what is meant by those perplexing, enigma-
tical things, the square root, and the cube root. Then
would the boy " with shining morning face," no long-
er be seen " creeping like snail, unwillingly to school,"
but tripping with a light heart, and singing for joy.

Should the present treatise contribute anything to-
ward bringing about such results, it will furnish a true
response to what I trust has been the leading stimu-
lus to carry me through the labor of its preparation ;
— for the end and aim of all knowledge should be to
do good — to elevate both the giver an^ the receiver.
Saint Paul beautifully said, " Though I have the gift
of prophecy, and understand aU mysteries and aU know-
ledge, and though I have all faith, so that I could re-
move mountains, and have not charity, I am nothing."
And the same sentiment is so finely expanded and so
happily expressed by Lord Bacon in the following pas-

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THE PHILOSOPHY OF GEOMETRY. 73

sage, that I am sure its quotation cannot fail to give
pleasure and profit to the reader :

'' The greatest error of all is the mistaking or mis-
placing the last or farthest end of knowledge ; — for
men have entered into a desire of learning and know-
ledge, sometimes upon a natural curiosity and inquisi-
tive appetite ; sometimes to entertain their minds with
variety and delight ; sometimes for ornament and re-
putation ; and sometimes to enable them to victory of
wit and contradiction ; and most times for lucre and
profession ; and seldom #incerely to give a true ac-
count of their gill of reason, to the benefit and use of
men ; as if there were sought in knowledge a couch
whereupon to rest a searching and restless spirit ; or
a terrace, for a wandering and variable mind to walk
up and down with a fair prospect; or a tower of state,
for a proud mind to raise itself upon ; or a fort or
commanding ground, for strife and contention ; or a
shop for profit and sale ; and not a rich storehmisey for
the glory of the Creator^ and the relief of marCs estate.^^

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PART SECOND.

DEMONSTRATIONS IN GEOMETRY.

Remark. — ^Before entering upon this part of our
work, it may be well to apprise the critical reader,
that it has not been deemed at all necessary to adopt
the rule apparently followed by Euclid, viz., never to
suppose anything done, till the manner of doing it has
been shown or explained. The rule was a very Safe
one, in the early progress of the science, to prevent
the possibility of error, or the danger of resting on un-
warrantable assumptions ; but it also led to much un-
necessary labor and tedious prolixity. The rule which
I have rather endeavored to follow in these demon-
strations, is, to give under each proposition aB that is.
necessary to produce perfect conviction of the truth stated,
and not to encumber the demonstration with anything
more. So that if it should appear to the critical geo-
meter, that links are sometimes omitted which he may
think ought to be brought into the chain of reasoning,
he may understand the reason of the omission. He
may sometimes miss the repetition of an axiom or a
well-known and established principle of geometry,
which might have served to lengthen out a demon-

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7« PART SECOND.

stration, but would not make the truth any more ap-^
parent. So also in these demonstrations many figures
are required to be drawn when no rule or mode of
drawing them has been given. But, says Sir Isaac
Newton, " geometry does not teach us to draw these
figures, but requires them to be drawn." The con-
struction of them is entirely a mechanical operation.

DEFINITIONS.

1. J\ru7nber8 are the signs or representatives of things, or of
whatever has existence.

2. Arithmetic is the science of numbers, and regards things
only as they are numerable, or may be counted.

3. Geometry is the science of magnitude, and measures and
compares extension ^xA forms.

4. Arithmetic has but one language or mode of expression,
which is by numbers.

5. Geometry has two languages or modes of expression, one by
numbers, and one by material substances, or pictures representing
material substances.

6. The unit in arithmetic is the sign or representative of any-
thing considered as one and indivisible, without regard to form or
magnitude.

7. The unit in geometry is the sign or representative of any
assumed magnitude, considered as one and indivisible, and in the
form of a cube. A unit in geometry, therefore, is always one in
length, one in breadth, and one in thickness.

8. A unit in geometry may be of any positive magnitude, from
magnitude infinitely diminished, to magnitude infinitely extended.

9. A straight line is composed of a succession of single and
equal units. A line therefore always has a breadth of one.

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DEMONSTRATIONS IN GEOMETRY. T7

10. A line, or length, is measured by the application of one
dimension only of the unit, yiz., its linewr ec^.

11. ^ surface or plane is composed of a succession of single
lines. A surface therefore, always has a thickness of one.

12. Plane figures J or formsy are those in which extensi<Mi is
measured in two directions only, length and breadfti, without re-
gard to thickness.

13. The elements of plane figures consist of area, perimeter,
circumference, and diameter.

14. The area of a plane figure is the quantity of extension or
space inclosed by its circumference ; and is measured by the ap-
plication of two dimensions of the unit, its length and breadth.

15. The perimeter of a plane figure is the distance around it,
measured upon the extreme limits of the figure.

16. The circumference of a plane figure is a line, or lines,
touching and inclosing it, having a breadth equal to one^ and a
length equal to the perimeter of the figure.

17. The diameter of a plane figure is the diameter of its in-
scribed circle.

18. A circle is a plane figure, which has an equal extension in
erery direction from its center to its circumference.

19. The diameter of a circle is a straight line passmg through
its center, and extending in length to the extreme limits of the
circle.

20. A circle is said to be inscribed in any plane figure, when
the circle touches every side or line of the circumference of the
figure ; and circumscribed when the circumference of the circle
touches every comer or angle of the figure.

21. A plane figure is said to be circumscribed about a circle,
when every side of its circumference touches the circle ; and in-
scribed when every comer or angle touches the circumference of
the circle.

22. The base of a plane figure is the side on which it is sup-
posed to rest, when oonsidered in a vertical position.

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78 PART SECOND.

23. 7%e perpendicular^ or height, of a plane figure is the
shortest distance from any point in the base to a line dra^m
parallel to the base and resting on the highest point of the figure.

24. Paralld lines are those which everywhere preserve an
equal distance between them. Parallel lines therefore can never
meet each oth^, however far they may be produced.

25. Solid figures^ or bodies, are those in which extension ift
measured in three directions, length, breadth, and thickness.

26. The elements of solid figures consist of solidity, face, sur-
face, and diameter*

27. The solidity of a solid, in geometry, is the quantity of
extension, or the amount of bulk, inclosed by the surface. The
solidity is measured by the application of the unit in its three di-
mensions, length, breadth, and thickness.

28. The faces of a solid are the planes by which its extension
is terminated.

29. The surfaccj [^tfper-facies,] of a solid is the sum of all
the planes supposed to perfectly cover all its faces, and everywhere
having a thickness of one.

30. The diameter of a solid, with plane faces, is the diameter
of its inscribed sphere.

31. Jl sphere is a solid figure which has an equal extension in
every direction from its center to its surface. Its surface there-
fore is a perfect curve, everywhere returning into itself.

32. The diameter of a sphere is a straight line passing through
its center, and extending in length to the extreme limits of the
sphere.

33. A sphere is said to be inscribed in a solid with plane faces,
when the sphere touches every plane of the surface ; and ctrcum-
scribed when the surface of the sphere touches every solid angle.

34. A solid is said to be circumscribed about a sphere, when
every plane of its surface touches the sphere ; and inscribed when
every solid angle touches the surface of the sphere.

Lines are of two kinds, straight and curved.

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DEMONSTRATIONS IN GEOMETRY. 79

35. Ji straight line is one which never changes its direction
in any part of its length. A straight line therefore is the shortest
distance between two points*

36. A curved line is one which cantinnally changes its direction
in every part of its length.

37. The straight line is a measure of extension in one direc-
tion only.

38. The curved line is a measure of extension in every possible
direction.

39. Straight lines are used in the composition of diameters,
circumferences, and surfaces.

40. Curved lines are used only for circumferences and surfaces.

41. A plane surface is composed of straight lines.

42. A curved surface is composed of curved lines.

43. The radius of a circle is a straight line drawn from the
center to the circumference ; — ^and because a circle has an equal
extension in every direction from the center to the circumference,
every radius of the circle is equal to every other radius of the same
circle.

Xhe radius of a sphere is a straight line from the center to the
surface ; — and because a sphere has an equal extension in every
direction from the center to the surface, every radius of a sphere
is equal to every other radius of the same sphere.

44. An angle is the opening between two
lines which meet each other at a point.

45. One straight line is perpendicular to
another, when the angles on each side of the
perpendicular are eqiml to each other.

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PART SECOND.

46. Angles made by lines meeting each
other perpendicularly, or crossing each other
perpendicularly, are called Hght angles.

47. An acute angle is one that is
smaller or sharper than a right angle.

48. An obtuse angle is one that is
, larger or more open than a right angle.

49. A triangle is a plane figure inclos-
ed by three straight lines of circumference.

50. An equilateral triangle is one whose
three sides are all of equal length.

51. An isosceles triangle is one which has two
sides equal to each other.

52. ^ A scalene triangle is one whose
three sides are all unequal in length*

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DEMONSTRATIONS IN GEOMETRY.

81-

53. A right angled triangle has one right
Migle.

54. The hypothenuse is the longest side of a right angled tri-
angle, or the side opposite to the right angle.

55. An obtuse angled triangle has
one obtuse angle.

56. All other triangles have three acute angles, and are called
actUe angled.

57.^ A rectangle is a plane figure having
four sides, and four right angles*

58.. A square is a rectangle whose four sides are
equal to each other.

59. A rhombus is a plane figure, having
four equal sides, and two obtuse and two
acute angles.

- 60. A parallelogram, is a plane figure with four sides, having
the opposite sides parallel to each other. And because parallel
lines everywhere preserve an equal distance betweenthem, the op-
posite sides of parallelograms are equal.

The rectangle, the square, and the rhombus, are species of
parallelograms.

61. Equal plane figures are such as contain an equal amount
of area ; also such as, being supposed applied to each other, would
manifestly coincide in their whole extent.

62. Equal solid figures are such as contam an equal amount
of solidity or bulk.

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63. Similar plane figures are such as have the same number
of angles, and each angle in the one equal to an angle in the other,
and the sides adjacent to any angle in the one proportional to the
sides adjacent to the equal angle in the other. And such proper*-
tional sides in two similar figures are called homologous sides.

64. Similar solid figures are such as have an equal number of
similar plane faces. 1 he angles in the one solid being respect-
ively equal to the angles in the other.

65. The diagonal of a parallelogram, or of a cube, or of an oc-
tahedron, is a straight line passing through the center and extend-
ing to two opposite angles.

66. As an angle in a plane figure is formed by the meeting of
lines, which constitute part of a circumference, so an angle in a
solid figure, [commonly called a solid angle,] is formed by the
meeting of planes, which constitute part of a surface.

6T. •^ polygon is the general name applied to plane figures
with any number of sides.

68. ^ regular polygon is one which has all its sides equal to
each other, and all its angles equal to each other. If either the
sides or the angles are unequal, the polygon is called irregular.

Polygons with but few sides are generally designated by names
expressive of the number of their sides or angles. A pentagon has
five sides, « hexagon has six, a heptagon seven, an octagon eight,
a nonagon nine, a decagon ten, an undecagon eleven, a dodecagon
twelve, &c.

69* •/} cylinder is a regular round solid, having a plane circle
for its base, a plane circle for its top or side opposite and paraUel
to the base, and having every point of its curve surface in the cir-
cumference of a circle equal to the base, and also in a straight line
perpendicular to the base.

70. ^ cone is a regular round solid, having a plane circle for
its base, a point for its top or apex, and having every point of its
curve surface in tihie circumference of a circle parallel to the base,
and also in a straight line extending from the apex to the perime-
ter of the base.

Remark. — The last definitioa applies strictly only
to the right cone, in which the apex is in the perpen-

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DEMONSTRATIONS IN GEOMETRY.

dicular from the center of the base. If the cone is
oi/t^ue, sections parallel to the base would not be per-
fect circles.

71* An arc of a circle is any part of its circumference.

72. The chord of an arc is a straight line joining the two ex-
tremities of the arc.

73. The segment of a circle is the part of the area cut off by
a diord, or the part inclosed by an arc and its chord.

74. An cueiom is a self-evident truth, or one so manifest that
it cannot be made more clear by any demonstration ; such as,

First. Magnitudes which are equal to the same thing, are equal
to each other.

Second. Magnitudes which are double, triple, &c., of the same,
or of equal magnitudes, are equal to each other.

Third. Magnitudes which are each one-half, one-third, &c., of
the same or of equal magnitudes, are equal to each other.

Fourth. If equals be added to, or taken from equals, the results
will be^ equal.

Fifth. The whole is greater than a part.

Susth. The whole is equal to the sum of all its parts.

75. A theorem is a truth which is made manifest by a course
of reasoning ; and that course of reasoning is called a demonstr ac-
tion.

76. A problem presents some operation to be performed.

77. A proposition is a general term applied either to a theorem
or a problem.

78. A corollary is an obyious truth, resulting from a demon-
stration.

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84 PART SECOND.

SIGNS.

This sign, + [called plusy or more], when placed after a num-
ber, or series of numbers, denotes that something more is to be
added in order to complq|;e the perfect quantity intended to be re-
presented by the number or numbers.

This sign, — [called minus, or less], when placed after a num-
ber, denotes that something must be subtracted from the number
in order to make it represent the perfect quantity intended.

This sign, =^, placed betweeu two numbers or quantities, de-
notes that they are equal.

This, X, placed between two numbers or quantities, denotes that
they are to be multiplied, the one by the other.

This, -r-, placed between two numbers or quantities, denotes that
the one is to be divided by the other.

This, V, placed before a number, denotes that the square root is
to be extracted.

This, y^ , denotes that the cube root is to be extracted.

RULES.

1. To obtain the area of any rectangle or parallelogram ; mul-
tiply the base by the perpendicular height.

2. To obtain the area of ai^y tdftogle.; multiply half the b&se
by the perpendicular height.

8* To obtain the side of an equilateral triangle, when the per-
pendicular is given ) square the perpendicular, add to the square
one-third of the square, and then extract the square root.

4. To obtain the perpendicular of an equilateral triangle, when
the side is given ; square the side, take three-fourths of the square^
and then extract the square root*

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DEMONSTRATIONS IN GEOMETRY. 85 .

5. To obtain the area of any plane figore from the circumfer-
ence and diameter ; multiply half the circumference by half the
diameter, (diameter always being the diameter of the inscribed
circle.)

6. To obtain the circumference of any plane figure from the
area and diameter; multiply the area by 4, and divide by the dia-
meter.

7. To obtain the diameter of any plane figure from the area
and circumference ; multiply the area by 4, and divide by the cir-
cumference.

8. To obtain the circumference off any circle from the diame-c
ter ; multiply the diameter by the circumference of a circle whose
diameter is 1, viz., by 3.14169+.

9. To obtain the surface of any sphere whose diameter is
given ; first obtain the circumference by the last rule, and then
multiply the circumference of the sphere by its diameter for the
surface.

10. To obtain the solidity of any sphere ; multiply one-third
of the surface by the radius, or half the diameter.

11. To obtain the solidity of a regular pyramid ; multiply the
area of the base by one-third of the perpendicular height.

12. To obtain the solidity of a right cone ; multiply the area
of the base by one-third of the perpen£cular hei^t.

13. To obtam the curve surface of a right cone ; multiply the
perimeter of the base by one-half the side, or half the slant height
of the cone.

14. To obtain the solidity of a cylinder ; multiply the area of
the base by the perpendicular height.

15. To obtain the curve surface of a cylinder ; multiply the
perimeter of the base by the perpendicular height.

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86 PART SECOND.

PROPOSITION I.

The circumference of any magnitude^ assumed as a
geometrical unit^ is a quantity four times as large as
the assumed magnitude ; and it is impossible that it
should be anything greater or anything less. And the
surface of a magnitude^ assumed as a geometrical
Unit, is a quantity six times as large as the assumed
unit ; and it is impossible that it should be anything
greater or anything less.

Let the light square, in the center of
the diagram, be assumed as a geometri-
cal unit, [Definition 7.] It is then a
fixed magnitude, indivisible, invariable
in quantity, and invariable in form. It I
may be assumed of any positive size,!
from magnitude infinitely diminished,!
to magnitude infinitely extended. But'
whatever may be the positive size of
the magnitude assumed, when it is fixed
and bound up into one, and made the
standard of measure, it is a unit in
every particular. Its solidity is 1, its diameter is 1, its length is
1, its breadth is 1, and its thickness is 1. This unit is the only in-
strument with which geometry works. In the hands of geometry,
this little unit, of simple and perfect form, becomes the magician's
wand, decomposing all magnitudes and all forms, and moulding
them into quantities of its own perfect likeness, thus furnishing
the material with which geometry constructs all its perfect works,
and out of which it manufactures all its diameters, its areas, its
circumferences, its solidities, and its surfaces.

Now let geometry commence with the white unit in the center of
the diagram, and supply it with a circmnference ;. that is, surround
it with something which shall touch and entirely cover the four
faces of the unit that look in a horizontal direction. It is the
business of geometry to make this circumference of such form and
such magnitude, that it can perfectly measure it and give an exact
account of it. The only possible way in which geometry can ac-
complish this, is by applying four magnitudes, Uke the four shaded
squares in the diagram, of exactly the same form and size as the
assumed unit. Because, in the first place, it has no other mate*

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DEMONSTRATIONS IN GEOMETRY. 67

rials to work with but just such units ; and in the second place, if
it had other materials of different form or size, they would not an-
swer the purpose. If they were smaller, it is manifest they would
not entirely cover the four faces of the unit, and therefore would not
form a perfect circumference. And if they were larger, it is ma-
nifest they would crowd each other from their position of contact
"with the unit, and therefore would not in that case form a perfect
circumference. And again, since geometry has but one standard
of measure, the indivisible unit, invariable in form and invariable
in size, if a circumference could any way be patched up from ma-
terials of different form or size, it would be utterly impossible for
geometry to measure it and render an exact account of it ; for geo-
metry can measure nothing which does not perfectly agree with the
assumed unit. Therefore it is impossible that the circumference
of a magnitude, assumed as a geometrical unit, should be anything
greater or anything less than four times the quantity of the as-
sumed unit.

If the unit be regarded as a solidy instead of a plane figure, it
tiien has a solidity instead of an area, and must be entirely inclosed
by a surface^ instead of only being surrounded by a circumference.
In that case, there are two more faces to be covered, one at the
top and one at the bottom of the unit, making six faces requiring a
surface. It is the business of geometry to make a surface for this
unit, and to make it of such dimensions that she can measure it
and give an exact account of it. The only possible way that she
can accomplish this, is to apply six magnitudes of exactly the same
form and size as the assumed unit ; for the same reasons and ne-
cessities govern in this case, that governed in the formation of the
circumference. If they were less magnitudes, they could not en-
tirely cover the faces of the unit, and if they were greater, they
would crowd each other from their position of contact with the imit,
so that they could not form a perfect surface, entirely inclosing it
and everywhere touching it. And if they were either greater or
less, or of a different form, geometry could not measure them by
the unit. It is therefore impossible that the surface of a magni-
tude, assumed as a geometrical unit, should be anythmg j^eater or
anything less than six times the quantity of the assumed magnitude.

What is true of the unit of any determined size or magnitude,
is true of units of every possible size ; for numbers know no differ-
ence between small and great, and nature, in her divisibilities and
extensions, as Sir Isaac Newton has well remarked, ^^ is not con-
fined to any bounds."

It may appear to the student, before he has well reflected upon
the matter, that in order to complete the circumference of the
square, as represented in the diagram, there should be applied four

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PART SECOND.

more units to fill the vacant spaces at the four corners of the dia-
gram. But examination will show, that the moment four more
such units are added we have a new square to deal with. It is no
longer a square with a single unit for its diameter, but a square
wil£ three units for diameter, and requiring for its circumference
a line of three units on each of its four sides, makmg twelve for
circumference, and leaving again a vacant space at each of the four
comers, as in the annexed diagram.

Again, if these
four vacant spaces
at the comers were
filled up by placing
ft unit in each of
them, the figure
would no longer be
a square with three
units for diameter,
nine for area, and
twelve for circumfe-
but

rence,

a new

square with five units
for diameter, twenty-
five for area, and re-
quiring twenty more
units, five on each
side, for a new cir-
cumference*

PROPOSITION n.

In the square whose diameter equals two units, the
area equals half its circumference.

The principles already established render but little more neces-
sary, for the proof or illustration of this proposition, than to pre-
sent the diagram for inspection.

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DEMONSTRATIONS IN GEOMETRY.

Let the li^t square in the
center be composed of four
units. Its diameter then is
two units, its area four, and its
shaded circumference eight.
The area therefore equals half
the circumference.

In the arithmetical calctda^
tion of the same square, the
diameter or side of the square
is represented by 2, which be-
ing squared, or multiplied by
itself, produces 4, for the area.
And the side of the square be-
ing 2, and there being 4 sides, 4 multiplied by 2 produces 8, for
the circumference. Area is 4, and circumference 8. Area there-
fore equals half the circumference in numbers, as well as in mag-
nitude or quantity of extension.

PROPOSITION III.

In the square whose diameter is three units, the
area equals three-fourths of its circumference.

In the diagram, the white square
is seen to have a diameter of three
units, and an area of nine units.
The shaded circumference is seen
to consist of twelve units. And
nine is three-fourths of twelve ;
therefore the area equals three-
fourths of the circumference.

In the arithmetical calculation of
the same square, the diameter or

side of the square is 3, and 3 squared

or multiplied by itself gives 9 for the area. And 3, one side,
multiplied by 4, the number of sides, gives 12 for circumference.
Area therefore equals three-fourths of the circumference, in num-
bers as well as in magnitude.

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•0

PART SECOND.

'!■ 'ii,

• ll

I!'

. '■

:.,l

PROPOSITION IV.

In the square whose diameter is four units, the area

equals the circumference.

The white square in the diagram Is
seen to have four units for diameter
and sixteen units for area. The shaded
circumference is also seen to consist of
sixteen units. The area therefore
equals the circumference.

In the arithmetical calculation cfi
the same square, the side is 4, which '
being multiplied by itself produces 16
for area. And 4 for one side, being
multiplied by 4, the number of sides,
gives 16 for circumference. The area therefore equals the cir-
cumference in numbers as well as in magnitude or geometrical
quantity.

Remark. — ^Mathematicians have always been able
to tell us that a square^ whose diameter or side is 4,
has an area of 16, and a circumference of 16. But
while they have always seen this perfect agreement
in numbers, they have always denied that there could
be any agreement in geometrical quantity or magni-
tude between area and circumference, because of the
fundamental error in which they have always rested
in supposing that a mathematical line has no breadth.

Before proceeding to demonstrate the general prin-
ciples of area, circumference, and diameter, applied to
other forms besides the square, it will be necessary
to present two or three preliminary demonstrations,
especially for the help of readers who are not already
familiar with the established truths of geometry.

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DEMONSTRATIONS IN GEOMETRY.

PROPOSITION V.
The diagonal of a rectangle divides the area into
two equal parts.

The rectangle ABCD is divided by the dia- ^^
gonal AC into two triangles, 1 and 2. And,
because the opposite sides of a rectangle or
parallelogram are equal, the sides of one trian-
gle are severally equal to the sides of the other
triangle. AD is equal to BC, DC is equal to
AB, and AC forms the third side of both tri-
angles. If, therefore, we suppose the triangle
1 to be removed and turned round, so as to
bring the angle D upon the angle B, both being right angles, the
side AD woiud coincide with BC, and the side DC would coincide
with AB, and the two triangles would manifestly coincide in their
whole extent, and would therefore be equal, by Definition 61.

The same proposition is true with regard to all parallelograms,
as well as rectangles.

PROPOSITION VI.
The area of a triangle equals the area of a rec-
tangle of the same height and half the base of the
triangle.

Let ABC be an equilateral or isosceles
triangle. CD drawn from the vertex per-
pendicularly to the base AB will divide the
base into two equal parts, and also divide
the area of the triangle into two equal parts.
From C draw CE parallel and equal to AD,
and join A£. Then AD£C will be a rec-
tangle of the same height and half the base
of the triangle ABC. But the right angled triangle ADC is
half the triangle ABC. And, by the last proposition, the same
triangle ADC is half the rectangle ADEC. Therefore the whole
triangle ABC is equal to the whole rectangle ADEC, which is of
the same height and half the base of the triangle.

The same proposition is true wiUi regard to all parallelograms,
M well as rectangles.

Corollary. — If a triangle and a rectangle have the same or
equal bases, and the same or equal heights, the area of tiie triangle
equals half the rectangle. And the same is true with regard to
triangles and parallelograms.

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PART SECOND.

PROPOSITION VII.

The diameter of a circle inscribed in an equilateral
triangle equals two-thirds of the perpendicular of the
triangle.

From the center A, with
the radius AB, equal 1, [in
this diagram the unit is half
an inch,] draw the circle
BLD. With the radius AC,
equal 2, or double AB, draw
the circle CEF. From the
point C draw the straight
line CE touching the inner
circle at L, and draw the
straight line CF touching the
inner circle at G. Then will
the straight line joining E
and F also touch the inner
circle at D, and CEF will
be an equilateral triangle,
because none but an equilateral triangle can have an inscribed and
a circumscribed circle drawn from the same center. Bj construc-
tion the radius AB is 1, therefore the radius AD is 1, [Def. 43.]
AC is double AB, therefore BC is 1. And DB, the diameter of
the circle equals 2, and DC, the perpendicular of the triangle^
equals 3. Therefore the diameter of a circle inscribed in an equi-
lateral triangle equals two-thirds of the perpendicular of the tri-
angle.

Corollary. — From this demonstration it follows, that the per-
pendicular of an equilateral triangle inscribed in a circle is three-
fourths of the diameter of the circle. For AC, the radius rf the
outer circle, is 2 ; therefore the diameter of the circle is 4. But
the perpendicular CD of the inscribed triangle is 3, and is there-
fore three-fourths of the diameter of the circle.

Remarks. — ^We may now proceed to apply the laws
of area, circumference, and diameter to the equilate-
ral triangle, and other forms both regular and irre-
gular.

The student will find great convenience and facility,
in constructing and demonstrating diagrams, by assum-

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DEMONSTRATIONS IN GEOMETRT.

ing units of definite measures, as given on the rule or
scale which he uses, and which he can take in his
compasses and transfer tO' paper. As the rules are
generally marked in inches and parts of inches^ the
unit adopted in most of the following diagrams will
be found to be one inch, or where that would make
the diagram too large, half an inch, or a third, or a
quarter, is sometimes taken for unit.

It may perhaps appear to the reader, in perusing the
following pages, that more demonstrations are given,
than were needed, to illustrate and demonstrate a few
general and simple principles. But when it is remem-
bered that these principles are in direct opposition to
principles laid down and maintained in all works on
geometry from Euclid's time to the present day, some
latitude and diffusiveness in this particular may well
be excused. For as Dr. Barrow has very justly re-
marked, ** no diligence or solicitude should be thought
too much, which is spent in establishing the first prin-
ciples of sciences. It is far better that many demon-
strations be redundant, than for one to seem defec-
tive."

But I feel that there is another argument in favor
of these diffuse and familiar demonstrations. By hold-
ing a principle up to view in three or four different
lights, by a different dress, in several different demon-
strations, the reader becomes master of it with less
labor and fatigue, and with more profit and de-
light. And thus the general reader, as well as the
professed student, may be led on by an easy and
pleasant pathway into the broad and beautiful fields

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PART SECOND.

of geometry, where he may gather the rich fruits of
its discipline and enjoy its perfect and delightful har-
monies.

PROPOSITION VIII.

In the equilateral triangle, whose diameter is (me,
the area equals one-fourth of the circumference.

Let the diame-
ter of the circle,
HK, be oney (one
inch.) Then the
diameter of the
equilateral trian-
gle circumscribed
about the circle is
also one, [Def. IT.]
Let the diameter
of the circle coin-
cide with the per-
pendicular of the
triangle, CH, then
will the base AB
be equally divided
at H ; and the rect-
angle HBIC, will
equal the area of
the triangle, [Prop.
6.] Then because
the radius of the

circle, HO, is half the diameter, HK, and the diameter, HK, is
two-thirds the perpendicular, CH, [Prop. 7,] CH is divided into
three equal parts at the points and K, and lines drawn from
these points parallel to HB manifestly divide the rectangle HBIC
into three equal rectangles, each having a length equal to HB, and
a breadth equal to the radius, HO, or half of one.

On AB construct the rectangle ABDE, having a breadth,
AD, equal to one. Then if this be divided into four equal, rect-
angles, each will have a length equal to HB and a breadth equal
to half of one, and they will, therefore, be severally equal to the
three rectangles contained in HBIC. But the rectangle ABDE
is one-third of the circumference of the triangle, for its length,
AB, is one-third of the perimeter, and its breadth, AD, is equal

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DEMONSTRATIONS IN GEOMETRY.

0§

to one^ [Def. 16.] Therefore the three sides, or whole circuinfer*
ence of the triangle, must be equal to twelve such rectangles as the
four contained in ABDE. It has been shown that the area pf
the triangle is equal to three such rectangles, and three is one-
fourth of twelve. Therefore in the equilateral triangle, whose dia-
meter is one, the area equals one-fourth of the circumference,
agreeably to tlie proposition.

Arithmetical calculation. In the arithmetical calculation of
the same triangle, in decimal numbers, the perpendicular is 1.5.
(Once and a-half, or once and five-tenths of the diameter of the
circle.) To obtain the side of an equilateral triangle, add one-third
to the square of the perpendicular, and extract the square root.
Therefore, 1.5, squared, produces 2.25. A third of this is, .75,
which added to 2.25 produces 3, and the square root of 3, viz.,
1.732+, is the side of the triangle, or one-third of the circumfer-
ence, and multiplied by 3, makes the whole circumference, 5.196-j-.
The area of an equilateral triangle is obtained by multiplying the
perpendicular by half the base or side. Half of 1.T32-J- is .866-f ,
and this multiplied by 1.5 produces for area 1.299-|-, and this
area multiplied by 4, produces 5.196-j-, equal the circumference.
Therefore the area is one-fourth the circumiference in numbers, as
well as in magnitude or geometrical quantity.

PROPOSITION IX.

In the equilateral triangle whose diameter is two,

the area equals half the circumference.

In the present diagram,
which is tibe same as the
last, except that the line of
circumference is but half
the breadth of the last, the
unit is taken at half an inch.
The radius of the circle
therefore is 1, the diame-
ter of the circle and of the
triangle is 2, and the per-
pendicular of the triangle is
3. By the proofs exhibited
in the last proposition, the
rectangle HBIC equals
the area of the triangle
ABC. And this area in
the rectangle HBIC is
divided, as in the last pro-

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PART SECOND.

position, into three Bmaller rectan^eB, eqiuJ to each oth^. The
shaded line of circumference on one side of the trianglle, contains
two such rectangles, its lengdi being equal to one-third of the peri-
meter of the tnangle, and its breadth equal to 1. It is manifest,
therefore, that the whole circmnference would equal six such rect-
angles. And since the area is equal to three, the area of the
triangle whose diuneter is two, equals half its circumference,
agreeably to the proposition.

Arithmetical calculation of the same triangle. — The perpendi-
cular is 3, and 3 squared makes 9. One-third of the square, or 3,
added to the square makes 12. The side of the triangle therefore
is the square root of 12, viz., 3.464-|- ; and this, multiplied by 3,
gives the three sides or circumference of the triangle, viz., 10.392+.
Half the side or base [half of 3.464+] is 1»732+, which being
multiplied by the perpendicular, 3, produces 5.196+ for area.
And 6.196-j- multiplied by 2 produces 10.392+, equal the cir-
cumference. Therefore the area of an equilateral triangle with
two for diameter, equals half its circumference in numbers, as well
as in magnitude or geometrical quantity.

PROPOSITION X.

In the equilateral triangle whose diameter is three,
the area equals three^fourths of the circumference.

In the present diagram, the
diameter of the circle being
three units, the perpendicular
of the triangle, CH, must be
four and a half units, [Prop,
vii.] Therefore if CH be
divided into nine equal parts,
each part will equal half the
unit, and six ef the parts will
be contained in the diameter
of the circle. From these /
points of division in CH let --.
lines be drawn parallel to
HB, and it is manifest they
will divide the rectangle
HBIC into nine rectangles,
equal to each other. Now let AD equal one-third the diameter
of the circle, that is, equal the unit. If AD be divided in the
center, each part would equal half the unit, and AB being twice

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DEMONSTRATIONS IN GEOMETRY.

the length of HB, it is manifest that the rectangle ADEB con-
tains four rectangles, each equal to each of the nine rectangles
contained in HBIC. But the four rectangles constitute the
shaded line of circumference on one side of the triangle, having a
breadth of 1, and a length equal to the side of the triangle. There-
fore the three sides, or the whole circumference, must be composed
of twelve such rectangles. The nine rectangles in HBIC equal
the area of the triangle ABC [Prop, vi.], and nine is three-fourths
of twelve. Therefore the area of an equilateral triangle, whose
diameter is three, equals three-fourths of its circumference, agree-
ably to the proposition.

Arithmetical calculation. — The perpendicular of the triangle
being 4.5, the square of it is 20.25, and one-third of the square is
6.T6, which added to 20.26, makes 27. The square root of 27,
viz., 5,196-j-, therefore equals one side of the triangle, and multi-
plied by 3 produces 15.688+ for circumference. Half the base,
or half of one side, viz., 2.698-j-, multiplied by 4.5, the perpendi-
cular, produces 11.691+ for area, and this is three-fourths of
15.588+ ; for the last number divided by 4 and multiplied by 3
produces 11.691+. Therefore the area of an equilateral triangle,
whose diameter is 3, equals tln*ee-fourths of its circumference in
numbers, as well as in magnitude or geometrical quantity.

PROPOSITION XI.

In the equilateral triangle, whose diameter is fouVy

the area equals the circumference.

In this diagram the circle
and the triangle are still of
the same 'positive si^e as in
the preceding, that is, the
positive length of the diame-
ter of the circle is one inch ;
but one inch is not the unit
of lie calculation. The va-
lue of the unit has been di-
minished in each successive
triangle, to avoid the neces-
sity of making the diagrams
of larger size. The unit in
the first triangle was one inch,
in the second it was half-an-
inch, in the third it was a third of an inch, and, in the present, it
7

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98 ' PART SECOND.

is the fourth part of an inch. And the diameter of the circle con-
taining four units, the perpendicular of the triangle, CH, must
contain six units, [Prop, vii.] Therefore let the perpendicular he
divided into six equal parts, and lines be drawn parallel to HB,
dividing the rectangle HBIC into six equal rectangles, which will
together equal the area of the triangle ABC, [Prop, vi,] The
shaded line of circumference being drawn with a breadth equal to
1, it is manifest that the rectangle ADEB, which constitutes the
circumference on one side of the triangle, contains two rectangles,
severally equal to those contained in HBIC. Therefore the
whole circumference would be composed of six such rectangles ;
and the area of the equilateral triangle, whose diameter is 4, is
therefore equal to its circumference, agreeably to the proposition.

Arithmetical calculation. — The perpendicular of the triangle
being 6, the square of it is 36, and a third of the square added,
makes 48. Therefore the square root of 48, viz., 6.928+j equals
one side of the triangle ; and multiplied by 3, produces 20.784+ for
the three sides or whole circumference. And half of one side, that
is, half the base, 3.464, multiplied by 6, the perpendicular, pro-
duces 20.7844- for area, which is the same that was obtained for
circumference. Therefore the area of an equilateral triangle, whose
diameter is 4, is equal to its circumference in numbers, as well as
in magnitude or geometrical quantity.

Remark. — ^Thus it has been seen in the last four
propositions, that in equilateral triangles, as well as
squares, there is a perfect agreement in geometrical
quantity between the area of the triangle and its line
of circumference. It is seen also that the law of agree-
ment is precisely the same in the triangle as in the
square — viz., if diameter is 1 area equals one-fourth
of the circumference ; if diameter is 2, area equals
half the circumference ; if diameter is 3, area equals
three-fourths of the circumference ; and if diameter is
4, area and circumference are equal. We shall soon
see how this same simple and beautiful law extends
to all other forms.

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DEMONSTRATIONS IN GEOMETRY.

PROPOSITION xn.

In the regular pentagon^ whose diameter is one^ the
area equals one-fourth of the circumference.

Let the circle in the diagram be described with a diameter of
one, [one inch,] and draw aromid it a regular pentagon. The
diameter of the pentagon will then be one, [Definition 17.] Be-
cause it is a regular pentagon, the sides are all equal, and the
angles are all equal, [Definition 68.] Therefore it is manifest that
lines drawn from the center to the five angles of the pent^^on di-
vide the area into five equal isosceles triangles ; for the base of
each triangle is one side of the pentagon, as AB, and the height or
perpendicidar of each triangle is the radius of the circle, as CD.
The triangle ABC therefore is one-fifth of the area of the penta-
gon ; and let DBCE be a rectangle, and the triangle and rectangle
are equal to each other, [Prop. 6.] The area of the pentagon,
therefore, is equal to five such rectangles as DBCE. Let the ra-
dius of die circle CD be produced to H, making DH equal to
one — that is, equal to the diameter of the circle, or twice CD, and
draw the rectangle DBHG. This rectangle then will manifestly
be equal to two such rectangles as DBCE, for it has the same base

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100

PART SECOND.

and double the height. And ABFG is manifestly comp^ed of
four such rectangles. But ABFG is one-fifth of the circumference
of the pentagon, for it has the length of one-fifth of the perimeter,
AB, and a breadth of one, DH, [Definition 16.] Therefore the
whole circumference would be composed of twenty such rectangles
as DBCE. The area equals five such rectangles, and five is one-
fourth of twenty ; therefore in the regular pentagon whose diame-
ter is one, the area equals one-fourth of the circumference, agree-
ably to the proposition.

PROPOSITION XIII.

In the regular pentagon, whose diameter is two, the
area equals one-half the circumference.

* The circle and pentagon
in the present diagram be-
ing the same as in the last,
if we take the unit at half
an inch, diameter will be
two, CD, the radius, will
be one, and DH, made
equal to CD, will be the
breadth of the line of cir-
cumference. As was shown
in the last proposition, the
area of the pentagon is
equal to five such rectan-
gles as DBCE. The two
rectangles contained in
ABFG are each equal to
the rectangle DBCE, for by construction they each have an equal
base and equal height. But ABFG is one-fifth of the circum-
ference of the pentagon ; therefore the whole circumference must
be composed of ten such rectangles. And the area of the penta-
gon being equal to five such rectangles, the area is equal to half
the circumference, agreeably to the proposition.

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DEMONSTRATIONS IN GEOMETRY.

101

PROPOSITION XIV.

In the regular pentagon, whose diameter is three,
the area equals three-fourths t)f the circumference.

Jn the present diagram, diameter being three, the radius CD is
one and a half. Therefore if DH be made equal to one, it will
equal two-thirds of CD. Then let CD be divided into three equal
parts, and DH into two equal parts, and it is manifest that the
three rectangles drawn in DBCE, and the four rectangles drawn
in ABFG are severally equal to
each other, for by construction
tliey all have equal bases and
equal heights. But the area of
the pentagon is equal to five such
rectangles as DBCE; therefore it
is equal to fifteen of the smaller
rectangles. And the circumfer-
ence of the pentagon is equal to
five such rectangles as ABFG;
therefore it is equal to twenty of
the smaller rectangles. Fifteen
is three-fourths of twenty, there-
fore the area of the pentagon,
whose diameter is three, equals three«fourths of its circumference*
agreeably to the proposition.

PROPOSITION XV.

In the regular pentagon, whose diameter is four^
the area equals the circumference.

In the present diagram, diameter
being four, the radius, CD, is two.
If CD be produced to H, making DH
equal to one, the breadth of the line
of circumference, it is mimifest that
the two rectangles drawn in DBCE
are severally equal to the two rect-
angles drawn in ABFG, for they all
have equal bases and equal heights
by construction. But DBCE equals
one-fifth of the area of the pentagon ;
therefore the area of the pentagon
equals ten such rectangles as the two

u (I

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PART SECOND.

contained in DBCE. ABFG is one-fifth of the circumference of
the pentagon, therefore the circumference equals ten such recU
angles as the two contained in ABFO. Therefore the area of the
pentagon, whose diameter is four, equals its circumference, agree-
ably to the proposition.

PROPOSITION XVI.

In the regular hexagon, whose diameter is one, the
area equals one-fourth of the circumference.

Let the diameter of the circle be unit, or one, [one inch ; ] then
the diameter of the regular hexagon, circumscribed about it, is
one ; [Def. 17.] The sides of the hexagon being equal, [Def. 68,]

lines drawn from the center to the six angles divide the area into
six equal triangles, for they all have equal bases and equal heights.
One triangle, as ABC, is equal to the rectangle, DBCE, [Prop. 6.]
Therefore the area of the hexagon equals six such rectangles as
DBCE.

Let CD be produced to H, making DH equal to one, that is,
equal to the diameter of the circle. Then DH will be double CD,
and AB being double DB, it is manifest that the rectangle ABFG

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DEMONSTRATIONS IN GEOMETRY.

103

contains four rectangles, each equal to DBCE, for they have equal
bases and equal heights by construction. But ABF6 is one-sixth
of the circuinf^renoe of the hexagon, haying a length of one-sixth
of the perimeter, AB, and a breadth, DH, equal to one, [Def. 16.]
Therefore the whole circumference must be composed of twenty-
four such rectangles as the four contained in ABFO. And the
area of the hexagon being equal to six such rectangles, therefore
the area of a hexagon, whose diameter is one, equals one-fourth of
its circumference, agreeably to the proposition.

PROPOSITION XVII.

In^the regular hexagon, whose diameter is two, the
area equals one-half the circumference.

Diameter being two, the
radius, CD, equals one, and
DH or AF, made equal
to one, is the breadth of
the line of circumference.
Lines drawn from the cen-
ter to the angles of ike
hexagon divide the area in-
to six equal and equilateral
triangles. One of these
triangles, ABC, [Prop. 6.]
is equal to the rectangle
BDCE. Therefore the
whole area of the hexagon
must be equal to six such
rectangles. The two rect-
angles contained in ABFO
are each equal to BDCE, for by construction they have equal bases
and equal heights. The rectangle ABFG is one-sixth of the cir-
cumference of the hexagon, for the length, AB, is one-sixth of the
perimeter, and the breadth, AF, is one. Therefore the whole cir-
cumference must be equal to twelire such rectangles as the two
contained in ABFG. The area is equal to six such rectangles.
Therefore in the regular hexagon whose diameter is two, the area
equals one-half the circumference, agreeably to the proposition.

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104

PART SECOND.

PROPOSITION xvni.

In the regular hexagon, whose diameter is threCy the

area equals thre^e-fourths of the circumference.

In this diagram, the diameter
being three, the radius, CD, equals
one and arhalf. And DH being
drawn equal to the breadth of the
line of circumference, and there-
fore equal to one, is consequently
equal to two-thirds of CD. There-
fore let CD be divided into three
equal parts, and DH into two
equal parts, and let the small rect-
angles be completed, and it is mani-
fest that the four rectangles con-
tained in the rectangle ABFG are
severally equal to each of the three rectangles contained in DBCE.
But DBCE equals one-sixth of the area of the hexagon ; therefore
the whole area of the hexagon must equal eighteen such rectangles
as the three contained in DBCE. And ABFG is one-sixth of the
circumference ; therefore the whole circumference must be com-
posed of twenty-four such rectangles as the four contained in
ABFG. Eighteen is three-fourths of twenty-four, therefore the
area of a hexagon, whose diameter is three, equals three-fourths of
its circumference, agreeably to the proposition.

PROPOSITION XIX.

In the regular hexagon, whose diameter i&fouVy the

area equals the circumference.

In the present diagram, the diame-
ter is divided into four units^ there-
fore CD, the radius, is two, and DH,
the breadth of the line of circumfer
ence, is made equal to one. Cb bef
double of DH, let CD be dividei
in the center, and the small rectangles
completed. Then it is manifest that
the two rectangles contained in ABFG
are severally equal to the two con-
tained in DBCE, for they all have
equal bases and equal heights by con-

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DEMONSTRATIONS IN GEOBiETRT.

106

struotion. But DBCE is equal to one-sixth of the area of the
hexagon ; therefore the whole area of ike hexagon is equal to twelve
such rectangles as the two contained in DBCE. And ABFO is
one-sixth of the circumference of the hexagon ; therefore the whole
circumference must be composed of twelve such rectangles as the
two contained in ABFO. Area and circumference, each being
equal to twelve such rectangles, are equal to each other. There-
fore the area of a hexagon, whose diuneter is four, is equal to its
circumference, agreeably to the proposition.

PROPOSITION XX.

In the regular octagon, whose diameter is one, the
area equals one-fourth of the circumference.

Take the diameter of the circle for unit, and then the diameter
of the octagon is one, [Def. 17.] And the radius, CD, is hatf of
one. Let CD be produced to H, making DH equal to one, or
double CD ; and AB being double DB, it is manifest that the
rectangle DBCE is equal to each of the four rectangles contained

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106

PABT SECOND.

in the rectangle ABFG^ for they haye equal bases and equal hd^ts
by construction. Lines drawn from the center to each of the eight
angles of the octagon, divide the area into eight equal triangles, for
t^e base of each triangle is an equal side of the octagon, and the
height or perpendicular of each triangle equals the radius CD.
One of the triangles, ABC, is equal to tlie rectangle DBCE, [Prop.
6.] Therefore the area of the octagon equals ei^t such rectangles
as DBCE. AB being one-eighth of the perimeter of the octag(m,
and DH being equal to one, me rectangle ABFO constitutes one-
eighth of the circumference, FDef. 16.] Therefore the whole cir-
cmnference must be composed of thirty-two such rectangles as the
four contained in ABFG. And the area being equal to eight such
rectangles, therefore the area of an octagon, whose diameter is one,
is equal to one-fourth of its circumference, agreeably to the propo-
sition.

wwm

/"\

.■•^

PROPOSITION XXI.

In the regular octagon, whose diameter is two, the

area equals half the circumference.

Let the diameter of the cir-
cle and of the octagon be two ;
then the radius, CD, equals
one. DH or AF also equals
one, being the breadth of the
line of circumference. And
AB being double DB, it is
manifest that each of the two
rectangles contained in ABFG
is equal to the rectangle
DBCE, for they have an equal
breadth and equal height by
construction. ABFG is one-
eighth of the circumference of
the octagon ; therefore the
whole circumference must be
equal to sixteen such rectangles as the two contained in ABFG. The
area of the octagon equals eight such rectangles as BDCE, [Prop.
6.] Therefore the area of an octagon, whose diameter is two,
equals one-half its circumference, agreeably to the proposition.

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DEMONSTRATIONS IN GEOMETRY.

107

PROPOSmON XXII.

In the regular octagon, whose diameter is three^ the
area equals three-fourths of the circumference.

In the present diagram^ diameter
being three, the radius CD is one
and a half, and DH, the breadth of
the line of circumference, is one,
and therefore equal to two-thirds of
CD. Let CD be divided into three
equal parts, and DH into two equal
jyarts, and the small rectangles
drawn upon the equal parts will be
equal, having equal bases and equal
heights. As was shown in the two
last propositions, the area of the oc-
tagon is equal to eight such rect-
angles as DBCE, therefore it must be equal to twenty-four such
rectangles as the three contained in DBCE. And ABFG being
one-eighth of the circumference of the octagon, the whole circum-
ference must be equal to thirty-two such rectangles as the four
contained in ABFG. Twenty-four is three-fourths of thirty-two,
therefore the area of an octagon, whose diameter is three, equals
three-fourths of its circumference, agreeably to the proposition.

PROPOSITION XXIII.

In the regular octagon, whose diameter is four, the

area equals the circumference.

In this diagram, diameter is di-
vided into four units ; therefore the
radius CD is two, andDH, the breadth
of the line of circumference, is one.
CD being divided into two equal
parts, and AB being double DB, it
is manifest that the two rectangles
contained in ABFO are equal to the
two rectangles in DBCE, for they
all have equal bases and equal heights.
As already shown in the preceding
propositions, the area of the octagon
IS equal to eight such rectangles as

''lU^

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108 PABT SECOND.

DBCE ; it is therefore equal to sixteen sucli rectangles as the two
contained in DBCE. The rectangle ABFG is one-eighth of the
circumference of the octagon, therefore the whole circumference
must be equal to sixteen such rectangles as the two contained in
ABFO. Therefore the area of an octagon, whose diameter is four,
is equal in geometrical quantity to its circumference, agreeably to
the proposition.

Remark. — ^In all the preceding propositions, diame-
ter has been the given or assumed quantity, and it
has been seen to control the relation between area
and circumference by one simple and uniform law,
whatever may be the form or figure in which the area
is presented. We shall now see that when circum-
ference is the given or assumed quantity, it controls
the relation between area and diameter by precisely
the same simple and uniform law, so rigidly enforced
by diameter in the preceding cases.

PROPOSITION XXIV.

In the square, whose circumference is one^ the area
equals one-fourth of the diameter.

Remarks. — ^The terms of the proposition may seem
to require a few words of explanation, to make them
appear consistent with the principles already laid
down. Since the unit in geometry always has the
form of a cube, and when once assumed, is always
indivisible, and invariable in form or size, the question
may arise in the mind of the student, how can one
constitute a circumference, or inclose area ? It is
manifest that the one, assumed as the unit, cannot in-
close area. But we may assume smaller units, a cer-.

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DEMONSTRATIONS IN GEOMETRY. 109

tain number of which shall together be equal in size
and value to the first assumed unit, and out of these
smaller units we may form a circumference that will
inclose an amount of area having a certain and fixed
relation to the first assumed unit. When therefore
we speak of a square whose circumference is one, it
simply means a square whose circumference is equal
in quantity to an assumed unit, that is, a square whose
side is equal to the fourth part of an assumed unit.

Now, in pursuing the proposition, we will assume
the same quantity for the unit which we have been
using in the previous demonstrations, viz., one inch.
In comparing diameter and area, we have nothing
whatever to do with circumference, except to con-
sider its extension simply in length, for it is this
length which controls the relation between diam<iter
and area.

Therefore let AB be one [one inch]. Let ^ ,

it be divided into four equal parts, [which f — I I V^
simply means, let four smaller units be assumed, [.^ | | . ^-4

each of which shall be equal to a fourth part
of AB,] and out of these four parts form the square BCDE. Then
the square BCDE, having a circumference which is equal in quan-
tity to AB, may be said to have a circumference of one, and by the
terms of the proposition, the area of this square must equal one-
fonrth of its diameter. The diameter of the square is the diameter
of its inscribed circle, [Def. 17.] And this diameter must be
measured by a perfect /ine, whose breadth is fixed by the assumed
unit. The diameter, passing through the center of the circle, must
extend in length to the extreme limits of the -circle, and will there-
fore in length be equal to BE or AF, [Def. 19.] But the breadth
of the diameter is oncy that is, equal to the length of AB ; for a
line is always one in breadth, and varies not, whether it be to mea-
sure a quarter of an inch or a thousand iriches. Therefore the
rectangle ABEF is the diameter of the square BCDE. And AB
being divided into four equal parts, the square BCDE is manifestly
a fourth part of the rectangle ABEF, for it has the same height,
and a fourth part of the base of the rectangle. Therefore, m the

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PART SECOND.

square whose cireamferenee is oney the area equals <Hie-f<mrih of
the diameter^ agreeably to the proposition*

PROPOSITION XXV.

In the square, whose circumference is two, the area
equals one half the diameter.

Let AB be one, [one inch.] Divide it •»

into two equal parts at C, and on AC con- /^ "\
struct the square ACDF. Then each side
of the square will be half of one, and the ^ J
circumference of the square ACDF equals a^ ^ : *-^j ,
two. On CB also construct the square
CBDE, which will also have a circumference of two, and the two
squares will be equal to each other. The whole rectangle ABEF
is the diameter of the square ACDF, for the breadth of this rect-
angle, AB, is oncj and the length, AF or CD, extends to the
heidit or extreme limits of the inscribed circle, [Definitions 17 and
19.T But this rectangle ABEF, which constitutes the diameter^
is double the area of the square ACDF. Therefore in the square
whose circumference is two, the area equals one-half the diameter,
agreeably to the proposition.

PROPOSITION XXVI.

In the square, whose circumference is three, the
area equals three-fourths of the diameter.

Again, let AB be one. Divide it into four
equal parts, and on three of these parts CB,con-
struct the square BCDE. Each side of this
square is composed of three parts, each equal
to a quarter of BA9 making twelve quarters.
Twelve quarters equal three units, therefore
tiie circumference oi the square BCDJS equals
three. BE is the length of the diameter,
and AB is the breadth of the diameter ; therefore the rectangle
ABEF is the diameter of the cunole and of the square BCDE.
Divide the rectangle ABEF into equal squares on the four equal
piirts of AB^ and the rectangle ABEF, or the diameter, will be
seen to contam twehre of these equal squares* The square BCDE
will be seen to contam nine of these small equal squares ; and nine
is three-fourths of twelve. Therefore in tiie square, whose cir-

JL (

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DEMONSTRATIONS IN GEOMETRT. Ill

ouioferenGe is three, the area equals three-fourths of the dianketMr^
agreeably to the proposition.

PROPOSITION XXVIL
In the square, whose circumference is four, the area
equals the diameter.

Again, let AB be one, and upon it construct ^
the square ABCD. Each side of the square
is equal to one, [Def. 58,] tlierefore the cir-
cumference is equal to four. The length of
the diameter is BC, which now equals one,
and the breadth of the diameter is AB, which
is one ; therefore the diameter is one in length
and one in breadth, or one square. But the ^
area also is one square. Therefore in the
square, whose circumference is four, the area equals the diameter,
agreeably to the proposition.

Corollary. — The diameter of a circle of one diameter is

Precisely equal to the circumscribed square of thq same circle,
Definitions 19 and 21.]

Remark. — ^The student may now be prepared to
see why, in numbers, 1 can never be made anything
but 1 by any operation performed upon it. If 2 be
squared, it becomes 4, and if it be cubed it becomes
8. But I squared, is still 1, and 1 cubed is still 1,
and the root of 1 is still 1, and sometimes to the no
small embarrassment and perplexity of the student
or mathematician. The mystery and difficulty seem
to vanish when we understand that 1 represents a
magnitudcy which is 1 in every particular ; a magni-
tude which is 1 in length, 1 in breadth, and 1 in thick-
ness; a magnitude whose diameter is 1, whose area is
1, and whose solidity is 1 ; a magnitude whose square
is 1, and whose cube is 1, and whose square root is 1,
and whose cube root is 1. As the unit is one in all
these particulars, the number which represents each

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112 PART SECOND.

and all of these particulars must always be 1, and can
never be made anything else.

i PROPOSITION xxvin.

In the equilateral triangle, whose circumference is
one, the area equals one-fourth of the diameter.

Let AB be the unit, [one inch.] Divide
AB into six equal parts, and on two of the xl.
parts, CB, equal to one-third of AB, erect the j-
equilateral triangle BCD. Each side of the •«-
triangle being equal to a third of AB, or a third of onej the whole
circumference of the triangle is equal to one. And DH being the
perpendicular of the triangle BCD, the area of the triangle is
equal to the rectangle BHDE, [Proposition 6.] The rectangle
BHDE is divided into three equal smaller rectangles by lines drawn
parallel to the base, one through the center of the circle and one
touchmg its upper limit, because the diameter or height of the cir-
cle is equal to two-thirds of the perpendicular of the triangle, [Pro-
position 7.] And these two lines being produced parallel to AB
and equal in length to AB, and perpendiculars being drawn from
the several points of division in AB, it is manifest that the whole
figure will be divided into small rectangles equal to each other.
The length of the diameter of the circle and of the triangle is AF,
and the breadth of the diameter is AB, therefore the diameter is
seen to contain twelve of the small equal rectangles. And BHDE,
which equals the area of the triangle, contains three such rectangles.
And three is one-fourth of twelve. Therefore the area of an equila-
teral triangle, whose circtimfere^e is one, equals one-fourth of its
diameter, agreeably to the proposition.

Arithmetical calculation of the same triangle. The side of
the triangle being a third of 1, the decimal expression is .333338+,
which being squared gives .111111+. Three-fenorths of this
square is .0833333+, tiie square root of which is .288675+. And
this last number is the perpendicular of the triangle. But the
diameter of a circle inscribed in an equilateral triangle is two-thirds
the perpendicular. Therefore the diameter of the circle, and of
the triangle, is two-thirds of .288676+, viz., .19246+ ; and one-
fourth of this last number, by the proposition, must equal the area.
Multiply the perpendicular, .288675+^ by half the base or side,
viz., .166666+, and it gives for the area of the triangle .048112+,
and this multiplied by 4 gives .1924+, equal to the diameter.

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DEMONSTRATIONS IN GEOMETRY. 113

Therefore in the equilateral triangle whose oircnmference is 1, the
area equals onoTfourth of the diameteir in numbers as well as in
geometrical quantity.

PROPOSITION XXIX.

In the equilateral triangle, whose circumference is
two, the area equals one-half the diameter.

Let AB be the unit, divided into thtee equal
parts at C and H. On CB erect the equila-
teral triangle BCD. Each side of the triangle
gquals two-thirds of one, therefore the whole
circumference equals six-thirds q( one. or
equals two. The area of the triangle BCD
equals the rectangle BHDE, [Proposition 6.]
And the rectangle BHDE is divided as in the last proposition into
three equal rectangles. The diameter has a length, AF, equal to
two-thirds of DH, and a breadth, AB, equal to one. Therefore
by lines drawn as in the last proposition, the diameter is seen to be
divided into six rectangles, each similar and equal to the three rect-
angles in BHDE. Area is equal to three rectangles, and diameter
to six. Therefore in the equilateral triangle, whose circumference
is twoy the area equals half the diameter, agreeably to the propo-
sition.

Arithmetical calculation of the same triangle. The side of
the triangle being two-thirds' of 1, is .666666-f-, which being
squared is .4444444-. ^^^ three-fourths of tbe square is
•8333334-, the square root of which is .57735+, and this is the per-
pendicular of the triangle. And the perpendicular multiplied by
half the base, viz., .333334- gives for area .192454-* And two-
thirds of the perpendicular, which is the diameter of the circle, gives
.3849+, which is double .19245+. Therefore in the equilateral
triangle, whose circumference is two, the area equals one-half the
diameter in numbers as well as in geometrical quantity.

PROPOSITION

In the equilateral triangle, whose circumference is
three, the area equals three-fourths of the diameter.

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PART SECOND.

Take AB equal to one, and upon it con-
struct the equilateral triangle ABC. Each
side of the triangle then is one, and the cir-
cumference is 3. The area oj^ the triangle
is equal to the rectangle BDCE, [Prop, vi.]
AF is the length of the diameter of the
circle, and AB the breadth, therefore the
rectangle BAFO is the diameter, and it is
seen to be divided into four equal rectangles.
And BDCE, which is equal to the area, is divided into three simi-
lar equal rectangles. Therefore, in the equilateral triangle whose
circumference is three, the area equals three-fourths of the diame-
ter, agreeably to the proposition.

Jlrithmetical calculation of the same triangle. — The side qf
the triangle being 1, the square of the side is 1, and three-quarters
of the square is .75. Therefore the ^uare root of the decimal .75.
viz., .8660254+, is the perpendicular of the triangle. And two-
thirds of the perpendicular, viz., .5773502+, is the diameter, [Prop,
vii.] Now for the area, multiply the perpendicular, .8660264+,
by half the base, that is, by half of 1, or the decimal .5, and it
gives .4330127+, and this last sum is three-fourths of .5778502+,
which equals the diameter. Therefore in the equilateral triangle,
whose circumference is three, the area equals three-fourths of the
diameter in numbers, as well as in geometrical quantity.

PROPOSITION XXXI.

In the equilateral triangle, whose circumference is
four, the area equals the diameter.

Take AB equal to one atud one-third.
Divide AB into four equal parts, and
vHB, being three of the parts, will be
equal to one, or the unit. On AB erect
the equilateral triangle ABC ; then
each side of the triangle being one and
one-third, the whole circumference will
be four. The area of the triangle
equals the rectangle BDCE, [Prop. vi. 1 .
And the diameter of the triangle and a
;the circle is the rectangle BHFL, hav-
ing a breadth of one, HB, and extending in length to the extreme
limits of the circle, [Definitions 17 and 19.] HB is divided into
three equal parts, and CD is also divided into three equal parts at
.the center and circumference of the circle. Therefore, lines drawn

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DEMONSTRATIONS IN GEOMETRY.

115

through these points of division, divide the whole figure into small
rectangles equAl to each other. BDCE, equal to area, contains
six of these rectangles ; and BHFL, the ^ameter, also contains six
of these rectangles. Therefore, in the equilateral triangle, whose
circumference is four, the area equals the diameter, agreeably to
the proposition.

Arithmetictd calculation of the same triangle.-rThe side of
the triangle being one and one-third, viz., 1.333333-f-> the square
of the side is 1.777777+ ; and three-quarters of the square is
1.3333333+, [equal to the side.] The square root of this last
sum, viz., 1.1547+,is the perpenoicular of uie triangle ; and two-
thirds of the perpendicular, viz.. .7698+ is the diameter, [Prop,
vii.] The perpendicular, 1.1647+, multiplied by half the base,
•666666-I-, gives also .7698+ for area. Therefore in the equi-
lateral triangle, whose circumference is four, the area equals the
diajneter in numbers, as well as in geometrical quantity.

PROPOSITION XXXII.

In the regular pentagon, whose circumference is one,
the area equals one fourth part of the diameter.

Let AB be the unit. [In
this instance the unit is
taken at a length of two
inches instead of one inch,
in order to give sufficient
distinctness to the dia-
gram.] Then if AB be one, the side of a regular pentagon, whose
circumference id one, must be one-fifth of AB. Therefore divide
AB into ten equal parts, and upon two of those parts, CB, con-
struct a. regular pentagon. The pentagon will then have a cir^
cumference equal to one. The diameter of the pentagon and of
the inscribed circle is the rectangle ABFG, for AB is the breadth
of the diameter, being one, and AF or BG is the length of the dia-
meter, as they extend the line of diameter in length to the extreme
limits of the circle. AB being divided into ten equal parts, per-
pendiculars drawn from those points of division, and a line drawn
parallel to AB through the center of the circle, manifestly divide
the rectangle ABFG into twenty equal rectangles. The area of
the pentagon is divided into five equal triangles by the lines drawn
from the center to the five angles. One triangle, BCD^ is equal to
one rectangle, BEDH, [Prop, vi.] Therefore the five triangles,
or the whole area of the pentagon, is equal to five of the rectangles.

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PART SECOND.

And five is one-fourth of twenty ; therefore in the regular penta*
gon, whose circumference is one, the area equals one-fourth of the
diameter, agreeably to the proposition*

(

7^.

PROPOSITION xxiiii.

In the regular pentagon, whose circumference is
twoy the area equals one-half the diameter.

Let AB be one. [In this diagram it is
again one inch;] Divide AB into five equal
parts, and on two of the parts, CB, construct a
regular pentagon* The circumference of the
pentagon will then be equal to ten such parts,
or two units. The diameter is the rectangle
ABFG, afiid it is seen to be divided j as was shown in^e last dia-
gram, into ten equal rectangles. The area of the pentagon is di-
vided into five equal triangles, the base of each triangle being a
side of the pentagon. And one triangle, BCD, is equal to one
rectangle, BEDH, of the same height and half the base of the tri-
angle, [Prop. 6.] Therefore the five triangles, or the whole area,
is equal to five of the rectangles, of which the diameter contains
ten. Therefore in the regular pentagon, whose circumference is
two, the area equals half the diameter, agreeably to the j^ropo-
sition.

PROPOSITION XXXIV.

In the regular pentagon, whose circumference is '
three, the area equals three-fourths of the diameter.

Let AB be one. Divide it into ten
equal parts, and on six of the parts, CB,
construct a regular pentagon* Each side
of the pentagon then being equal to six-
tenths of one, the five sides, or whole
circumference, will be equal to thirty-
tenths, or three units. AB being one,
and AF the height of the circle^ the
rectangle ABFG constitutes the diame-
ter. And by the equal divisions of AB, and the line through the
center of the circle parallel to AB, the diameter is seen to, be di-
vided into twenty equal rectangles. The area of the pentagon is
divided mto five equal triangles, and one triangle, BCD, is equal

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117

to three rectangles, contained in BEDH, [Prop. 6.] Therefore
the five triangles, or the whole area, most be equal to fifteen of the
rectangles ; and fifteen is three-fourths of twenty. Therefore in
the regular pentagon, whose circumference is three, the area equals
three-fourths of l£e diameter, agreeably to the proposition.

PROPOSITION XXXV.

In the regular pentagon, whose circumference is
/bur, the area equals the diameter.

Let AB be one, and divide it into five
equal parts. On four of the parts, CB,
construct a regular pentagon. Each
side of the pentagon will then be equal
to four-fifths of one, and the five sides,
or whole tsircumference, will be equal to
twenty-fifths of one, or equal to four
units. AB is the breadth of the diame-
ter, and AF the length, and the rect-
angle ABFO, which constitutes the dia-
meter, is seen to be divided into ten
equal rectangles, having equal bases — ^viz., one-fifth <^ AB, and
equal heights — ^viz., the radius of the circle, DE. The area of the'
pentagon is divided into five equal triangles, one of which, BCD,
is equal to two of the rectangles, contained in BEDH, [Prop. 6.]
Therefore the five triangles, or the whole area of tl^e pentagon,
must be equsd to ten such rectangles, and consequently equal to
the diameter. Therefore in the regular pentagon, whose circum^
ference is four, the area equals the £ameter, agreeably to the pro-
position.

PROPOSITION XXXVI.

In the regular hexagon, whose circumference is one^
the area equals one-^fourth of the diameter.

Let AB be the unit.
[It is taken at the length
of two inches, in order to
be large enough to give
distinctness to the dia-
gram.] Then if AB be
one, let it be divided into twelve equal parts, and upon two of the
parts, CB, construct a regular hexagon. The circumference of the

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PART SECOND.

hexagon will then be equal to AB^ for each of the six sides is equal
to two of the divisions of AB. The circumference therefore is
equal to one. AF is the length of the diameter^ being equal to the
height of the circle, and AB is the breadth of the diameter because
it is the unit. Therefore the rectangle ABFG constitutes the dia-
meter, and it is seen to be divided into twenty-four equal rectangles
by the equal divisions' of AB, and the line parallel to AB drawn
through the center of the circle. The area of the hexagon is di-
vided into six equal and equilateral triangles, the base of each
triangle being a side of the hexagon, and its perpendicular the
radius of the circle, as DE. One triangle, BCD, is equal to one
of the rectangles, BEDH, [Prop, vi.] Therefore the six triangles,
or the whole area of the hexagon, must be equal to six of the rect-
angles. But the diameter is equal to twenty-four such rectangles.
Therefore in the regular hexagon whose circumference is one, the
area equals one-fourth part of the diameter, agreeably to the pro-
position.

PROPOSITION XXXVII.

In the regular hexagon, whose circumference is two,

the area equals one-half the diameter.

Let AB be one, and divide it into six
equal parts. On two of the parts, CB,
construct a regular hexagon. The six sides,
or circumference of the hexagon, will then
be equal to twelve such parts, or equal to
two units, and circumference will be two.
The rectangle ABFG is the diameter, as previously shown, and it
is seen to be divided into twelve equal rectangles. The area of the
hexagon is divided into six equal triangles. And one triangle, BCD,
is equal to one of the rectangles, BEDH. Therefore the whole
area of the hexagon is equal to six of the rectangles, and diameter
being equal to twelve, area equals half the diameter. Therefore
in the regular hexagon, whose circumference is two, the area equals
half the diameter, agreeably to the proposition.

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PROPOSITION XXXVIII.
In the regular hexagon, whose circumference is
threSy the area equals three-fourths of the diameter*

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119

Let AB be the unit, divided into four
equal parts. On two of the parts, CB,
construct a regular hexagon. The cir-
cumference of the hexagon is then equal
to three, for each of the six sides is equal
to half of one, liiat is, half of AB. The
diameter is the rectangle ABFG, as be-
fore shown, and it is seen to be divided
into eight equal rectangles. — The hexagon is divided into six equal
triangles, each triangle having a side (? the hexagon for its base.
One triangle, BCD, is equal to one of the rectangles, BEDH, [Prop.
6.] Therefore the six triangles, or the area of the hexagon, is
equal to six of the rectangles. But diameter is equal to eight rect-
angles, and six is three-fourths of eight ; therefore in the regular
hexagon, whose circumference is three, the area equals three-fourths
of the diameter, agreeably to the proposition.

PROPOSITION XXXIX.

In the regular hexagon^ whose circumference is
fcur^ the area equals the diameter.

Let AB be one, and divide it into
three equal parts. On two of the parts,
CB, construct a regular hexagon. One
side of the hexagon, CB, being two-
thirds of one, the six sides, or whole
circumference, will be equal to twelve-
thirds, or four units. Therefore cir-
cumference is four. The diameter has
a breadth of one, equal to AB, and a
length extending to the height of the
circle, equal to AF. Therefore the rectangle ABFG constitutes
-the diameter, and it is seen to be divided into six equal rectangles.
The hexagon is divided into six equal triangles, having the sides of
the hexagon for their bases, and one of these triangles, BCD, is
equal to one of the rectangles, BEDH, [Prop. 6 ;] therefore the
six triangles of the area are equal to the six rectangles of the dia-
meter. Therefore in the regular hexagon, whose circumference is
four, the area equals the diameter, agreeably to the proposition.

PROPOSITION XL.

In the regular octagon, whose circumference is om^
the area equals one-fourth of the diameter.

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1 —

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KT-

tnt ji

Let AB be the tmit,
[taken at a length of two
inches on account of the
smalhiess of the octagon J
and let AB be divided
into sixteen equal parts.
Then if two of the parts
be made the side of an octagon, the circumference of the octagon
will be one, that is, equal to AB, Therefore upon two of the
equal parts, CB, construct a regular octagon, and its circumference
will be one. The breadth of the diameter is AB, and its length
is AF. Therefore the rectangle ABFG constitutes the diameter,
and it is seen to be divided into thirty-two equal rectangles ; equal,
because the base of each is the sixteenth part of the tLnit AB, and
the height of each equals the radius of the circle, DE.

The area of the octagon is divided into eight equal triangles,
each having a side of the octagon for a base, and a perpendicular
equal to the radius of the circle. One of tiiese triangles, BCD, is
equal to one of the rectangles, BEDH, [Prop. 6.] Therefore the
eight triangles, or the whole area of the octagon, must be equal to
eight rectangles. But the diameter equals thirty-two rectangles,
and eight is one-fourth of thirty-two. Therefore in the regular oc-
tagon whose circumference is one, the area equals one-fourth of the
diameter, agreeably to the proposition.

PROPOSITION XLI.

In the regular octagon, whose circumference is two,
the area equals half the diameter.

Let AB be one, divided into eight equal
parts, and on two of the equal parts, CB,
construct a regular octagon. Each side of
the octagon then is equal to a fourth part
of AB, or a fourth of one, and the eight
sides therefore equal eight-fourths, or two
units ; and circumference is therefore two. The rectai^le ABFG,
which constitutes the diameter, is seen to be divided into sixteen equal
rectangles. The area of the octagon is divided into eight equal tri-'
angles, one of which, BCD, is equal to one of the rectangles, ^EDH,
[Prop. 6.] Therefore the eight triangles, or the whole area of the
octagon, is equal to eight rectangles. But the diameter is equal to
sixteen such rectangles ; therefore in the regular octagon, whose
circumference is two, the area equals half the diameter, agreeably
to the proposition.

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PROPOSITION XUI.

In tne regular octagon, whose circumference is
threey the area equals three-fourths of the diameter.

Take AB equal to one, (one inch,) and f^
divide it into eight equal parts. On three
of the parts, CB, constmct a regular octa-
gon. The circumference of £e octi^on
irill then be equal to three ; for txod side
being three-eighths of one, the eight sides
ynSi be equal to twentj-four-eighths, or
twelve quarters, or six halves, or three
wholes or units. Therefore circumference is three.

The rectangle 413FG, which constitutes the diameter, is divided
into sixteen equal rectangles by the perpendiculars from the points
of division in AB and the line drawn parallel to AB through the
center of the circle. The area of the octagon is divided into eight
equal triangles, each bavins a side of the octagon for its base.
One of these triangles, BCI), is seen to have the same height ot
perpendicular as one of the rectangles, and to have a base, CB,
equal to the base of three of the rectangles. Therefore the trian-
gle is equal to one rectangle and a-half, [Prop. 6 ; ] and the eight
triangles, or the whole area of the octagon, must be equal to twelve
rectangles. Diameter is equal to sixteen rectangles, and twelve is
three-fourths (^ sixteen. Therefore in the regular octagon, whose
circumference is three, the area equals three-fourths of the diame-
ter, agreeably to the proposition.

PROPOSITION XLIII.

In the regular octagon, whose circumference is four,

the area equals the diameter.

Let AB be one, and divide it intq
four equal parts. On two of the parts,
CB, construct a regular octagon. One
side of Ihe octagon, CB, being half of
one, two sides mil be equal to one, and
the whole eight sides equal to four.
Therefore the circumference of the oc-
tagon is four. The rectangle ABFQ
is the diameter of the octagon and of
the inscribed circle, for it has a breadth
of one, AB, and a length, AF, equal to -^
the height or extreme lunits of die circle,

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[Definitions 17 and 19.] The diameter is divided into eight equal
rectangles by the perpendiculars drawn from the points of equal
cSvisions in AB, and the line drawn parallel to AB through the
center of the circle. The area of the octagon is divided into eight
equal triangles, each having a side of the octagon for its base. One
of these triangles, BCD, is equal to one of the rectangles, BEDH,
[Prop 6^] therefore the eight triangles of the octagon must be
equal to the eight rectangles of the diameter. Therefore in the
regular octagon, whose circumference is four, the area equals the
diameter, agreeably to the proposition.

PROPOSITION XLIV.
In any triangle, regular or irregular, whose diame-
ter is one, the area equals one-fourth W the circum-
ference.

Let the diame-
ter of the circle in
the diagram be one.

SThe unit in this
emonstration is
taken at half-an-
inch.] Now draw
around it the tri-
angle ABH, with-
out any regard to
the relative length
of the sides. The
diameter of the triangle is then one, [Def. 17.] On each side of
the triangle construct a rectangle of the breadth of one, or a breadth
equal to the diameter of the circle. The three rectangles then
constitute the circumference of the triangle, for their whole length
equals the perimeter of the Jriangle, and their breadth is one, [Def.
16.] Lines drawn from the center, C, to each of the angles of the
triangle, divide the area into three new triangles. Let each of
these triangles be compared separately with that jwrtion of the cir-
cumference adjacent to it. Take the triangle ABC. Its base is
AB, and its perpendicular or height is CI, the radius of the circle,
equal to halt" of one. The portion of circumference adjacent to this
triangle is the rectangle ABFO, having a breadth of one ; and if
divided into two equal rectangles, AD will be half of one. The
triangle ABC and the rectangle ABDE have the same base, AB,
and equal heights, CI and AD ; therefore the triangle equals half
the rectangle ABDE, [Corollary, Prop. 6,] and consequently the

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triangle ABC equals one-fourth of the rectangle ABFG, In the
same manner it may be shown, that the triangle AHC equals one-
fourth of the rectangle AHKL, and that the triangle HBC equals
one-fourth of the rectangle HBMN. Therefore the three triangles
together, equal one-fourth of the three rectangles. Therefore in
any triangle, whose diameter is one, without regard to the relative
length of the sides, the area equals one-fourth of the circumference,
agreeably to the proposition,

PROPOSITION XLV.

In any triangle whatever, whose diameter is two,
the area equals one-half of the circumference.

Let the diameter 'Of the circle be two, making the unit half-an-
inch. Circumscribe around it the triangle ABH, without regard
to the relative length of the sides. Divide the area into three tri-
angles by lines drawn from the center of the circle to the three
angles of the triangle ABH. Diameter being two, the radius, CI,
is one ; therefore the perpendicular or height of the triangle ABC
is one. The shaded lines of circumference on each side of the
triangle being drawn with a breadth of one, AD is equal to CI.
And the rectangle ABDE and the triangle ABC having the same^
base, AB, and equal heights, AD and CI, the triangle equals half
the rectangle. In the same manner it may be shown that the tri-

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angle AHC equals half the rectangle AHKL, and that the triangle
HBC equals half tae rectangle HBMN. Therefore the three
triangles together eq[ual half of the three rectangles together. But
the three triangles constitute the area of the triangle ABH, and
the three rectangles constitute the circumference. Therefore in
any triangle whose diameter is two, the area equals one-half the
circumference, agreeably to the proposition.

PROPOSITION XLVI.

In any triangle whatever, whose diameter is f(Air^
the area equals the circumference.

.^-ri

Let the diameter of the circle be four, and ABH be a triangle
circumscribed around it without regard to the relative length of &e
sides. And let the triangle be divided into three new triangles by
lines drawn from the center to the three angles. Diameter being
four, the radius, CI, is two ; and the shaded line of circumference
having a breadtfi of one, AD equals half of CI. Therefore tiie
triangle ABC is equal to the rectangle ABDE ; for if AD were
equal to CI, the rectangle would be double the triangle, as appeared
in the last proposition.

In the same manner it may be shown, that the triangle AHC
equals the rectangle AHKL, and that the triangle HBC equals the
rectangle HBMN. Therefore the three triangles, which constitute
the area of the triangle ABH, are equal to the three rectangles,
which constitute the circumference. Therefore. in any triangle
whatever, whose diameter is four, the area equals the circumfer-
ence, agreeably to the propositioti.

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PROPOSITION XLVII.

In any quadrilateral, or four-sided figure, regular or
irregular, whose diameter is owe, the area equals one-
fourth of the circumference.

/ /

Let the diameter of the circle be one^ and ABDE a quadrilate-
ral figure circumscribed around it, without any regard to the rela-
tive length of the sidea. On each of the four sides construct a
rectangle, each having a breadth of one. The four rectangles then
mil constitute the circumference of the figure. Lines drawn from
the center of the circle to the four angles divide the area of the
figure into four triangles, each having a side of the figure for its
base, and the radius of the circle for its height or perpendicular.
Then if AH be equally divided at F, the triangle ABC will equal
half the rectangle ABFG, for they have the same base, AB, and
equal heights, CV and AF, [corollary, Prop. 6.] And ABFG be-
ing half of ABHI, the triangle ABC is equal to one-fourth of the
rectangle ABHI. In the same manner it may be shown that each
of the three remaining triangles is equal to one-fourth of that por-

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PART SECOND.

tion of the circumference adjacent to it. Therefore the
four triangles which constitute tiie area are together equal to
one-fourth of the four rectangles which constitute the circumfer-
ence* Therefore in any quadrilateral figure whatever, whose
diameter is one, the area equals one-fourth of the circumference,
agreeably to the proposition.

PROPOSITION XLVIIL

In any quadrilateral figure whatever, whose diame-
ter is tivOy the area equals one-half of the circum-
ference.

Let the diameter of the
circle be two, and ABDE
a quadrilateral figure cir-
cumscribed around it, with-
out any regard to the rela-
tive length of the sides.
On each of the four sides
construct a rectangle, each
having a breadth of one.
Diameter being two, the
radius, CH, is one, and
equal to the breadth of
the circumference, AF.
Therefore, the triangle
ABC equals one-half 3ie
rectangle ABFG, for they
have the same bd.se, AB, and equal perpendiculars, CH and AF.
[cor. Prop. 6.] In the same manner it may be shown that the
three remaining triangles of the area are equal to half the three
remaining rectangles of the circumference. Therefore the whole
area id equal to half the circumference. Therefore in any quad-
rilateral figure whatever, whose diameter is two, the area equals
half of the circumference, agreeably to the proposition.

PROPOSITION XLIX.
In any quadrilateral figure whatever, whose diame-
ter is ybwr, the area equals the circumference.

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Let the diame-
ter of the circle
be four. Then
the radius, CH,
is two, and the
breadth of the
line of circum-
ference, AF, is
one. Therefore
the triangle ABC
is equal to the
rec tangle ABFG,
for they have the
same base, AB, and the perpendicular of the triangle, CH, is
double the perpendicular of the rectangle, [corollary. Prop. 6.] In
the same manner it may be shown that the triangle AEC is equal
to the rectangle adjacent to it, and that the triangle EDC is equal
to the rectangle adjacent to it, and that the triangle DBC is equal
to the rectangle adjacent to it. But the four triangles constitute
the area, and the four rectangles constitute the circumference of
the quadrilateral figure ; therefore the whole area is equal to the
whole circumference. Therefore in any quadrilateral figure what-
ever, whose diameter is four, the area equals the circumference,
agreeably to the proposition.

Remark. — The demonstrations thus far are deemed
sufiSicient to establish the general law, that in all poly-
gons, regular or irregular, of any number of sides
whatever, when diameter is one, area equals one-
fourth of the circumference ; when diameter is two,
area equals one-half the circumference ; when diame-
ter is three, area equals three-fourths of the circum-
ference ; when diameter is four, area and circumfer-
ence are equal. And as the law remains in all poly-
gons, even to an infinite number of sides, it manifestly
remains when the sides at last vanish and the polygon
dissolves into the perfect circle. So that the law ne-
cessarily applies to the circle as well as to all plane
figures bounded by straight lines.

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PROPOSITION L.
A rectangle and a parallelogram of the same base
and equal heights^ or of equal bases and the same
height, are equal to each other.

Let ABOD be a rectan-
gle, and ABCE a parallelo-
gram. Then the opposite
sides of the two figures are
respectively equal to each
other— viz., AB to CD, AB
to CE, BD to AC, and BC
to AE, [Def. 60.] The
rectangle ABCD is divided
by the diagonal BC into two
equal triangles, marked 1 and 2, and the parallelogram ABCE is
divided by the diagoiml AC into two equal triangles, marked 1 and
3, [Prop. 5.] But the triangle 1 is half of the rectangle, aiid it
is also half of the parallelogram. Therefore the whole rectangle
is equal to the whole parallelogram, [Axiom Third.] Both figures
have the same base, AB, and the same or equal heights, ^C.
Therefore a rectangle and parallelogram, of the same base and
equal heights, are equal to each other, agreeably to the proposition.

Also, if the diagram be turned the other side up, the figures will
be seen to have equal bases, DC, and C£, and the same height,
AC, according to the proposition.

PROPOSITION LL
In any quadrilateral figure, whose circumference is
four, the area equals the diameter.

Let ABCD be a parallelo-
gram, whose circumference is
four, each side bein^ one.
The rectangle ABEF is the
diameter of the inscribed cir-
cle, and therefore the diame-
ter of the parallelogram, for it
has a breadth of one, AB, and a
length, AE) or BF, extending to the height of the circle. But the
rectangle ABEF is equal to the parallelogram ABCD, [Prop. 50.]
Therefore the area of the parallelogram is equal to its diameter,
agreeably to the proposition.

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Again: let ABCD be a quadri-
lateral figure, whose circumference
is four. If the circumference be s«
divided as to make AB equal one
and a half, AC and BD each one,
and CD equal half of one, and paral-
lel to AB, the figure then can receiye
an inscribed circle. Take BE equal
to one, and the rectangle BECF is

the diameter of the circle and of the quadrilateral figure ABCD.
Draw the perpendicular DG and the diagonal CO, and the whole
diagram is seen to be divided into five ri^t angled iariangles, which
are equal to each other, for they have equal bases, and equal
heights. Four of these triangles are contained in the quadrilateral
ABCD, and four are contained in the rectangle BECF. There-
fore the area of the quadrilateral, whose circumference is four^
equals its diameter, agreeably to the proposition.

PROPOSITION LIL '
In all triangles whatever, the whole circumference
bears the same proportion to the base as the perpen-
dicular of the triangle bears to the radius of the in-
scribed circle.

First. Let ABC be a triangle, whose
circumference is three [three inches], and
the base, AB, one. Draw the perpendicular,
C£, and divide it into three equal parts, and
one part, EI, will be the radius of the in-
scribed circle. Circumference is to the base
as three to one, and the perpendicular is to
radius as three to one, agreeably to the pro-
position.

Second. Let ABC be a triangle, whose
circumference is four, and base one, making
the other two sides together equal to three ;
and if they are equal, or the triangle is isos-
celes j the two legs will each be one and a half.
Draw the perpendicular CE, and divide it
into four equal parts, and one part will be the
radius of the inscribed circle. Circumference
is to the base as 4 to 1, and the perpendicular
is to radius of the inscribed circle as 4 to 1,
agreeably to the proposition.
9

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PART SECOND.

Third. Let the circumference of the tri-
angle be five, and the base, AB, cme. ' Then
the sides AC and BC, being equal, will each
be two. Draw the perpendicular, CE, and
divide it into five equal parts, ^uid one part
will be the radius of the inscribed circle.
Circumference is to the base as 5 to 1, and
the perpendicular is to the radius of the in-
scribed circle as 5 to 1, agreeably to the pro-
position.

Fourth. Let the circumference of the
triangle be six, and the base one. The other
two sides will then be two and a half each.
Draw the perpendicular, CE, and divide it
into six equal parts, and one part will be the
radius of the inscribed circle. Circumfer-
ence is to base as 6 to 1, and perpendicular
to radius as 6 to 1, agreeably to the propo-
sition.

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Fifth. Let the circumference of the tri-
angle be seven, and the base one, making the
other two sides each three. If the perpendi-
cular, CE, be divided into seven equal parts,
one part will be the radius of the inscribed
circle. Circumference is to base as 7 to 1,
and perpendicular to radius as 7 to 1, agree-
ably to the proposition.

■^ I

Sixth. Let the cir-
cumference of the triangle
be six, and the base two.
Then if the other two sides
are equal, the triangle will
be not only isosceles, but
equilateral, each side being
two. If the perpendicular,
CE, be divided into three
equal parts, one part will be
the radius of the inscribed
circle. Circumference is to .
base as 6 to 2 or 3 to 1, and
the perpendicular is to the
radius of the circle as 3 to
1, agreeably to the proposition.

Corollary. — The diameter of a circle inscribed in an equila-
teral triangle equals two-thirds the perpendicular of the triangle.

R£MA]^e:. — The truth contained in this cdi-oUary
WW attempted to be proved in the seventh proposi-

c -2,

E li

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PART SECOND.

tion, as the principle was needed in subsequent de-
monstrations. It is, however, clearly developed as a
corollary from the demonstrations under the present
proposition, and presents itself with great clearness
and simplicity.

Seventh. Let the circumference of the tri-
angle be eight, and the base one, the other two
sides being three and a half each. If the per-
pendicular, CE, be divided into eight equal
parts, one part will be the radius of the in-
scribed circle. Circumference is to base as
8 to 1, and perpendicular to radius as 8 to 1,
agreeably to the proposition.

Eighth. Let the cir-
cumference of the triangle
be five, and the base too,
making the other two sides
one and a half each. If the
perpendicular, CE, be di-
vided into five equal parts,
two parts will be the radius
of the inscribed circle. Cir-
cumference is to base as 5 to
2, and perpendicular to radius as 5 to 2, agreeably to the propo-
sition.

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Jfinth* — Let the circum-
ference of the triangle be
seven, and the base two.
Then the other sides will
each be two and a half. If
the perpendicular CE be
divided into seven equal
parts, two of the parts will
constitute the radius of the
inscribed circle. Circum-
ference is to base as 7 to
2, and perpendicular to ra-
dius as 7 to 2, agreeably to
the proposition.

c p

Tenth. — Let the circum-
ference of the triangle be
eight, and the base two, mak-
ing the other two sides each
three. If the perpendicular
CE be divided into four equal
parts, one part will be the
radius of the inscribed circle.
Circumference is to base as
8 to 2, or 4 to 1, and perpen-
dicular is to radius as 4 to 1,
agreeably to the proposition.

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134

PART SECOND.

Eleventh, — Let the circumference of the triangle be seyen, and
the base threes making each of the other sides two. If the per-

) ^.

A I

— rA

pendicular CE be divided into seven equal parts, three of the parts
will constitute the radius of the inscribed circle. Circumference
is to the base as 7 to 3, and the perpendicular to the radius as 7 to
3, agreeably to the proposition.

Twelfth. — Let the circumference of the triangle be eight, and.
the base three, making the other sides two and a half each. If the

perpendicular CE be divided into eight equal parts, three of the
parts will constitute the radius of the inscribed circle. Circum-
ference is to the base as 8 to 3, and perpendicular to radius as 8
to 3, agreeably to the proposition.

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DEMONSTRATIONS IN GEOMETRY.

13^

Thirteenth. — Let the cir-
cumference of the triangle be
four, and the base one and a
half, making the other sides
each one and a quarter. If
the perpendicular CE be di-
vided into four equal parts, one
part and a half will constitute
the radius of the inscribed cir-
cle. Circumference is to base as four to one and ahalf, or as 8 to
8, and perpendicular is to radius as four to one and a half, or as 8
to 8, agreeably to the proposition.

Four^een/A.— Let the circumference of the triangle be eiffht,
and the base two and a-half, making each of the oAer sides two
and three-quarters. If the perpendicular be divided into eight
equal parts, two parts and a-half wiU constitute the radius of the

/ ^

/ \.^

A ^^ I

I B

inscribed circle. The circumference of the triangle is to the base
as eight to two and a-half, or as 16 to 5, and the perpendicular is
to the radius as eight to two and a-half, or as 16 to 5, agreeably
to the proposition.

Remark. — ^Thus far the examples given under this

proposition have been isosceles triangles ; but the law

applies universally to all triangles, as will appear from

the following additional examples.

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136

PART SECOND.

Fifteenth.— Let ABC be a
right angled triangle, whose
circumference is 8 (inches),
and base 2. Then the per-
pendicular, AC, will be
2.6666+- an(J the hypothe-
nuse, BC', will be 8.3383+.
If the perpendicular be di-
vided into four equal parts,
one part will equal the radios
of the inscribed circle.

Sixteenth. — Let ABC be an obtuse angled triangle, whose cir-
cumference is 8, and base 2. Let AC be two and a-half, and BC
three and a-half. On the base produced, draw the perpendicular

DC, and divide it into four equal parts,
radius of the inscribed circle.

One part will equal the

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DEMONSTRATIONS IN GEOMETRY.

137

Seventeenth. — Let the circumference of the triangle ABC be 8,
the base 2, AC two and two-tenths, and BC three and eight-
tenths. On the base produced, draw the perpendicular CD, and
divide it into four equal parts. One part will equal the radius of
the inscribed circle.

Eighteenth, — Again let circumference be 8, the base 2, AC two
and one-tenth, and BC three and nine-tenths. On the base pro-
duced, draw the perpendicular CD, and divide it into four equal
parts. One part will equal the radius of the inscribed circle.

Remark. — In each of the last four exapiples, the
circumference of the triangle is to the bas6 as 4 to 1,
and the perpendicular is to the radius of the inscribed
circle also, as 4 to 1, agreeably to the 52d Proposition.

All these demonstrations under the 52d Proposition
can be readily tested with the rule and compasses,
with sufficient accuracy to establish their truth.

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138

PART SECOND.

PROPOSITION LIIL

In any triangle, wKose circumference is four, the
area equals the diameter.

Let the base AB be one, and the
AC and BC each be one and a half. Then
the circumference is four, and the base one,
and the radius of the inscribed circle is one-
fourth of the perpendicular CE, [Prop. 62.]
The rectangle BECD equals the area of the
triangle, [Prop. 6,] and this rectangle, by
the radius of the circle as a measure, is di-
vided into four equal rectangles. The dia-
meter, BAFG, is also divided into four
rectangles equal to each other and equal to
the four contained in BECD. Therefore
the area equals the diameter, agreeably to the propositioQ.

Again: Let the base AB be
one inch, the side BC one inch
and nine-tenths, and AC one
inch and one-tenth. Then the
circumference of the triangle
ABC will be four. On the base,
produced to E, draw the perpen-
dicular CE, and divide it into
four equal parts. One part will equal the radius of the inscribed
circle, [Prop. 62.] The rectangle BAFG is the diameter, for it
has a breadth of one, AB, and a length, AF, equal to the height
of the circle. The rectangle and the triangle have the same base,
AB, and if they had equal heights, the rectangle would be double
the triangle, [Prop. 6.] But the rectangle has just half the height
of the triangle, and its area is therefore equal to the triangle.
Therefore the area of any triangle, whose circumference is four,
equals its diameter, agreeably to the proposition.

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DEMONSTRATIONS IN GEOMETRY.

139

, the

PROPOSITION LIV.

In any triangle, whose circumference is five

diameter equals four-fifths of the area.

Let the base AB be one, and the sides
AC and BC each two. Then the circum-
ference of the triangle ABC will be five ;
and the iradius of the inscribed circle is one-
fifth of the perpendicular CE, [Prop. 62.]
If CE is divided into five equal parts, it is
manifest that the rectangle BE CD is divided
into five equal rectangles, which are to-
gether equal to the triangle ABC, [Prop. 6.]
The rectangle BAFG is the diameter, be-
<!ause it has a breadth of one, AB, and a
length, AF or BG, extending to the height
of the circle. The diameter BAFG is seen
to be divided into four equal rectangles, sev-
erally equal to the five contained in BECD.
Therefore iix the triangle whose circumfer-
ence is five, the diameter equals four-fifths
of the area, agreeably to the proposition.

Again: let the base AB
be two, and AC and BC
each one and a half. Then
the circumference of the
triangle will be five, and the
radius of the inscribed circle
is two-fifths of the perpen-
dicular CE, [Prop. 52.]
The rectangle BECD, which
equals the area of the tri-
angle, is seen to be divided

into five equal rectangles, and the diameter, BEFG, contains four
of them. Therefore the diameter equals four-fifths of the area,
agreeably to the proposition.

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140

PART SECOND.

PROPOSITION LV.
In any triangle whose circumference is six, the
diameter equals two-thirds of the area.

Let; the base, AB, be one, and AC and
BC each two and a-half . Then the circum-
ference of the triangle ABC will be six, and
the radius of the inscribed circle is one-sixth
of the perpendicular CE, [Prop. 62.] There-
fore if C£ be divided into six equal parts, it
is manifest that the rectangle BECD is di-
vided into six equal rectangles, which are to-
f;ether equal to the area of the triangle ABC,
Prop. 6.] The rectangle BAFG is the dia-
meter, because it has a breadth of one, AB,
and a length, AF or BG, extending to the
height of the circle ; and it is manifest that
BAFG contains four rectangles severally
equal to the six contained in BECD. There-
fore in any triangle whose circumference is
six, the cUameter equals two-thirds of the
area, agreeably to the proposition.

Again. — Let the base
AB be two, and AC and
BC each two. Then the cir-
cumference of the triangle
will be six ; and the radius
of the inscribed circle is
two-sixths or one-third of
the perpendicular CE. It
is manifest that BECD,
which equals the area of the
triangle, is divided into
three equal rectangles, and
that BEFG, the (Uameter,
contains two of the three
rectangles. Therefore the
diameter equals two-thirds of the area, agreeably to the proposition.

i- ■

i ' a

£ s

A E B

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DEMONSTRATIONS IN GEOMETRY.

141

PROPOSITION LVI.

In any triangle, whose circumference is seven, the
diameter eqifals four-sevenths of the area.

Let the base AB be one, and AC and BC
each three. Then the circumference of the
triangle will be seven, and the radius of the
inscribed circle is one-seventh of the perpen-
dicular CE [Prop. 62.1 Divide the perpen-
dicular into seven equal parts, and it is mani-
fest that BECD, which equals the area of the
triangle, is divided into seven equal rectangles,
and that the diameter, B AFG, is also divided
into four such rectangles. Therefore the
diameter is equal to four-sevenths of the area, jr
agreeably to die proposition.

Again. — Let the base
AB be two, and AC and
BC each two and a-half.
Then the circumference of
the triangle is seven, and
the radius of the inscribed
circle is two-sevenths of the
perpendicular CE. Divide
the perpendicular into seven
equal parts, and it is mani-
fest that BECD, which is
equal to the area, is divided
into seven equal rectangles,
and that the diameter,
BEFG, contains four of
them. Therefore the diame-
ter equals four-sevenths of
the area, agreeably, to the
proposition.

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142

PART SECOND.

PROPOSITION Lyil.

In any triangle, whose circumference is eight, the
diameter equals half the area.

Let the base AB be one, and AC and BC each three and a-half.
Then the circumference of the triangle is eight, and the radius
of the inscribed circle is one-eighth of the perpendicular CE, [Prop.
52.] Therefore if the perpendicular be divided into eight equal
parts, it is manifest that BECD, which equals the area of the tri-
angle, is divided into eight equal rectangles, and that the diameter,
BAFG, is divided into four such equal rectangles. Therefore the
diameter equals half the area, agreeably to the proposition.

Again. — Let the base AB be two, and AC and BC each three*
Then the circumference of the triangle is eight, and the radius of
the inscribed circle is two-eighths, or one-fourth of the perpendicu-
lar CE, [Prop. 62.] Therefore divide the perpendicular into four
equal parts, and it is manifest that BECD, which equals the area

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DEMONSTRATIONS IN GEOMETRY.

143

of the triangle, is divided into four equal rectangles, and that the
diameter, BEFG, contains two of these rectangles. Therefore the
diameter of any triangle, whose circumference is eight, equals half
the area, agreeably to the proposition.

Corollary. — In all plane figures whatsoever, which can receive
an inscribed circle, four times the area, divided by the diameter,
equals the circumference.

PROPOSITION LVIII.

In a circle of one diameter, one side of the inscribed
square equals the area of the inscribed octagon.

Let the diameter of the circle be one, [it is
drawn at one inch,] and let ABCD be an in-
scribed square, and the dotted lines a regular
inscribed octagon. One side of the square, as
AB, has the same value in area as the octagon.
The octagon is seen to be composed of the square,
ABCD, and eight similar and equal right-apgled
triangles lying outside of the square and within
the dotted lines. The value in area of any line
is its length, whatever that is, and a breadth of
one. Therefore the value of the line AB is the rectangle EFGH,
for GH [equal to AB] is the lengthy and GE is the breadth^ be-
cause it is equal to the diameter of the circle, that is, equal to one^
and all lines have a breadth of one. Now the rectangle EFGH,
which is the value of the line AB in area, is seen to be composed
of the square ABCD and eight right-angled triangles similar and
equal to the eight contained in the octagon. They are manifestly
equal, because the rectangle ABGH is divided into two equal rect-
angles by IK, and each of these rectangles is divided into two equal
right-angled triangles by the dotted diagonals KA and KB. And
the rectangle CDEF is seen to be divided in the same manner into
four equal triangles. Therefore the whole rectangle EFGH is
equal to the octagon, [Axiom 4.] Therefore in the circle whose
diameter is one, one side of the inscribed square equals the area of
the inscribed octagon, agreeably to the proposition.

In the arithmetical calculation of. the same square and octagon,
the side of the square will be found to be half the square root of 2,
viz., the decimal expression .7071+. And the area of the octagon
will also be found to be half the square rooi of 2, viz., .7071+.
Therefore the side of the square equals the area of the octagon in
numbers, as well as in geometrical quantity.

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144

PART SECOND.

PROPOSITION LIX.

In the circle whose diameter is two^ one side of the
inscribed square equals half the area of the inscribed
octagon.

Let the diameter of the circle
be two inches, and let a square
and a regular octagon be in-
scribed, as in the last proposi-
tion. Then the octagon will
equal the rectangle CDGH,
[Prop. 58.] The value of
one side of the inscribed square,
as AB,is the rectangle CDEF ;
for the length of AB equals
CD, and the breadth, being
one, is equal to CE, that is,
half of CG, which is two, being
equal to the diameter of the
circle. Therefore the rect-
angle CDEF, which is the
value of the line AB, equals half the rectan^e CDGH, and conse-
quently equals half the octagon. Therefore m a circle whose dia-
meter is two, one side of the inscribed square equals half the area
of the inscribed octagon, agreeably to the proposition.

In the arithmetical calculation of the same square and octagon,
the side of the square will be found to be the square root of 2, that
is, 1.4142+ , and the area of the octagon will be found to be the
square root of 8, that is, 2.8284+. This last number is double
the former ; therefore the side of the square equals half the area of
the octagon, in numbers as well as in geometrical quantity.

•

.^^^^^^^^

"•-«!^/*""*N^

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(

PROPOSITION LX.

In the circle whose diameter is four, one side of the
inscribed square equals one-fourth of the area of the
inscribed octagon.

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DEMONSTRATIONS IN GEOMETRY.

Let the diagram be the same
as in the last proposition, and
make diameter four. Take
CK equal to one-fourth of CG ;
then CK will be one, and the
rectangle CDKL will be the
value in area of AB, one side
of the square ; for the length
of AB is CD, and its breadth
is CK, equal to one. But the
rectangle CDKL is one-fourth
of the rectangle CDGH, and
therefore equal to one-fourth
of the octagon. Therefore in
the circle whose diameter is
four, one side of the inscribed
square equals one-fourth of the inscribed octagon, agreeably to the
proposition.

In the arithmetical calculation of the same square and octagon,,
the side of the square will be found to be the square root of 8, viz.,
2.828427+, and the area of the octagon will be found to be the
square root of 128, viz., 11.3137+ ; and this last number divided
by 4 produces 2.82842+, equal to the side of the square. ^ There-
fore in the circle whose diameter is 4, one side of the inscribed
square equals one-fourth the area of the inscribed octagon, in num-
bers as well as in geometrical quantity.

y/^.''''

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' 13

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1

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""D

PROPOSITION LXI.

In the circle, whose diameter is one, one side of
the inscribed equilateral triangle is equal to the area
of the circumscribed hexagon.

Let the diameter of the circle be
one, and ABC an inscribed equila-
teral triangle ; and let a regular
hexagon be circumscribed about the
circle. Then if each side of the tri-
angle ABC be divided into three
equal parts, and two straight lines
drawn through each point of division
parallel to the sides of the hexagon,
the hexagon will be divided into
10

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146 PART SECOND.

twenty-four equal and equiln^ral triangles ; for the triangle ABC
being equilateral, and its sides equally divided, the lines drawn
through the points of division must necessarily be equally distant
from each other. AB, the base of the inscribed triangle, is seen
to equal the bases of three of the small triangles. From D let the
side or base of the hexagon be produced to E, and the side of the
triangle CB produced to meet DE in E. Then D£ will equal the
bases of three of the small triangles, and will therefore equal AB.
Again, produce the side of the hexagon from C to F, making FC
equal to AB or DE. Then FC will also equal the bases of three
of the small equal triangles. And D and F being joined, the par-
allelogram DEFC will be seen to contain twenty-four of the small
equal triangles, and is therefore equal to the hexagon. But the
parallelogram DEFC is equal to the line AB ; for the line AB is
equal to a rectangle whose length is AB, and whose breadth is one.
The base of the parallelogram, DE, is equal to AB, and the per-
pendicular height of the parallelogram is equal to the diameter of
the circle, and therefore is one ; and a rectangle and a parallelo-
gram of the same base and equal heights are equal, [Prop. 50.]
Therefore the parallelogram DEFC is equal to AB, and the line
AB is equal to the area of the hexagon. Therefore in a circle of
one diameter, one side of the inscribed equilateral triangle equals
the area of the circumscribed hexagon, agreeably to the proposition.

In the arithmetical calculation of the same triangle and hexa-
gon, the side of the triangle will be found to be half the square
root of 3 — ^viz., .866+, and the area of the hexagon will also be
found to be half the square root of 3 — that is, .866+. Therefore
the side of the triangle is equal to the area of the hexagon in num-
bers as well as in geometrical quantity, agreeably to the propo-
sition.

PROPOSITION LXII.

In the circle, whose diameter is two, one side of the
inscribed equilateral triangle equals half the area of
the circumscribed hexagon.

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DEMONSTRATIONS IN GEOMETRY.

147

Let the diameter of
the circle be two, and
let a regular hexagon be
circumscribed around it.
Then if each side of the
hexagon be divided into
two equal parts, and two
lines be drawn from
each point of division
and parallel to the two
adjacent sides of the
hexagon, so as to meet
two opposite points of
division, and if lines be
drawn through the center
of the circle to each angle
of the hexagon, the whole
hexagon will be divided into twenty-four equal and equilateral tri-
angles, and ABC will be an equilateral triangle inscribed in the
circle. ^ The value or quantity of the line AB, one side of the tri-
angle, is the rectangle ABFG, for the length is AB, and the
breadth, AF, is one, being equal to half the diameter of the circle.
AF is the length of the perpendiculars of two of the small equal
triangles, and the diameter of the circle is manifestly the length of
four such perpendiculars, therefore AF equals half the diameter.
The rectangle ABFG is seen to contain twelve, or the value of
twelve, of the small triangles, that is ten whole triangles and four
halves. But the hexagon contains twenty-four triangles ; there-
fore the rectangle ABFG equals half the hexagon. But the rectr
angle ABFG is the value of the line AB, one side of the triangle
ABC, therefore the line AB equals half the hexagon ; and there-
fore in a circle, whose diameter is two, one side of the inscribed
equilateral triangle equals half the area of the circumscribed hex-
agon, agreeably to the proposition.

In the arithmetical calculation of the same triangle and hexa-
gon, the side of the triangle will be found to be the square root of
3 — ^viz., 1.782+, and the area of ihe hexagon will be found to be
the square root of 12 — that is, 3.464+, and half of this last num-
ber is 1.782+. Therefore one side of the triangle equals half the
area of the hexagon, in numbers as well as in geometrical quantity,
agreeably to the proposition.

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148

PART SECOND.

PROPOSITION LXIII.

In the circle, whose diameter is four, one side of
the inscribed equilateral triangle equals one-fourth of
the area of the circumscribed hexagon.

On the same diagram
as in the last proposi-
tion, let the diameter of
the circle be four, and
let ABC be the inscribed
equilateral triangle. The
diameter of the circle
being four, the perpen-
dicular of each of the
small triangles is one,
for four of these perpen-
diculars is seen to equal
the diameter of the cir-
cle. Therefore the line
AB, one side of the tri-
angle ABC, is equal to
the rectangle ABDE,
for the length is AB, and the breadth, AD, is one. The rectangle
ABDE is seen to contain six of the small equal triangles — that is,
five whole triangles and two halves. But the hexagon contains
twenty-four such triangles, and six is one-fourth of twenty-four.
Therefore the rectangle ABDE equals one-fourth of the hexagon ;
therefore the line AB equals one-fourth of the hexagon. Therefore
in the circle, whose diameter is four, one side of the inscribed equi-
lateral triangle equals one-fourth of the area of the circumscribed
hexagon, agreeably to the proposition.

In the arithmetical calculation of the same triangle and hexa-
gon, one side of the triangle will be found to be the square root of
twelve — viz*, 8.464+, and the area of the hexagon will be found
to be 13.856+, which is equal to four times the former number,
and is the square root of 192. Therefore the side of the inscribed
triangle equals one-fourth of the circumscribed hexagon, in num-
bers as well as in geometrical quantity, agreeably to the propo-
sition.

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DEMONSTRATIONS IN GEOMETRY. 149

PROPOSITION LXIV*

The area of an equilateral triangle inscribed in a
circle is one-fourth of the area of the equilateral tri-
angle circumscribed about the same circle.

Let ABC be an equilate-
ral triangle, each side equal
to one, and let the circle be
described about the triangle.
Upon each side of the tri-
angle ABC construct an-
other equilateral triangle,
then will DEF be an equi-
lateral triangle circumscrib-
ed about the circle. The
whole diagram contains four
equilateral triangles, which
are equal to each other ; for
the triangle ABC is equila-
teral, and one side of this
triangle forms a side of each of the other triangles, which are also
equilateral by construction, therefore all the sides of the four tri-
angles are equal to each other ; and the angles also being equal,
the triangles would coincide if placed one upon another, therefore
their areas must be equal. But the circumscribed triangle DEF
contains the area of the whole four triangles, and the inscribed
triangle, ABC contains the area of one triangle. Therefore the
area of an equilateral triangle inscribed in a circle is one-fourth of
the area of the equilateral triangle circumscribed about the circle,
agreeably to the proposition.

In the arithmetical calculation of the same triangles, the area
of the circumscribed triangle, DEF, will be found to be the square
root of 3, viz., 1.732+, and the area of the inscribed triangle,
ABC, will be found to be .433-f-, which is one-fourth of 1.732-47.
Therefore the area of the inscribed triangle is one-fourth the area
of the circumscribed, in numbers as well as in geometrical
quantity.

PROPOSITION LXV.

The area of a square inscribed in a circle is one-half
the area of the square circumscribed about the same
circle.

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150

PART SECOND.

Let ABCD be a square,
each side equal to two, and
let a circle be inscribed in
the square. Let each side
of the square be divided in-
to two equal parts in the
points £, F, G, and H, and
draw the straight lines EG
and HF. Then the whole
square will manifestly be
divided into four smaller
squares, equal to each other,
and each equal to one, for
each side of all the four
squares is one. Let each
of the four small squares be
divided into two triangles by the diagonals EF, FG, GH, and HE.
Then the whole figure will be divided into eight right-angled tri-
angles, equal to each other, [Prop. 6.] But the four diagonals
constitute a square, EFGH, mscribed in the circle, and this in-
scribed square, EFGH, is seen to contain four of the right-angled
triangles. The circumscribed square ABCD contains eight such
triangles. Therefore the area of a square inscribed in a circle is
one-half the area of the square circumscribed about the same cir-
cle, agreeably to the proposition.

PROPOSITION LXVI.
The area of a regular hexagon inscribed in a circle
is three-fourths of the area of the hexagon circum-
scribed about the same circle.

With a radius equal to
one, describe a circle,
whose diameter will then
be two. Let a regular
hexagon be circumscribed
about the circle. Then
if each side of the hexa-
gon be divided into two
equal parts, and two lines
be drawn from each point
of division and parallel to
two sides of the hexagon,
BO as to meet two oppo-
site points of division, and
if lines be also drawn

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DEMONSTRATIONS IN GEOMETRY. 161

througli the center of the circle to each angle of the hexagon, the
whole circumscribed hexagon will be divided into twenty-four equal
and equilateral triangles. That they must be equal and equilate-
ral is manifest, because the hexagon being regular, its sides are
0qual, and each side being equally divided, and the lines from the
points of division being drawn parallel to the sides of the hexagon,
and therefore preserving everywhere respectively equal distances,
the small triangles between them must all have equal perpendicu-
lars and equal sides, and be equal to each other. The center of
each side of the circumscribed hexagon touches the circle, and if
these points of contact are joined by straight lines, as by the dot-
ted lines in the diagram, these lines will constitute a regular in-
scribed hexagon. And the inscribed hexagon is seen to contain
eighteen triangles, that is, twelve whole and twelve half triangles.
But the circumscribed hexagon contains twenty-four such triangles,
and eighteen is three-fourths of twenty-four. Therefore the area
of a regular hexagon inscribed in a circle is three-fourths of the
area of the hexagon circumscribed about the same circle, agreeably
to the proposition.

In the arithmetical calculation of these hexagons, the diameter
of the circle being two, the area of the circumscribed hexagon will
be found to be the square root of 12, viz., 3.464-f, and the area of
the inscribed hexagon will be found to be 2.598-]-, which is three-"
fourths of 3.464+. Therefore the area of the inscribed hexagon is
three-fourths of the area of the circumscribed hexagon, in numbers
as well as in geometrical quantity.

Remark. — ^From the principles established in these
demonstrations it appears that every mathematical
right line is a rectangle ^ whose breadth is one. And
as every rectangle has a diagonal, it follows that every
mathematical right line has a diagonal. Hence we
deduce the following proposition.

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1S3

PART SECOND.

PROPOSITION LXVII.

The diagonal of every right line is the square root
of a quantity exceeding the square of the line by one.

In the diagram, taking an
inch for the unit, AB is 1,
and AC is 4. Then the
rectangle ABCD is a mathe-
matical line of four inches.
The rectangle ABEl is one.
inch. Its diagonal, Al, is
the square root of 2. The
value of Al in area is the
rectangle A145. And the
quantity of this last rect-
angle is such, that if multi-^
plied into itself it would just
equal two square inches.
That is, it would exceed the
square of one inch by one.
Again, the rectangle ABF2
is a mathematical line of
two inches. This line
squared is 4 inches ; and
its diagonal, A2, squared,
equals 5 square inches.
Therefore the square of the
diagonal exceeds the square
of the line by one.

The rectangle ABG3 is a
line of three inches, whose
square is 9, and its diagonal
A3, is the square root of an
area, or the side of a square,
equal to 10 square inches.
And ABCD being a line of
4 inches, its square is 16,
and its diagonal AD,
squared, would equal 17
square inches. Thus the square of the diagonal of every mathe-
matical line exceeds the square of the line by one.

The dotted lines in the diagram show the value in area of each
diagonal. The general truth contained in this proposition, it wiU
readily be seen, results from the celebrated theorem of Pythagoras,

"-y

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DEMONSTRATIONS IN GEOMETRY.

108

that ^^ the square of the hjpothenuse of a right-angled triangle
equals the sum of the squares of the other two sides." For Sie
diagonal of a line divides it into two right-angled triangles, the
diagonal being the hypotheniise, as AD. The side of the triangle
representing Sie breadth of the line, as CD, is always 1, and its
square is 1. And since the square of the diagonal AD equals the
square of AC plus the square of CD, it will always exceed the
square of AC, whatever its length may be, by 1.

In the annexed diagram the rectangle
ABCD is a mathematical line, the unit
being one inch. AB is 1, and AC is
4, The whole line is divided into sec-
tions, which are the roots of quantities sue*
cessively increasing by unity. And the di-
agonals also are the roots of quantities succes-
sively increasing by unity. The first section,
ABEl, is the unit. It is one, and its root,
AB, is also one. Its diagonal, Al, is the
square root of 2. Take AF, equal to Al,
and the rectangle ABF2 will also be the
square root of 2. Then the diagonal of this
rectangle, viz., A2, will be the square root
of 3. Take AG, equal to A2, ,and the
rectangle ABG3 will also be the square root
of 3. In like manner A3 is the square root
of 4, and the rectangle ABH4 is also the
square root of 4. Each succeeding section
of the line is the square root of a quantity
larger by one square inch than the square of
the preceding section. The diagonal A15
is the square root of 16, and the rectangle
ABCD, being four square inches, is also tiie
square root of 16.

These quantities, as represented upon the
diagrams, may all be determined mik suffi-
cient accuracy to test their truth simply by
applying the rule and compasses.

PROPOSITION LXVin.

In every right-angled triangle, the square of the
hypothenuse equals four times the area of the trian-
gle, plus the square of the difference of the other two
sides.

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IW PART SECOND.

Remark. — I hardly know of a more beautiful de-
monstration in all geometry, than that which estab-
lishes the truth of this proposition. I discovered it in
the month of June, 1850, when the greater part of this
work was ready for the press. I was endeavoring to
discover some more simple method than that given by
Euclid and other geometers, of demonstrating the
celebrated theorem of Pythagoras — that the square of
the hypothenuse equals the sum of the squares of the
other two sides. I was not, at the time, even aware
of the existence of the principle stated, in the above
proposition, till it presented itself to me on the dia-
grams which I had drawn for another purpose. On
stating my discovery to a mathematical friend, he in-
formed me that this truth was known to mathemati-
cians in the forms of arithmetic and algebra, and that
a geometrical demonstration of it was considered a
great desideratum, but was not supposed to be pos-
sible.

Let ABC be a right-angled triangle, c

and BC the hypothenuse. Square BC /(""^""""^->^

on the diagram, and it gives the square /

BCDE. From the base of the triangle, /
AB, cut off BF, equal to the perpendi- /
cular AC. Then the remainder of the /
base, AF, will be the difference of the /

two sides AB and AC. Join FE, and jjL . h A I

take EG equal to AC. Join 6D, and take ^ ?/

DH equal to AC, and join HA. Then ^""^"'''"--■J^

AFGH will be a square, and it is the - ^

square of the difference of AB and AC. Besides this central
square, the diagram is seen to contain four right-angled triangles,
one of which is the original triangle ABC ; and each of the other
three is equal to this, because the hypothenuse of each triangle is
one side of the same squ?.re, BCDE ; these sides are therefore equal.
The shortest side of each triangle is equal to AC by construction ;
and the remaining side of each triangle is equal to AB, that is,

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DEMONSTRATIONS IN GEOMETRY.

355

eqnal to AC plus one side of the central square. The four triangles
are therefore equal to each other. And the square of the hypo-
thenuse, BCDE, equals four times the area of the triangle ABC,
plus the square of the diflference of the other two sides, agreeably
to the proposition.

Remark. — However the sides AB and AC may be
varied, the construction of the diagram will always be
the same, and will always afford the same demonstra-
tion, until the two sides become equal to each other,
and then the central square vanishes and the square
of the hypothenuse equals four times the area of the
triangle, as the following diagrams will show.

In the preceding diagram, the base AB is one inch, and the per-
pendicular AC a quarter of an inch.

In the present diagram AB is one
inch, and AC is half-an inch ; and
yet the construction of the diagram
proceeds in the same manner as in,
iJie last, and the demonstration is j^
precisely the same.

In the annexed diagram, AB
is one inch, and AC three-
quarters of an inch. BF is
taken equal to AC; FE joined,
and EG taken equal to AC;
GD joined, and DH taken equal .
to AC, and HA joined. The d4
results are seen to be the same
as in the preceding demo.nstra-
tion, except that the central
square becomes smaller.

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156

PART SECOND.

In this diagram AB is one
inch, and AC nine-tenths of
an inch, making the diflfer-
ence of the two sides, AF,
one-tenth of an inch, and the
square of AF one-hundredth
part of a square inch.

In this diagram AB and
AC are equal, each being one
inch, and the central square
has vanished. The square of
the hypothenuse, BCDE, is
seen to contain four right-
ai^gled triangles, equal to
ABC, for the hypothenuse d<<
of each triangle is one side
of the same square, and the
diagonals of this square bisect
each other equally at A, mak-
ing all the sides and angles
of the four triangles respect-
ively equal.

PROPOSITION LXK.
The grand theorem of Pythagoras. — ^In every right-
angled triangle, the square of the hypothenuse equals
the sum of the squares of the other two sides.

Remark. — More than tvro thousand years ago this
beautiful and important truth was discovered and de-
monstrated by the renowned philosopher of Samos.
So delighted and impressed was he by the discovery,
that he is said to have sacrificed a hundred oxen to

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DEMONSTRATIONS IN GEOMETRY.

157

the gods, in testimony of his joy and gratitude. And
certainly it was no small cause of joy ; for hesides the
great beauty and harmony of the changing forms
which it presents, in point of practical utility, geometry
can scarcely boast a more important demonstration.
But the demonstration of this proposition, as given by
Euclid and followed by most other geometers, is ra-
ther intricate and laborious, rendering it somewhat
difficult for the unpracticed student to retain in his
mind the different parts of the diagram, and see clearly
their connection from the beginning to the end of the
chain of reasoning. I was induced, therefore, to seek
for a more simple and direct method of demonstrating
this most valuable theorem ; and it was while engaged
in this attempt, as already intimated, that I discovered
the truth demonstrated in Proposition 68. My origi-
nal purpose, however, was also accomplished ; for the
same diagram, with a slight modification, gives a clear,
simple, and beautiful demonstration of the great theo-
rem of Pythagoras.

Let ABC be a right-
angled triangle. Square
the hypothenuse, BC,
on the diagram, and it
gives the square BCDE.
Square the base, AB,
•and it gives the square
ABFG. Take GI, equal
to the perpendicular AC,
and square it, and it
gives the square GHDI.
Take BL, equal to AC,
and join LE. AL is the
difference of AB and AC.
Square AL, and it gives
the square ALKL

The square of the hy-
pothenuse, BCDE, as rir

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158

PART SECOND.

was shown in the last proposition, contains four triangles, plus the
central square. The squares of the other two sides are, first, ABF6,
and second, GHDI. Let these two squares be considered together
as forming one figure, and take from it the central square. There
will then remain two parallelograms, LBFE, and EKDH. And
each of these parallelograms is seen to be divided into two right-
angled triangles, each equal to ABC, for each has a side of the
same square for its hypothenuse. Therefore the sum of the squares
of AB and AC, equals four times the area of the triangle, plus the
square of the difference of AB and ^C. The square of BC has
been shown to be equal to the same quantity ; and things which are
equal to the same are equal to each othe^. Therefore the square
of the hypothenuse of a right-angled triangle equals the sum of the
squares of the other two sides, agreeably to the proposition.

To show that the
construction of the
diagram and the
demonstration will
remain the same,
however the sides
AB and AC may
be varied, three
more diagrams are
here given. In the
preening diagram
AB is an inch and
a half, and AC is
three-quarters of an
inch. In the pre-
sent diagram, AB
is an inch and a
half, and AC an
inch.

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DEMONSTRATIONS IN GEOMETRY.

150

In this dia-
gram AB is an
inch and a half,
and AC an inch
and a quarter.
It will be ob-
served that in
all these dia-
grams, the sides
of the central
square*, produc-
ed, meet the an-
gles of the large
square, which is
the square of the
hypothenuse.

c
j/_ ^ \^»

dr '^l "^ T

. In this diagram AB and AC are equal, each being an inch and
a half, and the central square has vanished. The square of the

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XeO PART SECOND.

hypothenuse, BCDE, is seen to contain four right-angled triangles,
each equal to ABC. The square of the base AB, which is ABFE,
contains two such triangles, and AC, or its equal, AE, being
squared, gives the square AEGD, which also contains two such
triangles. The squares of AB and AC together equal four times
the area of the triangle ABC ; and the square o^ BC also equals
four times the same triangle. Therefore the square of the hypo-
thenuse of a right angle equals the sum of the squares of the other
two sides, agreeably to the proposition.

PROPOSITION LXX,

If from any circle there be cut a segment of one
diameter, the chord of half the arc of that segment is
the square root of the diameter of the circle.

The diagram presents eight circles — three perfect and five brok-
en-7-there not being room on the diagram to complete them. The
diameter of the first circle, AB, is one^ [one inch.] The diameter
of the second, AC, is 2. The diameter of the third, AD, is 3, and
so on, the diameters increasing by unity till the last circle, whose
diameter is 8. These circles all touch a common point at A, and
their centers are all in the same straight line, AD, produced. From
all these circles, except the smallest, the chord GH cuts a segment,
each segment having the same diameter, AB, which is 1.

From the second circle, whose diameter is 2, the segment cut off
by the chord GH is seen to be half the circle ; and the chord of
half the arc of that segment is seen to be Al, and Al is the square
root of 2 — viz., it is 1.4142+.

A2 is the chord of half the arc of the segment cut from the
third circle, whose diameter is 3 ; and A2 is the square root of 3 —
viz., it is 1.T32+.

A3 is the chord of half the arc of the segment of the fourth cir-
cle, whose diameter is 4, and A3 is the square root of 4 — that is,
the chord A3, with perfect drawing and perfect measurement,
would be just two inches.

A4 is the chord of the fifth circle, whose diameter is 5, and A4
is the square root of 5.

A6 is the chord of the sixth circle, whose diameter is 6, and A5
is the square root of 6.

A6 is the chord of the seventh circle, whose diameter is 7, and
A6 is the square root of T. %

A7 is the chord of half the arc of the segment cut off by GH
from the largest circle, whose diameter is 8, and A7 is the square
root of 8— viz., it is 2.8284+.

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1«2 PART SECOND.

PROPOSITION LXXI.
If from any circle there be cut any segment what-
ever, the chord of half the arc of that segment is the
square root of the diameter of the circle multiplied by
the diameter of the segment ; or the chord is a mean
proportional between the diameter of the circle and
the diameter of the segment.

AC, the diameter of the second circle, is 2. Therefore the chord
IK cuts segments from all the larger circles, each segment having
a diameter of 2.

A8 is seen to be the chord of half the arc of the segment cut
from the fourth circle. This circle has a diameter of four, which
multiplied by 2, the diameter of the segment, makes 8 ; and tt^e
chord AS is the square root of 8 — viz., it is 2.8284+.

A9 is seen to be the chord of half the arc of the segment cut
from the sixth circle. This circle has a diameter of 6, which mul-
tiplied by 2, the diameter of the segment, makes 12 ; and the chord
A9 is the square root of 12 — viz., it is 3.464+.

AD, which equals 3, is the diameter of the segments cut off by
the chord passing through D ; and AlO is seen to be the chord of
ha^ the arc of the segment cut from the sixth circle. This circle
had a diameter of 6, which multiplied by 3, the diameter of the
segment, makes 18 ; and AlO is the square root of 18.

Again : let the segments cut from all the circles each have a dia-
meter of half of one, that is, equal to A£. If lines were drawn
from A, terminating on the line £F at the points 1, 2, 3, 4, &c.,
they would be the chords of half the arcs of the segments thus cut
off. To make the lines more distinct they are drawn from the
point B, and are manifestly of the same length as they would be if
drawn from A. Now, these lines are respectively the square roots
of Aa//*the diameters of the circles on which they terminate : — that
is, Bi is the square root of half of 1, B2 is the square root of half
of 2, B3 is the square root of half of 3, B4, which terminates on
the circle whose diameter is 4, is the square root of half the diame-
ter, and B8, terminating on the circle whose diameter is 8, is the
square root of 4. So that, if the diagram were perfectly drawn
and perfectly measured, B8 would be just 2 inches.

In like manner, if the segments cut off had a diameter equal to
one-fourth of the unit, the chords would be the square roots respect-
ively of one-fourth of the diameters of the circles*

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1«4 PART SECOND.

PROPOSITION LXXII.

The curve line is the measure of extension in every
possible direction. [Def. 38.]

The truth of the proposition, and the propriety of the definition
on which it is based, may be illustrated on the annexed diagram,
upon which may be drawn lines representing the square roots of
any quantities whatever^ and series of lines representing the
square roots of quantities regularly increasing by any ratio what-
ever.

The series of chords drawn from A, and terminating on the line
BG, are severally the square roots of quantities which increase re-
gularly by a ratio of one ; that is, AB is the square root of 1, Al
is the square root of 2, A2 is the square root of 3, A3 is the square
root of 4, and so on, as far as the series may be carried.

The series of chords drawn from B, and terminating on the line
EF, [being the lengths of chords supposed to be drawn 'from A and
terminating at the same points,] are the square roots of quantities
increasing regularly by a ratio of half of one. Bl is the square
root of half of one, B2 is the square root of one, B3 is the square
root of one and a-half, B4 is the square root of two, B6 the square
root of two and a-half, &c.» Expressed in decimal numbers, Bl is
.the square root of .5 ; B2 is the square root of 1 ; B3 is the root
of 1.5 ; B4 is the root of 2 ; B5 is the root of 2.5, and so on, the
square of each succeeding line containing more area by half a square
inch than the square of the preceding line.

A series of lines drawn from A, and terminating on the second
chord, IK, at the intersections of the circles, would be the square
roots of quantities increasing regularly by a ratio of 2. Thus,
AC is the square root of 4 ; the chord of tibie next circle, if drawn,
woii,ld be the square root of 6 ; the next, A8, is the square root of
8 ; the next would be the square root of 10 ; and the next, A9, is
the square root of 12.

The series of lines drawn from B, and terminating on the line
CK at the intersections of the circles, are also the square roots of
quantities increasing regularly by a ratio of 2 ; that is, they are
the square roots of all the odd numbers^ as far as the series may
be carried. BC is the square root of 1 ; B9 is the square root of
3 ; BIO is the square of 5 ; Bll is the square root of 7 ; B12 the
square root of 9, and so on.

The series of lines drawn from C, and terminating on the next
chord at the intersections of the circles, are the square roots of
quantities increasing regularly by a ratio of 3. Thus CD is the
square root of 1 ; CI is the square root of 4 ; C2 is the square

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DEMONSTRATIONS IN GEOMETRY.

1«5

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19^ PART SECOND.

root of 7 ; C3 Is the square root of 10 ; and C4 is the square root
of 18.

Remark. — ^The examples thus far given on the dia-
grams under the last three propositions seem to be
sufficient to show that the circle is the measure of ex-
tension in every possible direction, and in every pos-
sible quantity. The diagrams afford the means of
proving the truth of every one of the examples given
under these three propositions, as well as innumerable
others which might be given, simply by the applica-
tion of the theorem of Pythagoras, that the square of
the hypothenuse equals the sum of the squares of the
other two sides. For instance, let it be required to
obtain the length of B9, terminating at 9 on the third
circle. This circle has a diameter of 3, and its center
is L. A line drawn from L to 9 would be the radius,
and therefore equal 1.5. This squared is 2.25. CL
is .5, which squared is .25. Subtract the square of CL
from the square of L9, and it leaves the square of C9
equal to 2. Now C9 square added to CB square equals
B9 square. But C9 square is 2, and CB square is 1,
making 3, therefore B9 square is 3, and the square
root of 3, viz., 1.732+, is the length of B9.

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187

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PART THIRD.

HARMONIES OF GEOMETRT.

Remark. — Of the following harmonies, numbering
something over a hundred, more than three-quarters
were original discoveries with me, and probably most
of them will be new to mathematicians in general.
These discoveries were reached entirely by the Greek
method of rule and compasses, and calculations by
arithmetical numbers ; and probably very few of them
would ever have been reached by anybody, by the
methods of algebra. The surd quantities which we
have to deal with in mathematics are very numerous,
while those quantities which have perfect roots are
comparatively few. Algebra is entirely blind to rela-
tions and agreements existing between surd quanti-
ties ; whilst arithmetical numbers, by carrying out the
roots to a few decimal places, can see and show these
relations and agreements as clearly and satisfactorily
as in quantities with perfect roots.

For instance, when arithmetic shows us that the
square root of 8 is 2.8284+, and that the square root
of 2 is 1.4142+, although it has not in either case
shown us what the perfect root is, but only an ap-
proximation to it, it has nevertheless shown us by sat-

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170 PART THIRD.

isfactory proof the perfect relation between those roots
when their quantities are made perfect, viz., that the
quantity of one root is precisely half the quantity of
the other. And so in a thousand cases the 'perfect re-
lations of surd quantities are shown by arithmetic, as
clearly as the relations of quantities with perfect
roots.

HARMONIES OP PLANE FIGURES.

CIRCLES AND SQUARES.

The circumference of any square whatever, divided by the cir-
cumference of its inscribed circle, produces the same quotient, viz.,
1.273+, and this is the square of the diameter of another circle
whose area equals one square. The square root is 1.1288+, and
this is the diameter of a circle whose area equals one square.

2.
The circumference of any circle whatever, divided by th6 cir-
cumference of its circumscribed square, produces the same quotient,
viz., .785394-, and this is the area of a circle whose diameter is
one square.

3.
The area of any square divided by the area of its inscribed
circle, also produces the square of the diameter of another circle
whose area equals one square ; viz., the quotient is always 1.273-f •

4.
The area of any circle, divided by the area of its circumscribed
square, always produces the area of a circle whose diameter is one
square, viz., .78539+.

6.
The circumference of one square, divided by the circumference
of a circle whose area equals one square, produces the diameter of
a. circle whose area equals one square, viz., 1.1288+.

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HARMONIES OF GEOMETRY. 171

6.
The circumference of a circle whose area equals one square, di-
vided by the circumference of one square, produces the square root
of tiie area of a circle whose diameter is one square, viz., .88622+.

7.
Twide the square root of the circumference of any given square,
produces the circumference of another square, whose area equals
the diameter of the given square.

Twice the square root of the circumference of any given circle,
produces the circumference of another circle whose area equals the
diameter of the given circle.

9
Twice the square root of the diameter of any given square is the
diameter of another square, whose area equals the circumference of
the given square. %

10
Twice the square root of the diameter of any given circle is the
diameter of another circle, whose area equals tibe circumference of
the given circle.

11
Four times the square root of the area of any given circle equals
the circumference of another circle, whose area is equal to the cir-
cumscribing square of the given circle.

12
The area of a square inscribed in a circle is half the area of a
square circumscribed about the same circle.

18
The area of a circle inscribed in a square is one-half the area of
a circle circumscribed about the same square.

14
Half the circumference of any circle, multiplied by half its
diameter, equals the area of the circle.

15
Half the circumference of any square, multiplied by half its dia-
meter, equals the area of the square.

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172 PART THIRD.

16
Half the circumference of any plane figure whatever, multiplied
by half its diameter, equals the area of the figure. (Diameter al-
ways bemg the diameter of the inscribed circle.)

The difference of the circumferences of any two squares, divided
by the difference of their diameters, produces the orcumference oi
a square of one diameter, viz., 4.

18
The difference ol the circumferences of any two circles, divided
by the difference of their diameters, produces the circumference of
a circle of one diameter, viz., 3.1415&4-.

19
The sum of the circumferences of any two squares, divided by the
sum of their diameters, produces the circumference of a square of
one diameter, 4.

20
The sum of the circumferences of any two circles, divided by the
sum of their diameters, produces the circumference of «a circle of
one diameter, viz., 8.14159+.

The square root of the circumference of any ^ven circle is the
circumference of another circle, whose area equals one-fourth g£
the diameter of the given circle.

22
The square root of the circumference of any ^ven square, is the
^circumference of another square, whose area equals one-fourth of
*ihe diameter of the given square.

To find a circle and a square whose areas shall be equal to each
other. Take any square and its inscribed circle, that is, a square
and a circle of the same diameter, and extract the square root
of the circumference of each. Double the root from the square
for the circumference of a new square^ and double the root from
the circle for the circumference of a new circle ; then shall the
areas of the new square and the new circle be equal to each other.

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HARMONIES OF GEOMETRY. 173

CIRCLES AND EQUILATEEAL TRIANGLES.

24.

The area of a circle inscribed in an equilateral triangle is one-
fourth of the area of a circle circumscribed about the same triangle.

26.
The area of an equilateral triangle inscribed in a circle is one-
fourth of the area of an equilateral triangle circumscribed about
the same circle.

26.
Twice the square root of the circumference of any given equila-
teral triangle is the circumference of another equilateral triangle
whose area equals the diameter of the given triangle.

27.
Twice the square root of the diameter of any given equilateral
triangle is the diameter of another equilateral triangle, whose area
equals the circumference of the given triangle.

28.
In any equilateral triangle, the square of the perpendicular, di-
vided by the square root of 3, equals the area of the triangle. And
double the perpendicular, multiplied by the square root of 3, equals
the circumference of the triangle.

29.

To find a circle and an equilateral triangle, whose areas shall be
equal to each other. Take any equilateral triangle and its in-
scribed circle — that is, a triangle and circle of the same diameter,
and extract the square root of the circumference of each. Double
the root from the triangle, for the circumference of a new triangle]
and double the root from the circle for the circumference of a new
circle; then shall the areas of the new triangle and the new circle
be equal to each other.

80.

In the equilateral Igriangle, if the perpendicular is one, the chr-
dunference is twice the square root of three j but if the area is
one, the perpendicular is the square root of three twice extracted^
or ^e biquadratic root of three.

Remark.— It has long been noticed by mathema-

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174 PART SECOND.

ticians^ as a particular and remarkable fact^ that if the
area of an equilateral triangle is one, the perpendicular
happens to be exactly the square root of three tmce
extracted. But in examining this partictdar facty and
the relations between the different parts of equilateral
triangles, I discovered that, like most other particular
and remarkable facts in geometry and mathematics,
it was but the expression of a general principle^ which
applies universally to all equilateral triangles. The
general principle is this :

81.

In all equilateral triangles j the biquadratic root of three times
the square of the area equals the perpendicular.

Thus, if the area of the triangle is 1, the square of the area is 1,
and three times the square of the area is 3, and the square root of
three, twice extracted, [the biquadratic root,] — viz., 1.816+> is
the perpendicular.

Again, if the area of the triangle is 2, \h& square of it is 4, and
three times the square is 12, and the square root of 12, twice
extracted — viz., 1.8612+, is the perpendicular.

It follows, that in every equilateral triangle whose area is a
whole number, the perpendicular twice squared will be a whole
number, as the following ten examples will show :

If area is 1, the perpendicular is the square root twice ex-
tracted from 3.

If area is 2, the perpendicular is the square root twice ex-
tracted from 12.

If area is 3, the perpendicular is the square root twice ex-
tracted from 27.

If area is 4, the perpendicular is the square root twice ex-
tracted from 48.

If area is 6, the perpendicular is the square root twice ex-
tracted from 75.

If area is 6, the perpendicular is the square root twice ex-
tracted from 108i

If area is 7, the perpendicular is the square root twice ex-
tracted from 147.

If area is 8, the perpendicular is the square root twice ex-
tracted from 192.

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HARMONIES OF GEOMETRY. 175

If area is 9, the perpendicular is the square root twice ex-
tracted from 248»

If area is 10, the perpendicular is the square root twice ex-
tracted from 800*

CIRCLES AND ALL POLYGONS.

82.
Twice the square root of the circumference of any given circle is
the circumference of another circle whose area equsJs the diameter
of the given 'circle.

88.

Twice the square root of the circumference of any given polygon
is the circumference of another similar polygon, whose area equals
the diameter of the given polygon.

84.
Twice the square root of the diameter of any given circle is the
diameter of another circle, whose area equals the circumference (^
the given circle.

85.

Twice the square root of the diameter of any given polygon is
the diameter of another similar polygon, whose area equals the cir-
cumference of the given polygon.

To find a circle, whose area shall equal the area of a polygon,
which is similar to any given polygon that can receive an inscribed
circle. Extract the square root of the circumference of the given
polygon, and also the square root of the circumference of the in-
scribed circle. Double the root from the polygon, for the circum-
ference of a new similar polygariy and double the root from the
circle for the circumference of a new circle ; then shall the area
of the new circle equal the area of the new polygon, which is simi-
lar to the given polygon.

HARMONIES OF SPHERES AND THE PLATONIC BODIES.

Remarks. — ^The harmonies and remarkable agree-
ments and coincidences of geometry, which are dis-

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176 PART THIRD.

cohered in the examination of solids^ are perhaps more
beautiful and interesting^ than even those which ap-
pear in the consideration of plane figures. And in the
study of the general principles of. the science, as Dr.
Barrow remarks in the Preface to his Euclid^ " the
noble contemplation of the^t?^ regylar bodies cannot,
without great injustice, be pretermitted."

Attempts to represent solids and parts of solids by
diagrams, in the page, or on a plane surface, generally
afford but little aid to the student, and sometimes per-
plex and embarrass more than they enlighten ; for it
often costs the unpractised student more labor to learn
and recollect the parts of the solid represented by
such diagrams, than it would to demonstrate half a
dozen propositions if the palpable solid in its proper
form were before him. Therefore instead of attempt-
ing to illustrate the principles and proportions of solid
figures by diagrams, I recommend to the student, in
examining and demonstrating the following harmonies,
by all means to have before him proper models of the
solids and parts of solids which he is considering.
They may readily be cut from soft wood, and even
temporary ones from a vegetable, a turnip or potatoe.
Perhaps the easiest and pleasantest mode however of
constructing them, with nearly accurate proportions,
is to cut them from paste-board, as represented in the
annexed diagrams. These may be easily and plea-
santly made by ladies; and I hope the day is not dis-
tant when this most perfect and beautiful science shedl
become a favorite study with ladies.

Let the plane figures, as represented in the dia-
grams, be cut out of common paste-board, and on the

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HARMONIES OF GEOMETRY.

177

dotted lines cut half through with a sharp knife. Then
by turning up the folds till the several edges meet,
and fastening them with paste or some adhesive sub-
stance, you will have what are called " the five regu-
lar solids of Plato,'^ viz., the tetrahedron, hexahedron,
octahedron, dodecahedron, and icosahedron.

Tetrahedron, — The diagram
presents an equilateral trian-
gle, each side being two in-
ches. Let each side be divided
in the center, and the points
of division joined by the dotted
lines, and the diagram then
presents four equilateral tri-
angles, each side of which is
one inch. If the whole 'diar-
gram be cut from pasteboard,
and the dotted lines cut half
through, then by turning up
three of the triangles till their edges meet we have a solid figure
with four triangular faces and four solid angles.

Hexahedron^ or Cube. — The surface of the hexahedron consists of
six squares. The diagram cut half through on the dotted lines, and
folded, makes the cul^, having the form of the geometrical unit.

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178

PART THIRD.

Octahedron*
— The surface
of the octahe-
dron is com-
posed of eight
equilateral tri-
angles.

Dodecahedron. — The sur-
face of the dodecahedron is
composed of twelve equilate-
ral pentagons.

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HARMONIES OF GEOMETRY. 179

Icosahedron. — The surface of the
icosahedron is composed of twenty e^ni*^
lateral triangles.

In each of these solids the planes, constituting the
surface, are all similar in form, equal in extension, and
meet each other at equal angles. They are therefore
called "regular solids;" and besides these five, it is
not possible for another regular solid to be formed ;
that is, a solid, the planes of whose surface are all si-
milar in form, equal in extension, and meet each other
at equal angles.

Legendre, the most distinguished French geometer,
in proposing some changes in definitions and names
applied to parts of these solids, remarks that, " as the
theory of those solids has hitherto been little investi-
gated, no great inconvenience could arise from intro-
ducing any new expressions which are called for by
the nature of the objects.*'

From the following enumeration o{ harmonies in the
Platonic bodies, and from the general principles laid

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180 PART THIRD.

down in the first part of this work, it will be seen that
the " theory of those solids^' is very simple, and that in
the relations of diameter, solidity and surface, one
simple and uniform law applies to them all. It will
be seen also, that the principles developed in these
new elements of geometry required some new defini-
tions, such as the distinction between the faces and
surface of a solid, and a new definition for diameter
both in plane figures and solids. For the reader must
not forget that the diameter of every solid is the dia-
meter of its inscribed sphere. Instead of hexahedron,
in what follows, we shall generally use the name cube,
'as being a more convenient word.

SPHERES AND CUBES.

37.
The surface of a cube of one diameter, divided by the surface of
its inscribed sphere, produces the square of the diameter of another
sphere, whose surface equals the surface of the given cube.

38.

The surface of any cube, divided by the surface of its inscribed
sphere, produces the square of the diameter of another sphere,
whose surface equals the surface of a cube of one diameter.

The surface of .a sphere of one diameter, divided by the surface
of its circumscribed cube, produces the solidity of the given sphere.

40.
The surface of any sphere, divided by the surface of its circum-
scribed cube, produces the solidity of a sphere of one diameter.

41.

The surface of any cube, divided by the surface of its inscribed
sphere, produces the cube or third power of the diameter of a
sphere, whose solidity is one, or equal to the solidity of a cube of
one diameter.

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HARMONIES OF GEOMETRY. 181

42.

The solidity of any cube, divided by the solidity of its inscribed
sphere, produces the cube or third power of the diameter of a
sphere, whose solidity is equal to a cube of one diameter.

43.

The solidity of any sphere, divided by the solidity of its circum-
scribed cube, produces the solidity of a sphere of one diameter.

44.

The solidity of any given sphere, divided by the solidity of a
sphere of one diameter, produces the solidity of the cube circum-
scribing the given sphere.

46.
The surface of a cube inscribed in a sphere, equals one-third of
the surface of the cube circumscribed about the same sphere.

46.
The surface of a sphere inscribed in a cube, equals one-third of
the surface of the sphere circumscribed about the same cube.

47.
If a sphere be inscribed in a cube and another sphere circum-
scribed about the cube, the square of the diameter of the inscribed
sphere equals one-third of the square of the diameter of the cir-
cumscribed.

48.
If a cube be inscribed in a sphere and another cube circum-
scribed about the sphere, the square of the diameter of the inscribed
cube equals one-third of the square of the diameter of the circum-
scribed.

49.
The cube root of the surface of a cube, whose diameter is six,
equals the surface of a cube whose solidity is one.

50.

The cube root of the surface of a sphere, whose diameter is six,
equals the surface of a sphere whose solidity is one, or equal to one
cube.

61.

In both the cube and the sphere, if diameter is six, solidity
equals the surface ; and if surface is six, solidity equals the dia-
meter.

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182 PART THIRD.

52.
In both the cube and the sphere, six times the solidity, divided
by the diameter, equals the surface.

63.
If 1 is the diameter of a cube, the diameter of its circum-
scribed sphere is the square root of 3.

64.

If 2 is the diameter of a cube, the diameter of its circum-
scribed sphere is the square root of 12.

66.
If 3 is the diameter of a cube, the diameter of its circum-
scribed sphere is the square root of 27.

66.
If 4 is the diameter of a cube, the diameter of its circum-
scribed sphere is the square root of 48.

67.
If 6 is the diameter of a cube, the diameter of its circum-
scribed sphere is the square root of 76*

68.
If 6 is the diameter of a cube, the diameter of its circum-
scribed sphere is the square root of 108.

69.
If 7 is the diameter of a cube, the diameter of its circum-
scribed sphere is the square root of 147.

60.
If 8 is the diameter of a cube, the diameter of its circum-
scribed sphere is the square root of 192.

61.
If 9 is the diameter of a cube, the diameter of its circum-
scribed sphere is the square root of 243.

62.
If 10 is the diameter of a cube, the diameter of its cir-
cumscribed sphere is the square root of 300.

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HARMONIES OF GEOMETRY.

SPHERES AND TETRAHEDRONS.

63.
The sorfaoe of a tetrahedron of one diameter, divided by the
surface of 4ts inscribed sphere, produces the square of the diameter
of another sphere, whose surface equals the surface of the given
tetrahedron.

64.

The surface of any tetrahedron, divided by the surface of its
inscribed sphere, produces the square of the diameter of another
sphere, whose surface equals the surface of a tetrahedron of one
diameter.

65.
The diameter of a sphere inscribed in a tetrahedron, equals half
the perpendicular or height of the tetrahedron.

66.

The perpendicular of a tetrahedron inscribed in a sphere, equals
two-thirds the diameter of the sphere.

6T.
If a sphere be inscribed in a tetrahedron and another sphere
circumscribed about the tetrahedron, the diameter of the inscribed
sphere equals one-third the diameter of the circumscribed, the
surface of the inscribed equals one-ninth of the surface of the cir-
cumscribed, and the solidity of the inscribed equals one twenty-
seventh of the solidity of the circumscribed sphere.

68.

If a tetrahedron be inscribed in a sphere and another tetrahe-
dron circumscribed about the sphere, the diameter of the inscribed
tetrahedron equals one-third the diameter of the circumscribed, the
surface of the inscribed equals one-ninth of the surface of the cir-
cumscribed, and the solidity of the inscribed equals one-twenty-
seventh of the solidity of the circumscribed tetrahedron.

69.
If the linear edge of a tetrahedron is 1, the surface equals the
square root of 8.

TO.

If the diameter of a tetrahedron is 1, the solidity equals the
square root of 8.

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184 PART THIRD.

The height or perpendicular of a tetrahedron equals the square
root of two-thirds the square of its linear edge.

72.
The surface of a tetrahedron of one diameter, divided by the
surface of its inscribed sphere, or sphere of one diameter, produces
the cube or third power of the diameter of a sphere whose solidity
equals the solidity of the given tetrahedron.

73.
The surface of Uny tetrahedron, divided by the surface of its
inscribed sphere, produces the cube or third power of the diameter
of a sphere whose solidity equals the solidity of a tetrahedron of
or^e diameter.

74.
The solidity of a tetrahedron, divided by the solidity of its in-
scribed sphere, produces the cube or third power of the diameter
of a sphere whose solidity equals the solidity of a tetrahedron of
one diameter.

76.
In both the sphere and tetrahedron, if diameter is six, solidity
equals the surface ; and if the surface is six, solidity equals the
diameter.

76.
In both the sphere and tetrahedron, six times the solidity, di-
vided by the diameter, equals the surface.

77.
If the surface of a tetrahedron is 6, the linear edge is the square
root of 12, twice extracted, or the biquadratic root of 12. The
linear edge also equals the diagonal of an octahedron, whose sur-
face is 6.

SPHERES AND OCTAHEDRONS.

78.
In any octahedron, the square of the diameter equals two-thirds
the square of tha linear edge.

79.
In any octahedron, the square of the linear edge equals one-half
the square of the diagonal.

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HARMONIES OF GEOMETRY. 185

80.
In any octahedron, the square of the diameter equals one-third
the square of the diagonal.

81.

In both the sphere and octahedron, if diameter is 6, the solidity
equals the surface; and if surface is 6, the solidity equals the
diameter.

82.
In both the sphere and octahedron, six times the solidity, divided
by the diameter, equals the surface.

83.
The surface of an octahedron of one diameter, divided by the
surface of its inscribed sphere, produces the square of the diameter
of another sphere, whose surface equals the surface of the given
octahedron.

84.

The surface of any octahedron, divided by the surface of its
inscribed sphere, produces the square of the diameter of another
sphere, whose surface equals the surface of an octahedron of one
diameter.

86.
The solidity of an octahedron of one diameter, divided by the
solidity of its inscribed sphere, produces the cube or third power
of the diameter of another sphere, whose solidity equals the solidity
of the given octahedron-

86.
The solidity of any octahedron, divided by the solidity of its
inscribed sphere, produces the cube or third power of the diameter
of another sphere, whose solidity equals the solidity of an octahe-
dron of one diameter.

87.
If an octahedron be inscribed in a sphere, and another circum-
scribed about the sphere, the square of the diametar of the inscribed
octahedron equals one-third of the square of the diameter of the
circumscribed ; the square of the surface of the inscribed equals
one-ninth the square of the surface of the circumscribed ; and the
square of the solidity of the inscribed equals one twenty-seventh of
the square of the solidity of the circumscribed octahedron.

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186 PART THIRD.

88.
If a sphere be inscribed in an octahedron, and another sphere
circumscribed about the octahedron, the square of the diameter of
the inscribed sphere equals one-third the square of the diameter
of the circumscribed ; the square of the surface of the inscribed
equals one-ninth the square of tl^e surface of the circumscribed ;
and the square of the solidity of the inscribed equals one twenty-
seventh of the square of the solidity of the circumscribed sphere.

89.
In the octahedron whose diameter is 1, the solidity equals half
the square root of 3 ; the linear edge equals the square root of one
and a-half, or 1.5 ; the diagonal equals the square root of 3 ; and
the surface equals the square root of 27.

90.
In the octahedron whose diameter is 2, the linear edge equals the
square root of 6 ; the diagonal equals the square root of 12 ; the so-
lidity equals the square root of 48 ; and the surface equals the square
root of 432. 48 is one-ninth of 432, and the square root of 48 is
one-third the square root of 432.

91.
In the octahedron whose diameter is 3, the linear edge equals the
square root of 13.5 ; the diagonal equals the square root of 27 ;
the solidity equals the square root of 546.75 ; and the surface
equals the square root of 2187. The square root of the last of
these numbers is double the square root of the preceding number ;
therefore when the diameter of the octahecbron is 3, £e solidity
equals half the surface.

92.
In the octahedron whose diameter is 4, the linear edge equals
the square root of 24 ; the diagonal equals the square root of 48 ;
and the solidity equals two-thirds the surface.

93.
In the octahedron whose diameter is 5, the linear edge equals the
square root of 37.5 ; the diagonal equals the square root of 75 ; and
the solidity equals five-sixths of the surface.

94.

In the octahedron whose diameter is 6, the linear edge equals the
square root of 54 ; the diagonal equals the square root of 108 ;
the soUdity is the square root of 34992, and the surface is also the

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HARMONIES OF 6E0METRT. 187

square root of S4992, ra., 18T.06148+. Therefore when diame-
ter is 6, the solidity equals the surface.

95.
In the octahedron whose surface is 6, the linear edge equals the
square root of 3, twice extracted ; the diagonid equals the square
root of 12, twice extracted ; die diameter equals the square root
of 1.383333-4-, twice extracted ; and the soUditj also equals die
square root ot 1.333333+, twice extracted. Therefore when sur-
face is 6) the solidity equals the diameter.

CUBES AND OCTAHEDRONS.

96.
In the cube whose diameter is 1, the diagonal equals the square
root of 3.

97.
In the octahedron whose diameter is 1, the diagonal equals the
square root of 3.

98.
In the cube whose diameter is 2, the diagonal is the square root
of 12.

99.
In the octahedron whose diameter is 2, the diagonal is the square
root of 12.

100.
And in all cubes and octahedrons of equal diameters, the diago-
nals are also equal.

TETRAHEDRONS AND OCTAHEDRONS.

101.
In the tetrahedron whose linear edge is 1, the surface equals the
square root of 3.

102.
In the octahedron whose diagonal is 1, the surface equals the
square root of 3.

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188 PART THIRD.

103.
In the tetrahedron whose linear edge is 2, the surface equals the
square root of 48.

104.

In the octahedron whose diagonal is 2, the surface equals the
square root of 48.

106.
And uniyersally, if the linear edge of a tetrahedron equals the
diagonal of an octahedron, the surfaces of the two bodies are equal.

106.
If four tetrahedrons, whose faces are severally equal to the faces
of an octahedron, be applied to four alternate faces of the octahe-
dron, the whole will constitute a regular tetrahedron.

107.
If the linear edges of a tetrahedron be all equally bisected, and
the four vertices or solid angles of the tetrahedron be taken away
by planes cutting through the points of bisection, the part that is
left will be a regular octahedron.

TETRAHEDRONS AND EQUILATERAL TRLiNGLES.

108.

If the perpendicular of a tetrahedron be 1, the solidity equals
one-sixteenth of the circumference of an equilateral triangle whose
perpendicular is 1.

109.
If the perpendicular of a tetrahedron be 2, the solidity equals
four-sixteenths of the circumference of an equilateral triangle
whose perpendicular is 2.

110.
If the perpendicular of a tetrahedron be 8, the solidity equals
nine-sixteenths of the circumference of an equilateral triangle
whose perpendicular is 3.

111.
If the perpendicular of a tetrahedron be 4, the solidity equals
sixteen-sixteenths, that is, it equals the circumference of an equi-
lateral triangle whose perpendicular is 4.

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HARMONIES OF GEOMETRY. 189

112-
And universa-Uy, the circumference of any equilateral triangle,
divided by 16, and multiplied by the square of its perpendicular,
equals the solidity of a tetrahedron of the same perpendicular.

THE THREE ROUND SOLIDS OF ARCHIMEDES.

Remarks. — The great geometer of Syracuse, who
has been styled the " Newton of antiquity/' discovered
and demonstrated the proportions and relations to
each other, of the cylinder, the sphere, and the cone ;
and at his decease the figures of these solids were
carved upon his tomb in honor of his distinguished
contributions to science. But the works of genius are
not commemorated by monuments of marble or brass.
A hundred and thirty-six years after the decease ot
Archimedes, Cicero, on visiting the Island, sought for
the monument of the great mathematician and philo-
sopher, but there was no one to point it out to him.
The light of science had almost become extinguished
in Syracuse, and the name of Archimedes nearly for-
gotten. After considerable exertion, however, Cicero
discovered the monument, overgrown with thorns and
briars, and was still able to read the half-effaced in-
scriptions, and to behold the figures of the cylinder
and the sphere. That monument has long, long since
crumbled to dust; two thousand years have passed
away ; but the beautiful theorems of Archimedes still
lire and flourish, undying evergreens in the gardens of
science.

Archimedes discovered and demonstrated, that the
perpendiculars of a cylinder and a cone, and the dia-
meters of their bases, and the diameter of a sphere,

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ItM) PART THIRD.

all being equal, the solidity of the cone equals one-
half the solidity of the sphere, and the solidity of the
sphere equals two-thirds the solidity of the cylinder.
And consequently that the cone equals one-third the
cylinder.

Had Archimedes discovered the simple law of so-
lidity, diameter, and surface of all solids, he would
have seen relations and harmonies existing in these
round bodies, more remarkable and more beautiful
than even those which he demonstrated. For the
surfaces of cylinders and cones being composed partly
of plane and partly of curved surfaces, it is indeed a
beautiful and wonderful illustration of the truth and
universality of the law of solidity, diameter, and sur-
&ce, already so fully explained in this work, to find
it governing the cylinder and the cone precisely as it
governs the cube and the tetrahedron, or any other
solid.

The geometrical diameter, or diameter of exten-
sioii, of all solids, it must be remembered, is the dia-
meter of an inscribed sphere. And as no cylinder
can have an inscribed sphere, except those in which
the altitude or axis equals, the diameter of the base,
it is manifest that no other cylinders, properly speak-
ing, have a geometrical diameter. But every right
cone, however the base or altitude may be varied, can
contain an inscribed sphere. Therefore all right cones
have a geometrical diameter.

A cylinder of one diameter, and a circle of one dia-
meter, are in every respect identically the same
figure ; precisely as the unit of a line or the unit of a

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HARMONIES OF GEOMETRY. 191

surface is identically the same figure with the unit of
a solid. One inch of area, and one inch of solidity,
ill geometry, have precisely the same value, and there
is no geometrical difference between a square inch
and a cubic inch. When we consider the unit in area,
or as a square inch, we consider only one face of the
unit, disregarding entirely its thickness. But when
we consider the unit in a solid, we look to all its di-
mensions, as contained under six faces.

Precisely in the same manner is explained the truth
stated above-7-that a cylinder of one diameter and a
circle of one diameter are in every respect the same
figure. As a circle, it is a plane figure, and belongs
to area ; and in considering it we look at one end or
one base of the cylinder only, disregarding its other
dimensions. As a cylinder, it becomes a solid, and is
considered in all its dimensions, as contained under a
curve surface and two plane bases.

The area of a circle of 1 diameter is .785894-» And the base
of a cylinder of 1 diameter being a circle of 1 diameter, the area
of the base of the cylinder is also .785394-* To obtain the solid-
ity of a cylinder we multiply the area of the base by the height.
Bnt the diameter of the cylinder being 1, the height is 1. And
multiplying the area of the base by 1, still gives .78539+ for the
solidity of the cylinder. Therefore the soUdity of a cylinder of
one diameter and the area of a circle of one diameter are precisely
the same thing.

SOLIDITY, DIAMETER, AND SURFACE OF CYLINDERS
AND CONES.

^ If the diameter of a cylinder or cone is oney solidity equals one-
sixth of the whole surface. ,

If diameter is two, solidity equals one-third of the surface.
If diameter is three, solidity equals one-hatf the surface.
If diameter is four, solidity equals two-tlirds of the surface.
If diameter is five, 'solidity equals fiye-sixths of the surface.

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IW PART THIRD.

If diameter is six, solidity and surface are equal.
If the surface of a cylinder or cone equals 07i€, solidity equals
one-sixth of the diameter.

If the surface is two, solidity equals one-third of tiie diameter.
If surface is three, solidity equals one-half the diameter.
If surface is four, solidity equals two-thirds of the diameter.
If surface is fiye, solidity equals five-sixths of the diameter.
If surface is six, solidity aiid diameter are equal.

EXAMPLES.

First. In the cylinder, whose diameter is oney the
solidity equals one-sixth of the surface.

It has already been shown that the solidity equals the area of a
circle whose diameter is one — ^viz., .78539-f-. The curve surface
of a cylinder is obtained by multiplying the perimeter or circum-
ference of the base by the height of the cylinder. This circum-
ference in a cylinder of one diameter, or a circle of one diameter,
is 3.14159+ ; and this quantity is not varied by multiplying it by
the height, for the height is 1.

Therefore the curve surface is - - 8.14159+

The surface of the base is - - - .78539+

The surface of the opposite base is - - .78539+

And. the «?Ao/6 surface is - - - 4.71237 +.

One-sixth of the last number is - - .78539+.

Therefore the solidity of a cylinder, whose diameter is one, equals
one-sixth of its surface.

Second. In the cylinder, whose diameter is six, the

solidity equals the surface.

The base of the cylinder is a circle, whose diameter is 6.
Therefore the area of the base is 28.2743-f-, and this area mul-
tiplied by 6, the height of the cylinder, gives for the solidity
169.6458+. The perimeter of the base is 18.8495+; and this,
multiplied by 6, the height of the cylinder, gives for the curve

surface 113.0972+

The area of the base is - - - 28.2743+

The area of the opposite base is - - 28.2743+

And the whole surface - - • . 169.6458+.

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HARMONIES OF GEOMETRY.

193

This last sum equals the solidity. Therefore in the cylinder
whose diameter is six, the solidity equals the surface.

Third. In the conCy whose diameter is one, the solid-
ity equals one-sixth of the surface.

If any right cone containing an inscrihed sphere be bisected by
a plane passing through the yertex and the c«iter of the base, the
section will present a triangle wilih an inscribed circle, like the
annexed diagram. And if the
slant height of the cone is equal
to the diameter of the base, the
triangle thus presented will be
equilateral. From this diagram
of the vertical section we can ob-
tain the dimensions of the cone.
The triangle being equilateral, the
diameter of the circle equals two-
thirds the perpendicular of the
triangle, [Prop. 7.] Let the dia-
meter therefore be one* [In the
diagram the unit is one inch.] ^
Then the perpendicular, CD, is one and a half — viz., 1.5 ; and
this squared is 2.25. Cfne-third t)f 2.25 is .75, which added to
2.25 makes 3. Then the square root of 8 equals one side of the
triangle, for the side of an equilateral triangle is the square root of
one-third added to the square of the perpendicular. Therefore
the square root of 3 — viz., 1.732+, equals the slant height of the
cone, AC or BC, and also the diameter of the base, AB.

The diameter of the base being 1.732+, the area of the base is
2.356+, and this multiplied by one-third the perpendicular height
— ^viz., .5, gives 1.178+ for the solidity of the cone. The peri-
meter of die base is 5.4412+, and this multiplied by half the slant
height — ^viz., .866+, gives 4.712+ for the curve surface. Add
to the curve siurface the area of the base, 2.356, and we have
7.0681+ for the whole surface of the cone. Divide this last num-
ber by 6, and it gives 1.178+, equal to the solidity. Therefore
in the cone whose diameter is oney the solidity equals one-sixth of
the surface.

Fourth. In the cone, whose diameter is six, the so-
lidity equals the surface.
13

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PART THIKD.

Make diameter 6, and divide
the perpendicular CD into nine
equal parts. Diameter being 6,
the perpendicular CD equals 9,
[Prop. 7.] The square of 9 is
81, and one-third added makes
108. Therefore the square root
of 108— viz., 10.3923+, equajs
the slant height of the cone AC or
BC, and also the diameter of the
base, AB. The diameter of the
base being 10.3928+, the area
of the base is 84.82285+, and
this area multiplied by 3, one-third of the perpendicular, gives
254.468+ for the solidity of the cone. The perimeter of t^e base
is 32.64834+, and this multiplied by half the slant height — ^viz.,
5.19615+, gives 169.64567+ for the curve surface. Add to the
curve suriface the area of the base, and we have 254.468+ for the
whole surface of the cone, which thus equals the solidity. There-
fore in the cone whose diameter is six, the solidity equals the
surEsLce.

Remark. — ^In tracing out these problems of the
cone I discovered another remarkable general princi-
ple. It is this :

In every right cone, whose slant height equals the
diameter of its base, if the square of the perpendi-
cular be made the diameter of a circle, the circum-
ference of that circle shall equal the surface of the cone.

For instance, in the last example the perpendicular of the c<me
is 9, and the square of 9 is 81. If 81 be made the diameter of a cir-
cle, and multiplied by the circumference of one— viz., 3.14159 +,
it gives for its circumference 264.468+, which equals the surface
of the cone.

And in the preceding example, the perpendicular is 1.5, the
square of which is 2.25. If 2.25 be made the diameter of a cir-
cle, its circumference will be T.068+, which equals the surface of
the cone.

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HARMONIES OF GEOMETRT.

Ids

Fifth. Let the diameter
of a right cone be one^ and
the diameter of the base two.
Then the diagram will present
a vertical section through the
center. The diameter of the
base, AB, is two inches, and
the diameter of the inscribed
circle or inscribed sphere one
inch. The slant height of the
cone, AC or BC, is one and a
two-thirds, 1.66666+, and the perpendicular, CD, is one and one-
third, 1.33333+ • Diameter of the base being 2, the area of the
base is 3.14159+, which being multiplied by one-third of the per-
pendicular, gives 1.8962+ for the solidity of the cone. The peri-
meter of the base is 6.28318+ , which being multiplied by half
the slant height ^ves 6.23598+ for the curve surface. Add to
the curve surface the area' of the base, 8.14159+, and it gives
8.37757+ for the whole surface of the cone. And this surface
divided by 6 gives 1.8962+, equal to the solidity. Therefore in
any cone whose diameter is one, the solidity equals one-sixth of the
surface.

That the sides AC and BC, in the last example, are each
1.6666+, and the perpendicular, CD, 1.3333-f-, may be proved
by applying the principle demonstrated in Proposition 52, viz., that
the whole circumference of a triangle is to the base as the peiT)en-
dicular is to the radius of the inscribed circle. Thus, AB is 2,
AC is 1.6666+, and BC is 1.6666+ ; and these being added to-
gether, the whole circumference of the triangle is 5.3333+. Then
we have the proportion 5.3333+ : 2 : : 1.33333+ : .5

6.8383+ 5 2.66666+ > .6
^2,66666+$
which gives .6 for the radius of the inscribed circle. But the dia-
meter was made 1 in the proposition ; therefore the radius is .5,
and agrees with the demonstration.

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NOTE.

The foregoing Treatise was commenced with a remark of Lord
Bacon, that ^' the invention [or discovery] of forms is of all other parts
of knowledge the worthiest to be sought, if it be posdble to be found ;"
and I feel constrained now to add, that in every stage of progress
through this work, and the researches and reflections to which it has led
me, the justice and importance of that remark have been more and more
deeply impressed on my mind. I am led strongly to conjecture that
geometrical forms underlie all the works of nature. The wonderful and
endless harmonies discoverable in those forms, show them to be capable
of infinite adaptations. We already know that every note of music
which strikes the ear, is governed by geometrical laws ; and the time
may yet come when every shade of color which delights the eye, and
every odor and every taste which regales the sense, may be referred to
different geometrical forms of matter. The time may yet come when
geometrical forms shall be found to be the mainspring of all the motions
and all the forces of nature— a mainspring receiving an eternal impress
from the finger of the Almighty ^ and forever and unceasingly doing
his will.

Sir Isaac Newton, in the preface to his Priruijpiay makes this re-
mark : ^M am induced by many reasons to suspect, that they [the
phenomena of nature] may all depend upon certain forces, by which the
particles of bodies, by some causes hitherto unknown, are either mutu-
ally impelled towards each other, Ad cohere in regular formsy or are
repelled and recede from each other ; which forces being unknown,
philosophers have hitherto attempted the search of nature in vain. "

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NOTE. 197

Let us fix our attention for a moment on one simple law of Geometry,
viz. , that in any given quantity of matter or space the relation of solidi-
ty, diameter and surface is an infinitely varying relation, changmg with
every change of form in the given quantity. As the equilateral triangle
and the circle are the two extremes of this infinitely varying scale in
plane figures, so the tetrahedron and sphere are the two extremes of the
infinitely varying scale in solid figures. If we take a ^ven surface
equal to six square inches, and measure the solidity or bulk of matter or
space inclosed by that surface under different forms, we shall find that
in the form of the tetrahedron it will inclose the least possible bulk that
it can inclose in any form whatever, and if 'brought into the form of a
perfect sphere, it will inclose the greatest possible bulk that a surface of
fidx inches can inclose under any form whatever.

For example : let the six square inches of surface be bro^ght into the
form of the tetrahedron, and it will contain a little more than three-
fourths of a cubic inch, viz., .759+.

Let the same surface be brought into the form of a hexaedron or cube,
and it will contam just one cubic inch.

Let the same surface be brought into the form of an octahedron, and
it will contain a little more than one cubic inch, viz., 1.074+ .

Let the same surface be brought into the form of a sphere, and it will
contain more than one inch and one-third, viz., 1.381 + .

And between the tetrahedron and the sphere, the same amount of sur-
face may contain an infinite series of magnitudes, all differing in quantity.
Even to our limited powers of comprehension it does not seem difficult
to suppose, ih&t this sample law of Geometry, in the hands of the Al-
mighty, may be made the baas of an infinite variety of attractive or
repulsive forces in matter.

Among the valuable publicatLons of the English Society for the Dif-
fusion of Useful Knowledge, is a paper on the '^ objects, advantages, and
pleasures of science," to which is attached the following note : — ^^ The
application of mathematics to chemistry has already produced a great

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198 NOTE.

change m tia^t science, and ia calctilated to produce stiQ greater im-
provements. It may be aln^ost cerfiunlj reckoned upon as the source of
new discoveries, made by induction after the mathematicail reasoninighas.
given the suggestion. The learned reader will perceive that we allude
to the beautiful doctrine of Definite or MaUiple Proportions, Take an
example : the probability of an oxide of arsenic being discovered, is im-
pressed upon us, by the composition of aisenious and arsenic acldd, in
which thd oxygen is as 2 to 3 ; and therefbre we may expect to find a
compound of the same base, with the oxygeii bs unity. ^^ This is
an interesting hint to the chemist and tiie natural philosopher. And
we may suggest fcuiher : If chemistry presents its products in matiie-
matical pn^rtions, represented by numbers, when we recollect that all
matiiematical numbers are but representatives of magnitudes, and that
all magnitudes have /arms, what other conclusion can we come to, but
tiiat all chemical changes ape amply changes in tiie forms of matter ?
Changes perhaps, in which diameters, solidities, and surfaces find new
relations, and become subject to Hew iniptdses.

It is supposed that all substances ar6 susceptible of crystallization by
nature or art. And all crystals, if produced in situations which allow
perfect freedom of motion in tiie pattides during the formation of the
crystal, are known to assume perfect geometrical forms. Thus, sea-salt
generally crystallises in the form of cub^s ; sometimes in the form of
octahedrons. Nitre takes the form of a hexaedral prism. Sugar ap-
pears in four or six-sided prisms, with trihedral terminations. Alum in
pure water crystallises in octahedrons, &c., &c. Some substances
assume difierent forms, according id the tempeniture in which the crys-
tallization occurs. Thus tiie carbonate of lime, for instance, takes a
great variety of forms. Heat and light both have a remarkable agency
in the formation of crystals. It is stated in the London Magazine of
Science, that ^^ prismatic crystals of sulphate of nickel, exposed to a
summer's sun in a close vessel, had their internal structure so com-
pletely altered, without any exterior change, that when broken open

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NOTE. 199

liey wero composed iutemally of octahedronSj with square bases."
And in the same way prismatio crystals of zinc iare said to be chai;iged in
a few )3econ4s by tihe heat pf the sun to octahedrqTis,

The difitiijKguifihed Hauy developed die ih^ory of crystals so far as to
show " that in every crystallized sujMStanqe, whatever may be tibe differ-
ence of figure which may arise from modifying circumstancies, there is in
all its crystals a primtive fornix the nu^eus^ as it were, of ihe crystal,
invariable in eadbi substance." The primitive fonns he reduced to six,
viz., the parallelopipedon, which includes the cul)e ; the rhomb, in-
eluding all the solids terminated by six faces, parallel two and two ; the
tetrahedron ; the octahedron ; the regular hexae^ral prism ; and the do-
dec^edron. But Hauy aiialyzed tl;e;^e primiiiive forms still farther,
cleaving off their parallel layers, till he discoyered theiir gepas, so to
i^eak, and found them to con^ of but three still more simple forms,
which he palled integraiit pfuiicles. These a^e the tetrah^gn;
the simplest of the pyramids ; the triangular prism, the simplest of the
prisms -y and the parallelopipedon, including the cube, the simplest of
solids, which have their faces parallel two and two.

Thus it would seem that the Almighty has been pleased to subject all
matter to the control of perfect geometrical laws. And while he has
endowed man with reason, and left him to search out these laws and
study their beauties and perfections, he has manifestly given to the
lower orders of animals, in many instances, and for aught we know, in
all instances, an instinctive faculty or power of being guided by these
laws in the pursuits adapted to their natures. The eagle, which fre-
quents high places, in descending from height to hei^t, or from the
mountain top to the plain, is believed by curious and philosophical ob-
servers to descend in that peculiar curve line called a cycloid, [A nail,
or any point in the rim of a wheel, moves in this curve through the air
as the wheel rolls along the plain.] Now^, mathematical demonstrations
prove, that if the eagle would descend from the mountain top to the

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200 NOTE.

plain in the least possUde time^ it must go, not in a straight line, nor in
any other onrre line, bat ezacdj in the cycloid.

Beavers are observed to bnild their dams nniformly on mathematical
principles. The side np stream is bnilt at a certain slope or angle ; and
mathematicians demonstrate (hat this angle is the very one which gives
to the dam the greatest possible power of resistance to the stream.

The wonderM honey-bee is a perfect geonieter in all his works. He
fills np his liive or his hollow tree with perfect geometrical figures, con-
structed on the truest principles. The walls'of the little cells to con-
tain his honey are perfect hexagons ; and tbey are closed at top and
bottom by triangular planes meeting in a point, and always at a certain
angle. Now the form of these cells is mathematically proved to be pre-
cisely that which combines the greatest possible capacity in the ^ven
Space for storing ihe honey, the greatest possible strength in the cells,
and the least pbssible quantity of material for their construction.

And the spider too, even " the villain spider " —

** Who tanght the gpider poiaUels dedgn,
^ Sure as De Moivre, without rule or line ?"

But I must close, lest what was intended only as a brief Note
should run into an Essay. I can but add my conviction that there
is yet a wide field for the future progress of human knowledge in the
investigation of geometrical forms and their relations to matter. When
we see all nature, both animate and inanimate, everywhere around us,
wearing the impress of perfect geometrical laws, well may we exclaim
with the pious Dr. Barrow — " O Lord ! how great a geometer art
thou!"

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