Full text of "The_philosophy_of_mathematics"

See other formats

THE

PHILOSOPHY OF MATHEMATICS

WITH SPECIAL REFERENCE TO

THE ELEMENTS OF GEOMETRY AND THE

INFINITESIMAL METHOD.

BY

ALBEET TAYLOR BLEDSOE, A.M.,LLJX,

LATE PROFESSOR OP MATHEMATICS IN THE UNIVERSITY
OF VIRGINIA

J. B. LIPPINCOTT COMPANY.
1886.

Entered according to Act of Congress, in the year 1867, l>y
J. B. LIPPINCOTT & CO.,

In tlie Clerk's Office of the District Court of the United States, for the Eastern
District of Pennsylvania.

IIPPINCOTT'S PR188,

CONTENTS.

CHAPTER I.

PAGE

FIRST PRINCIPLES OF THE INFINITESIMAL METHOD THE
METHOD or EXHAUSTION 9

CHAPTER II.

DEFINITION OF THE FIRST PRINCIPLES OF THE INFINITESI-
MAL METHOD 38

CHAPTER III.
THE METHOD OF INDIVISIBLES 56

CHAPTER IV.

SOLUTION OF THE MYSTERY OF CAVALIERI'S METHOD, AND
THE TRUE METHOD SUBSTITUTED IN ITS PLACE 73

CHAPTER V.
THE METHOD OF DESCARTES, OR ANALYTICAL GEOMETRY.. 93

CHAPTER VI.
THE METHOD OF LEIBNITZ 137

8 CONTENTS.

OHAPTEE YII.

PAGE

THE METHOD OP NEWTON 170

CHAPTEB VIIL

j

OF THE SYMBOLS - AND OX 00 209

NOTE A 247

NOTEB 247

THE PHILOSOPHY OF MATHEMATICS.

CHAPTER I.

FIRST PRINCIPLES OF THE INFINITESIMAL METHOD

THE METHOD OF EXHAUSTION.

THE student of mathematics, on passing from the
lower branches of the science to the infinitesimal
analysis, finds himself in a strange and almost wholly
foreign department of thought. He has not risen, by
easy and gradual steps, from a lower into a higher,
purer, and more beautiful region of scientific truth.
On the contrary, he is painfully impressed with the
conviction, that the continuity of the science has been
broken, and its unity destroyed, by the influx of prin-
ciples which are as unintelligible as they are novel.
He finds himself surrounded by enigmas and obscuri-
ties, which only serve to perplex his understanding
and darken his aspirations after knowledge. That
clearness of evidence, which is the boast of the mathe-
matics, and which has hitherto cheered and stimulated
his exertions, forsakes him as soon as he enters on the
study of the infinitesimal calculus, and the darkness of
doubt settles on his path. If, indeed, he does not

10 THE P&ILOSOPJSY OF MATHEMATICS.

abandon the study in disgust or despair, as thousands
have done, he pursues it for the sake of a diploma or
a degree, or from some less worthy motive than the
love of science. He certainly derives from it compara-
tively little advantage in the cultivation of his intel-
lectual powers; because the dark and unintelligible
processes he is required to perform scarcely demand fl
natural exercise of them.

These disadvantages of the study are due, for the
most part, to the manner in which the calculus is
usually taught. In most elementary works on the dif-
ferential calculus, the first principles of the science are
not set forth at all, or else so imperfectly defined as to
mislead the student from the clear path of mathemati-
cal science into a region of clouds and darkness. I
have frequently ma'de the experiment with some of
the best of such works. I have more than once put
them into the hands of a class of from ninety to a
| t hundred students, among whom there were mathema-

! l tical minds of no ordinary power, and required them

j to tell me what are the first principles of the infini-

jf : ; tesimal method or calculus. Yet, after having read

|fc and mastered the first chapter, which, of course, con-

[{ tained a discussion of " First Principles," not one of

* ; them had acquired the least notion of what those prin-

ciples are. Not one of them could even name the
r"! first principles of the science, much less define them.

M In this respect, the most capable and diligent members

ii of the class were on a perfect level with the most

stupid and indolent. Indeed, if the authors of the
i ; books themselves knew what the first principles of

> the calculus are, they were very careful not to unveil

their knowledge.

THE PHILOSOPHY OF MATHEMATICS. 11

Now, the very first condition of the existence of a
mathematical science as such is, that its first principles
shall be so clear and so perfectly defined that no
one could mistake them. But even this primary and
indispensable condition is not fulfilled by most of the
treatises or text-books on the infinitesimal analysis.
Hence this analysis, as usually developed in books for
the instruction of beginners, is still in a serai-chaotic
state. If, then, we would introduce anything like the
order, harmony, and beauty of real mathematical
science into the transcendental analysis, the first step
to be taken is to exhibit its first principles in a clear
and unmistakable light. My object in this work is
to contribute all in my power toward so desirable a
result ; or, in other words, to render as clear as pos-
sible the fundamental principles of the higher calculus,
from which the whole science should be seen to flow
in the form of logical consequence, and that, too, as
clearly as the light of day flows from the sun. Much
has already been done in this direction ; far far more
than has been appropriated by the so-called teachers
of the science. Hence I shall have frequent occasion
to avail myself of the labors of others ; but I shall
never do so without an explicit acknowledgment of
my obligation to them.

In the prosecution of this design, I shall trace the
rise and progress of the infinitesimal analysis from the
first appearance of its elements 111 the Greek geometry
to the present day. This will enable us to see, the
more clearly, the exact nature of its methods, by show-
ing us the difficulties it has had to encounter, and
the precise manner in which it has surmounted them.
It will also disclose, in a clear light, the merits of

12 THIS PHILOSOPHY OF MATHEMATICS.

the various methods of the calculus in tlie successive
stages of its development from Euclid and Archimedes
to Cavalieri and Pascal, and from Cavalieri and Pascal
to Xewton and Leibnitz. JSTor is this all. For such a
historical sketch will show us that, after all its wander-
ings through the dark undefined regions of the infinite,
the human mind will have to come back to the humble
and unpretending postulates of Euclid and Archimedes
in order to lay out and construct a satisfactory and
easy road across the Alpine heights of the transcen-
dental analysis. And besides, is there not a pleasure
is there not an inexpressible delight in the contem-
plation of the labors of the human mind by which it
has created by far its most sublime instrument of dis-
covery; an instrument, indeed, with which it has
brought to light the secrets of almost every depart-
ment of nature, and with which, above all, it has un-
veiled the entire system of the material universe to
the wonder and admiration of the world?

THE METHOD OF EXHAUSTION.

The ancient geometers, starting with the principle
of superposition, were enabled to compare triangles,
to ascertain their properties and the measure of their
surfaces. From triangles they proceeded to the inves-
tigation of polygons, which may be easily divided into
triangles, and thence to the consideration of solids
bounded by rectilinear figures, such as prisms, pyra-
mids, and polyedrons. Having ascertained the pro-
perties of these magnitudes, they were unable to pro-
ceed further without the aid of a more powerful or
searching method. Hence the method of exhaustion
was invented and used by them in their more difficult

THE PHIL OSOPHY OF MA THEM A TICS. 1 3

researches. This opened a new and brilliant career
to the ancient geometry. The theory of curved lines
and surfaces was partially developed, and the value
of the areas and volumes which they contain deter-
mined. It has more than a thousand times been
asserted, that the method of exhaustion, used by Euclid
and Archimedes, " contains the germ of the infinitesi-
mal analysis" of the moderns. But if we would see
this truth in a clear light, or comprehend the precise
meaning of what is so often and so vaguely asserted,
we must recur to the details or elements of the method
of exhaustion.

As the ancients, says Carnot, " admitted only de-
monstrations which are perfectly rigorous, they be-
lieved they could not permit themselves to consider
curves as polygons of a great number of sides ; but
when they wished to discover the properties of any
one of them, they regarded it as the fixed term, which
the inscribed and circumscribed polygons continually
approached, as nearly as they pleased, in proportion
as they augmented the number of their sides. In this
way, they exhausted in some sort the space between
the polygons and the curves ; which, without doubt,
caused to be given to this procedure the name of the
method of exhaustion. 77 *

This will, perhaps, be more distinctly seen in an
example. Suppose, then, that regular polygons of
the same number of sides arc inscribed in two circles
of different sizes. Having established that the poly-
gons are to each other as the squares of their homo-
logous lines, they concluded, by the method of exhaus-
tion, that the circles are to each other as the squares

* Reflexions sur la Mtflaphysiquc <lu Cahml Infinitesimal, p. 138.

14 THE PHILOSOPHY OF MA THEMA TICS.

of their radii. That is, they supposed the number of
the sides of the inscribed polygons to be doubled, and
this process to be repeated until their peripheries ap-
proached as near as we please to the circumferences of
the circles. As the spaces between the polygons and
the circles were continually decreasing, it was seen to
be gradually exhausted; and hence the name of the
method. But although the polygons, by thus continu-
ing to have the number of their sides doubled, might
be made to approach the circumscribed circles more
nearly than the imagination can conceive, leaving no
appreciable difference between them ; they would al-
ways be to each other as the squares of their homo-
logous sides, or as the squares of the radii of the cir-
cumscribed circles. Hence they conjectured, that the
circles themselves, so very like the polygons in the
last stage of their fulness or roundness, were to each
other in the same ratio, or as " the squares of the
radii." But it was the object of the ancient geometers,
not merely to divine, but to demonstrate. A perfect
logical rigor constituted the very essence of their
method. Nothing obscure, nothing vague, was ad-
mitted either into their premises, or into the structure
of their reasoning. Hence their demonstrations abso-
lutely excluded the possibility of doubt or controversy ;
a character and a charm which, it is to be lamented,
the mathematics has so often failed to preserve in the
spotless splendor of its primitive purity.

Having divined that any two circles (C and c) are
to each other as the squares of their radii (E and r), the
ancient geometers proceeded to demonstrate the truth
of the proposition. They proved it to be necessarily
true by demonstrating every other possible hypothesis

THE PHILOSOPHY OF MATHEMATICS. 15

to be false. Thus, said they, if C is not to c as II 2 :
r 2 ; then let as suppose that C:c: : E a :r /2 ; r' being
any line larger than r. By a process of reasoning,
perfectly clear and rigorous, they proved that this
supposition led to an absurdity. Then, again, they
supposed that C : c : : R 2 : r //2 ; r" being less than r ;
an hypothesis which, in like manner, was shown to
lead necessarily to an absurdity. Hence, as the line
which entered into the fourth term of the proportion
could be neither greater nor less than the radius r, it
was concluded to be that radius itself. This process,
by which every possible supposition, except the one
to be demonstrated, was shown to lead to an absurdity,
has always been called the reductio ad absurdum.
Hence this complex method, used by the ancient
geometers in their most difficult researches, has some-
times been called the reductio ad absurdum, as well as
the method of exhaustion a form of speech, in both
cases, in which a part is put for the whole. The
reductio ad absurdum is, indeed, generally included
in the meaning of those who simply speak of the
method of exhaustion, and vice versa.

By this method the ancients also demonstrated that
the volumes of spheres arc to each other as the cubes
of their radii; that pyramids having the same altitude
are to each other us their bases ; that a cone is one-
third of a cylinder with the same base and the same
altitude.

They used it also in regard to curved surfaces.
They imagined other surfaces to be inscribed and
circumscribed, of which they gradually increased the
number of sides and of zones, in such a, manner as to
continually approximate toward each other, and eon-

16 THE PHILOSOPHY OF MATHEMATICS.

sequently to close more and more upon the proposed
surface. The property of the mean figure was thus
indicated or inferred from the known property of the
figures which so nearly coincided with it ; and this
inference, or conjecture, was verified by the reductio
ad absurdum, which showed that every contrary sup-
position led infallibly to a contradiction.

It was thus that Archimedes,, the Newton of the
ancient world, demonstrated that the convex surface
of a right cone is equal to the area of the circle which
has for a radius the mean proportional between the
side of the cone and the radius of the circle of the
base ; that the total area of the sphere is equal to four
great circles ; and that the surface of any zone of a
sphere is equal to the circumference of a great circle
multiplied by the height of the zone. He likewise
demonstrated that the volume of a sphere is equal to
its surface multiplied by one-third of its radius. Hav-
ing determined the surface and the volume of the
sphere, it was easy to discover their relations to the
surface and the volume of the circumscribed cylinder.
.Accordingly, Archimedes perceived that the surface
of a sphere is exactly equal to the convex surface of
the circumscribed cylinder ; or that it Is to the whole
surface of the cylinder, including its bases, as 2 to 3 ;
and that the volumes of these two geometrical solids
are to each other in the same ratio ; two as beautiful
discoveries as were ever made by him or by any other
man.*

* When Cicero was in Syracuse he sought out the tomb of Archi-
medes, and, having removed the rubbish beneath which it had long
been buried, he found a sphere and circumscribed cylinder engraved
on its surface, by which he knew it to be the tomb of the great
geometer.

THE PHIL OSOPHY OF MA THEMA TICS. 1 7

Carnot has well expressed the merits of the method
of exhaustion. " That doctrine," says he, " is cer-
tainly very beautiful and very precious ; it carries with
it the character of the most perfect evidence, and does
not permit one to lose sight of the object in view; it
was the method of invention of the ancients; it is still
very useful at the present day, because it exercises the
judgment, which it accustoms to the rigor of demon-
strations, and because it contains the germ of the in-
finitesimal analysis. It is true that it exacts an
effort of the mind ; but is not the power of meditation
indispensable to all those who wish to penetrate into
a knowledge of the laws of nature, and is it not neces-
sary to acquire this habit early, provided we do not
sacrifice to it too much time?" *

Such were its principal advantages, some of which
it still enjoys in a far greater degree than the infini-
tesimal analysis of the moderns. But, on the other
tedious, slow and painful in its movements; and, after
all, it soon succumbed to the difficulties by which the
human mind found itself surrounded. It could not
raise even the mind of an Archimedes from questions
the most simple to questions more complex, because it
had not the KOU arco on which to plant its lever. Truths
were waiting on all sides to be discovered, and con-
tinued to wait for centuries, until a more powerful
instrument of discovery could be invented. Descartes
supplied the rcoi) aTco, the point d'appui, and Newton,,
having greatly improved the method of Archimedes,
raised the world of mind into unspeakably broader
and more beautiful regions of pure thought.

* Reflexions sur la Meta. physique du Ciilcul Infinitesimal, p. 138.
2*

18 2!H2? PHILOSOPHY OF MATHEMATICS.

The method of the ancients, says Carnot, " contains
the germ of the infinitesimal analysis" of Newton and
Leibnitz. But he nowhere tells us what that germ is,
or wherein it consists. It is certainly not to be found
in the reductio ad absurdum, for this has been banished
from the modern analysis. Indeed, it was to get rid
of this indirect and tedious process that Newton pro-
posed his improved method. But there are other ele-
ments in the method of the ancients : 1. In every
case certain variable magnitudes are used as auxiliary
quantities, or as the means of comparison between the
quantities proposed ; and these auxiliary quantities are
made to vary in such a manner as to approach more and
more nearly the proposed quantities, and, finally, to dif-
fer from them as little as one pleases. 2. The variable
quantities are never supposed to become equal to the
quantities toward which they were made to approach.
Now here we behold the elements of the modern
infinitesimal analysis in its most improved and satis-
factory form. The constant quantity, toward which
the variable is made or conceived to approach as
nearly as one pleases, is, in the modern analysis, called
"the limit" of that variable. The continually de-
creasing difference between the variable and its limit,
which may be conceived to become as small as one
pleases, is, in the same analysis, known as an " indefi-
nitely small quantity." It has no fixed value, and is
never supposed to acquire one. Its only property is
that it is a variable quantity whose limit is zero.
These are the real elements of the modern infinitesimal
analysis. If properly developed and applied, the in-
finitesimal analysis will retain all the wonderful ease
and fertility by which it is characterized, without

THE PHILOSOPHY OF MATHEMATICS. 19

losing aught of that perfect clearness of evidence
which constitutes one of the chief excellences of the
ancient method. But, unfortunately, such a develop-
ment of the infinitesimal analysis has demanded an
enduring patience in the pursuit of truth, and a capa-
city for protracted research and profound meditation
which but few mathematicians or philosophers have
been pleased to bestow on the subject. Indeed, the
true analysis and exposition of the infinitesimal method
is, like the creation of that analysis itself, a work for
many minds and for more ages than one. Although
a Berkeley, a Maclaurin, a Carnot, a D'Alembert, a
Cauchy, a Duhamel, and other mathematicians* of
the highest order, have done much toward such an
exposition of the infinitesimal analysis ; yet no one
imagines that all its enigmas have been solved or all
its unmathematical obscurities removed.

"When the true philosophy of the infinitesimal cal-
culus shall appear, it will be seen, not as a metaphysi-
cal speculation, but as a demonstrated science. It
will put an end to controversy. It will not only
cause the calculus to be all over radiant with the clear-
ness of its own evidence, but it will also reflect a new
light on the lower brandies of the mathematics, by re-
vealing those great and beautiful laws, or principles,
which are common to the whole domain of the science,
from the first elements of geometry to the last results of

* I have purposely omitted the name of Comic from the above list.
Mr. John Stuart Mill has, I am aware, in his work on Logic, ventured
to express the opinion that M. Comic "may truly be said to have
created the philosophy of the higher mathematics." The truth is,
however, that he discusses, with his usual verbosity, "the Philoso-
phy of the Trail sec u dental Analysis/' without adding a single notion
to those of his nredeo.essorfi. exeent n, few false ones of his own.

20

TEE PHILOSOPHY OF MATHEMATICS.

the transcendental analysis. Something of the kind is
evidently needed, if we would banish from the ele-
ments of geometry the indirect and tedious process of
the reductio ad absurdum. Accordingly, many at-
tempts have been made, of late, to simplify the dem )ii-
strations of Euclid and Archimedes, by introducing the
principles of the infinitesimal method into the elements
of geometry. But, unfortunately, from a misconcep-
tion of these principles, they have usually succeeded
in bringing down darkness rather than light from the
higher into the lower branches of mathematics. Thus,
the infinitesimal method, instead of reflecting a new-
light, is made to introduce a new darkness into the
very elements of geometry.

We find, for example, in one of the most exten-
sively used text-books in America, the following de-
monstration : * " The circumferences of circles are to
each other as tJieir radii, and the areas are as the squares

Let us designate the circumference of the circle
whose radius is C A by circ. C A ; and its area by
area C A ; it is then to be shown that

circ. C. A : circ t O B : : C A : O B, and that
area G A : area O B : : C A 2 : O B 2 .

* Davies' Legendre, Book V., Proposition XI. Theorem.

THE PHIL OSOPHY OF MA THEM A TICS. 21

Inscribe within the circles two regular polygons of
the same number of sides. Then, whatever be the
number of sides, their perimeters will be to each other
as the radii C A and O B (Prop. X.). Now if the
arcs subtending the sides of the polygons be continu-
ally bisected until the number of sides of the poly-
gons shall be indefinitely increased, the perimeters of
the polygons will become equal to the circumferences
of the circumscribed circles (Prop. VIII., Cor. 2), and
we have here

Giro. C A : ciro. O B : : C A : O B.

Again, the areas of the inscribed polygons are to
each other as C A 2 to O B 2 (Prop. X.). But when
the number of sides of the polygons is indefinitely in-
creased, the areas of the polygons become equal to the
areas of the circles, each to each (Prop. VIII., Cor. 1) ;
hence we shall have

area C A : area O B : : C A 2 : O B 2 . "

If this were an isolated case, or without any similar
demonstrations in the same work, or in other elemen-
tary works, it might be permitted to pass without
notice. But the principle on which it proceeds forms
the basis of the demonstrations of many of the most
important propositions in the work before us, and is
also most extensively used in other books for the in-
struction of the young. Hence it becomes necessary
to test its accuracy, or its fitness to occupy the position
of a first principle, or postulate, in the science of ma-
thematics.

The most scrupulous attention is, in the above in-
stance, paid to all the forms of a demonstration ; and

22 THE PHILOSOPHY OF MATHEMATICS.

this, no doubt ; has an imposing effect on the mind of
the beginner. But what shall we say of its substance?
The whole demonstration rests on the assumption that
an inscribed polygon, with an indefinite number of
sides, is equal to the circumscribed circle. Or, in
other words, as the author expresses it in a more re-
cent edition of his Geometry, " the circle is but a regu-
lar polygon of an infinite number of sides." * The
same principle is employed to demonstrate the propo-
sition that " the area of a circle is equal to the product
of half its radius by the circumference." Nor is this
all. All the most important and beautiful theorems,
relating to " the three round bodies," are made to rest
on this principle alone ; and if this foundation be not
valid, then they rest on nothing, except the too easy
faith of the teacher and his pupils. One would sup-
pose that if any portion of the science of geometry
should have a secure foundation, so as to defy contra-
diction and silence controversy, it would certainly be
the parts above indicated, which constitute the most
striking and beautiful features of the whole structure.
In another "Elementary Course of Geometry," f
extensively used as a text-book in our schools and
colleges, the same principle is made the foundation of
all the same theorems. Indeed, this principle of the
" infinitesimal method," as it is called, is even more
lavishly used in this last work than in the one already
noticed. " The infinitesimal system," says the author,
" has been adopted without hesitation, and to an ex-
tent somewhat unprecedented. The usual expedients

* Davies' Legendre, revised edition of 1856. Book V., Scholium to
Proposition XII.
f Hackley's Geometry.

THE PHIL OSOPHY OF MA THEM A TICS. 23

for avoiding this, result in tedious methods, involving
the same principle, only under a more covert form. The
Idea of the infinite is certainly a simple idea, as natural
to the mind as any other, and even an antecedent con-
dition of the idea of the finite." * Now the question
before us, at present, relates not to the use of " the in-
finite" in mathematics, but to the manner in which it
is used.

The author tells us that " the perimeter of the poly-
gon of an indefinite number of sides becomes the same
thing as the circumference of the circle." f Or, again,
" by an infinite approach the polygon and the circle
coincide." Now when he informs the student that
"the usual expedients for avoiding this" principle
" result in tedious methods, involving the same idea
only under a more covert form," he certainly requires
him to walk by faith, and not by sight or science. It
was, as we have said, precisely to avoid the principle
that any polygon ever coincides exactly with a circle,
that the ancient geometers resorted to the reductio ad
absurdum, which, from that day to this, has been
usually adopted for the purpose of avoiding that prin-
ciple. "As they admitted only perfectly rigorous
demonstrations," says Carnot, as well as every other
writer on the subject, "they believed that they could
not permit themselves to consider curves as polygons
of a great number of sides." Hence they resorted to
the indirect and tedious method of demonstration by
the reductio ad absurdum. This method was, in fact,
a protest against the principle in question, a repudia-
tion of it as false and spurious. If the ancient
geometers could have adopted that principle, which

* Hae^Icy's (loomctry, Preface. f Proposition LXXI.

modern teachers of the science. But they believed that
the fewness of its steps is not the only excellence of a
mathematical demonstration. Aiming at a clearness
and rigor which would not admit of controversy, they
refused to " consider a circle as a polygon of a great
number of sides," however great the number. Did
they, then, fail to escape the principle in question?
Does the reductio ad absurdum, their great expedient
for avoiding it, really involve that principle ? There
is certainly not the least appearance of any such thing,
and no such thing was ever before suspected. On the
contrary, it has hitherto been universally seen and
declared that the reductio ad absurdum does not in-
volve the principle from which it sought to escape.
Yet are we now gravely told, by a distinguished
teacher of geometry, " that the usual expedients for
avoiding" that principle only " result in tedious
methods involving" precisely the same thing ! That the
reductio ad absurdum, the one great expedient for this
purpose, is, after all, a miserable blunder, involving
the very principle from which its authors intended to
effect an escape ! But if that principle is false, then
the weak and tottering foundation of those portions of
geometry which it is made to support will require
something more than a mere assertion to bolster it up
and render it secure.

A third teacher of mathematics and compiler of
text-books has, in his " Elements of Geometry," made
a similar use of the principle that " a circle is identi-
cal with a circumscribed regular polygon of an infinite

THE PHIL 080PHY OF MA THEM A TICS. 25

number of sides." * Now I do not deny that very
high authority may be found for this principle, at least
among the moderns ; but then the foundations of ma-
thematical science rest, not upon authority, but upon
its own intrinsic evidence. Indeed, if there had not
been high authority for the truth of the principle in
question, it is believed that the more humble teachers
of geometry would scarcely have ventured to assert it
as one of the fundamental assumptions or first prin-
ciples of the science. It gets rid, it is true, of the
tedious and operose reductio ad absurdum, and seeks
to banish it from the regions of geometry. But will
not the stern and unrelenting reductio ad absurdwn
have its revenge on this modern pretender to its ancient
honors?

I object to the above so-called principle of " the
infinitesimal system," first, because it is obscure. It
neither shines in the light of its own evidence, nor
in the light of any other principle. That is to say, it
is neither intuitively clear and satisfactory to the
mind, nor is it a demonstrated truth. Indeed, the
authors above referred to do not even pretend to de-
monstrate it; they merely assume it as a fundamental
postulate or first principle. They profess to sec, and
require their pupils to sec, what neither a Euclid nor
an Archimedes could clearly comprehend or embrace.
Is this because they belong to a more advanced age,
and can therefore see more clearly into the first prin-
ciples of science than the very greatest minds of an-
tiquity ? I doubt if much progress has been made

* Elements of Geometry. By James B. Dodd, A.M., Morrison
Professor of Mathematics and Natural Philosophy in Transylvania
University. Book V., Theorem XXVIII.
3 B

26 THE PHILOSOPHY OF MATHEMATICS.

since the time of Euclid and Archimedes with respect
to the precise relation between a circle and an inscribed
or a circumscribed polygon with an infinite number
of sides. It is certain that the mathematicians of the
present day are not agreed among themselves respect-
ing the truth or the possibility of the conception in
question. Thus, for example, one of the teachers of
the science rejects the principle in question, " because,"
says he, " strictly speaking, the circle is not a polygon,
and the circumference is not a broken line." * An-
other teacher of the science says, after having alluded
to Euclid, that " modern writers have arrived at many
of his conclusions by more simple and concise methods ;
but in so doing they have, in most instances, sacri-
ficed the rigor of logical demonstration which so justly
constitutes the great merit of his writings." f Accord-
ingly, he rejects from the elements of geometry the
principle that a circle is a polygon of an infinite num-
ber of sides, and returns to the reductio ad absurdum
of Euclid.

Now, what right have the teachers of geometry to
require their pupils to assume as evident a principle
which the very masters of the science are utterly un-
able to receive as true? What right have they to
require the mere tyro in geometry to embrace as a first
principle what neither a Euclid nor an Archimedes
could realize as possible? Even if their principle
were true, what right have they to give such strong
meat to babes, requiring them to open their mouths,

* Ray's Plane and Solid Geometry, Art. 477.

| Elements of Geometry. By George R.Perkins, A. M., LL.D., Prin-
cipal and Professor of Mathematics in the New York State Normal
School; author of Elementary Arithmetic, Elements of Algebra, etc.,
etc., etc.

THE PHILOSOPHY OF MATHEMATICS. 27

If not to shut their eyes, and implicitly swallow down
as wholesome food what the most powerful veterans
are so often unable to digest ?

The greatest mathematicians and philosophers have,
indeed, emphatically condemned the notion that a
curve is ar can be made up of right lines, however
small. Berkeley, the celebrated Bishop of Cloyne,
and his great antagonist, Maclaurin, both unite in re-
jecting this notion as false and untenable, Carnot,
D'Alembert, Lagrange, Cauchy, and a host of other
illustrious mathematicians, deny that the circumference
of a circle, or any other curve, can be identical with
the periphery of any polygon whatever. This, then,
is not one of the first principles of the science of mathe-
matics. Even if it were true, it would not be entitled
to rank as a first principle or postulate, because it
admits of doubt, and has, in fact, been doubted and
denied in all ages by the most competent thinkers and
judges. Whereas, it is the diameter istic of all first
principles in geometry that they absolutely command
the assent of all sane minds, and rivet the chain of
inevitable conviction on the universal reason of man-
kind.

In the second place, I object to the above principle,
or rather the above conception, of the infinitesimal
analysis, because it is not true. Every polygon is, by
its very definition, bounded by a broken line. Now, is
the circle bounded by a broken line or by a curve?
Every line is that which, according- to its definition,
has length. How, then, can a right line, which never
changes its direction from one of its ends to the other,
coincide exactly with a curve line, which always changes
its direction ? The polygon and the circle are, indeed,

28 OF MATHEMATICS.

ornetry as dirtinot and different

h ' l >

be broken up and confounded, as if there

difference

t

is not th ls done in the darkness of the imagination
ather than in the pure light of reason ? If the c I

^

our ent

into the v nS ^ D< l thUS Ca " 7 *" first P"-iple down
mto the very foundations of the science ? Why dis

and then confound the m? The trol % the

e ro te

pnnc.ple that a curve is ma de up of indefinitely sLal
right hnes M one of those false conceptions of the in-

ha^lrf d T hich ' as sh ^ ] hereaft - ^

to fd:;t e ;itiiT e ' ind f ever *~*

THE PHILOSOPHY OF MATHEMATICS-. 29

the best way to refute an error), and by showing the
contradictions and absurdities in which it is involved.
The most celebrated of the above writers on the
elements of geometry does not seem, indeed, to have
been long satisfied with his own demonstration. Hence,
in a revised edition of his work* the principle in ques-
tion is not seen, and the word limit is substituted in
its place. I say the word limit, because this term is
not adequately defined by him. "The limit of the
perimeter" (of the inscribed polygon), says he, " is the
circumference of the circle ; the limit of the apothem
is the radius, and the limit of the area of the polygon
is the area of the circle. Passing to the limit, the
expression for the area becomes," and so forth. Now
what does the author mean by the expression " passing
to the limit?" Does he mean that the variable poly-
gon will ultimately become the circle or pass into its
limit ? If so, then he has made no change whatever
in the structure of his former demonstration, except
the substitution of an undefined term for an unintelli-
gible principle. Yet he evidently means that the
polygon will coincide with the circle ; for after saying
that " the circumference is the limit of its (variable)
perimeter," he adds, that " no sensible error can arise
in supposing that what is true of such a polygon is
also true of its limit, the circle." f No sensible error!
But can any error at all arise? If so, then the poly-
gon does not, strictly speaking, coincide with the |
circle. But he relieves the student from all hesitation 1
on this point by assuring him, in the next sentence,
that "the circle is but a polygon of an infinite number

i of I860.

* Davics' Logondrc, revised edition <
f Book V., Prop. XII., Scholium 2.
*

30 THE PHIL OSOPHY OF MA THEM A TICS.

of sides." Why, then, attempt to introduce the un-
nec.essary idea of limits ? If the polygon really coin-
cides with the circle, or if the circle is only one species
of the polygon, then, most assuredly, whatever is true
of every regular polygon is also true of the circle.
Why, then, introduce the wholly unnecessary notion
ception by the use of a hard term ?

It is certain that the author did not long continue
satisfied with this form of his demonstration ; for, in a
still later revised edition of his Geometry, he dismisses
the noticm of limits altogether, and returns still more
boldly to the use of " the infinite/' * Thus he builds
the demonstration of ail the same theorems on the
principle that " if the number of sides be made in-
finite, the polygon will coincide with the circle, the
perimeter with the circumference, and the apothem
with the radius." f Or? raore simply expressed, on
the idea that " the circle is only a regular polygon of
an infinite number of infinitely small sides." But
who can see what takes place in the infinite ? We are
told that two parallel lines meet at infinity, or if pro-
duced to an infinite distance. If so, it would be easy
to prove that two parallel lines may be perpendicular
to each other. We are also told that many other
things, equally strange and wonderful, happen at an
infinite distance. Hence I hope, for one, that it is the
destination of geometry to be rescued from the outer
darkness of the infinite and made to shine in the pure,
unmixed light of finite reason.

But if the circle is really a regular polygon with
an infinite number of sides, then let this be shown

* See edition of 1SG6. f Book V., Prop. XIV.

THE PHILOSOPHY OF MATHEMATICS. 31

once for all, and afterwards proceeded on as an estab-
lished principle. Why should constructions be con-
tinually made in every demonstration, and the same
process repeated, only to arrive at the conclusion that
a circle has the properties of a regular polygon with
an infinite number of sides? Why continue to estab-
lish that which is already supposed to be established ?
If a circle is really " but a regular polygon with an
infinite number of sides," then it is evident that the
cylinder is only a right prism, and the cone only a
right pyramid with such polygons for their bases, and
the sphere itself is only a solid generated by the revo-
lution of such a polygon around one of its diameters.
Hence all the theorems relating to the circle and the
"three round bodies," which are demonstrated in
Book VIII. of the work before us, are only special
cases of the propositions already demonstrated in re-
gard to the regular polygon, the right cone, and the
volume generated by the revolution of a regular poly-
gon around a line joining any two of its opposite ver-
tices. Why, then, after having demonstrated the gene-
ral truths or propositions, proceed, with like formality,
to demonstrate the special cases? Is this conformed
to the usage of geometers in other cases of the same
kind ? Do they prove, first, that the sum of the angles
of any triangle is equal to two right angles, and then
prove this of the isosceles triangle, or of any other
special case of that figure? If not, why prove what
is true of all regular polygons whatever, and then
demonstrate the same thing in relation to the special
case of such a polygon called the circle ? The only
reason seems to be that although they assume and
assert that "a circle is but a regular polygon of an

32 THE PHIL OSOPHY OF MA TSEMA TICS.

infinite number of sides," they are not clearly con-
vinced of the truth of this assumption themselves.

If this assumption may be relied on as intuitively
certain, or as unquestionably true, then how greatly
might the doctrine of the "three round bodies" be
simplified and shortened! All the theorems relating
to them would, indeed, be at the very most only sim-
ple corollaries flowing from propositions already de-
monstrated. Thus, the volume of the cylinder as a
species of the right prism would be equal to its base
into its altitude, and its convex surface' equal to the-
periphery of its base into the same line. In like
manner the volume of the cone, considered as a right
pyramid, would be equal to its base into one-third of
its altitude, and its convex surface equal to the peri-
phery of its base into one-half of its slant height. In
the same way we might deduce, or rather simply re-
state, all the theorems in regard to the frustum of a
cone, and all those which relate to the sphere. But what,
then, would become of Book VIII. of the Elements ?
Would it not be far too short and simple? As it is,
what it lacks in the substance it makes up in the form
of its demonstrations. It is now spread, like gold-
leaf, over twenty goodly octavo pages; and yet, if the
principle on which it is based be really true and satis-
factory, the whole book might be easily contained in
a few lines, without the least danger of obscurity.
Strip the demonstrations of this book, then, of all
their needless preparations and forms, and how small
the substance! Remove the scaffolding, and how
diminutive the edifice! It would scarcely make a
decent appearance in the market.

But if we reject the notion that the inscribed regu-

THE PHILOSOPHY OF MATHEMATICS. 33

lar polygon ever becomes equal to the circle, or coin-
cides with it, what shall we do ? If we deny that they
ever coincide, how shall we bridge over the chasm
between them, so as to pass from a knowledge of right-
lined figures and volumes to that of curves and curved
surfaces? Shall we, in order to bridge over this
chasm, fall back on the reductio ad absurdum of thp
ancients ? or can we find a more short and easy pas-
sage without the sacrifice of a perfect logical rigor in
the transit ? This is the question. This is the very
first problem which is and always has been presented
to the cultivators of the infinitesimal method. Is
there, then, after the lapse and the labor of so many
ages, no satisfactory solution of this primary problem ?
It is certain that none has yet been found which has
become general among mathematicians. I believe
that such a solution has been given, and that it only
requires to be made known in order to be universally
received, and become a possession for ever a xr7j/j.a ec
del more precious even than the gift of Thucydides.

But there are mighty obstacles to the diffusion of
such knowledge. The first and the greatest of these
is the authority of great names ; for, as was said more
than two thousand years ago, a With so little pains is
the investigation of truth pursued by most men, that
they rather turn to views already formed." Espe-
cially is this so in a case like the present, since the
great creators of the calculus, before whom we all bow
with the most profound veneration, are very naturally
supposed to have known all about the true analysis
and exposition of their own creation. But the fact is
demonstrably otherwise. Newton himself revealed
the secret of the material universe, showing it to be a
B*

34 THE PHILOSOPHY OF MATHEMATICS.

fit symbol of the oneness, the wisdom, and the powci
of its divine Author ; but he left the secret of his own
creation to be discovered by inferior minds. May we
not, then, best show our reverence for Newton, as he
showed his for God, by endeavoring, with a free mind,
to comprehend and clearly explain the mystery of his
creation ?

The second of these obstacles is, that few men can
be induced to bestow on the subject that calm, patient,
and protracted attention which Father Malebranche so
beautifully calls " a natural prayer for light. 7 ' Hence,
those who reject the solutions most in vogue usually
precipitate themselves and their followers into some
false solution of their own. Satisfied with this, al-
though this fails to satisfy others, their investigations
are at an end. Henceforth they feel no need of any
foreign aid, and consequently the great thinkers of the
past and of the present are alike neglected. Their own
little taper is the sun of their philosophy. Hence, in
their prayerless devotion to truth, all they do is, for
the most part, only to add one falsehood more to the
empire of darkness. I could easily produce a hundred
striking illustrations of the truth of this remark. But
with the notice of one in one of the books before me,
I shall conclude this first chapter of my reflections.

It is expressly denied in the book referred to that
a polygon can ever be made to coincide with a circle.
An inscribed polygon, says the author, " can be made
to approach as nearly as we please to equality with
the circle, but can never entirely reach it" * Accord-
ingly, he defines the limit of a variable in general to
be that constant magnitude which the variable can be

* Bay's Plane and Solid Geometry, Art. 475

THE PHILOSOPHY OF MATHEMATICS. 33

it " can never quite reach." Now this is perfectly
true. For, as the author says, the polygon, so long
as it continues a polygon, can never coincide with
a circle, since the one is bounded " by a broken line "
and the other by "a curve/' Here, then, there is
a chasm between the inscribed variable polygon and
its limit, the circle. How shall this chasm be passed?
How shall we, in other words, proceed from a know-
ledge of the properties of the polygon to those of the
circle? The author bridges over, or rather leaps, this
chasm by means of a newly-invented axiom. " What-
ever is true up to the limit/' says he, " is true at the
limit." * That is to say, whatever is true of the poly-
gon in all its stages, is true of the circle. Now is not
this simply to assume the very thing to be established,
or to beg the question? We want to know what is
true of the circle, and we are merely told that what-
ever is always true of the polygon is also true of the
circle ! In this the author not only appears to beg
the question, but also to contradict himself. For,
according to his own showing, the polygon is always,
or in all its stages, bounded by a broken line, and
" the circumference of the circle is not a broken line." f
Again, he says that the polygon is always less than
the circumscribed circle, and this certainly cannot 1x3
said of the circle itself. He appears to be equally
unfortunate in other assertions. Thus, he says, " what-
ever is true of every broken line having its vertices in 1}
a curve is true of that curve also."~| Now the broken
line has " vertices" or angular points in the curve ; has
the circumference of a circle any vertices in it? Again,

* Art. 1.98. f Art. 477. Art. 201.

36 THE PHILOSOPHY OF MATHEMATICS.

" whatever is true of any secant passing through a
point of a curve is true of the tangent at that point." *
Now every secant cuts the circumference of the circle
in two points, and, as the author demonstrates, the
tangent only touches it in one point. Thus, his
assumption or universal proposition is so far from
being an axiom that it evidently appears not to he
true.

The author does not claim the credit of having dis-
covered or invented this new axiom. " In explaining
the doctrine of limits/' says he, " the axiom stated by
Dr. Whewell is given in the words of that eminent
scholar." f Now Dr. Whewell certainly had no use
whatever for any such axiom. For, according to his
view, the variable magnitude not only approaches as
nearly as we please, but actually reaches its limit.
Thus, says he, " a line or figure ultimately coincides
with the line or figure which is its limit." J Now,
most assuredly, if the inscribed polygon ultimately
coincides with the circle, then no new axiom is neces-
sary to convince us that whatever is always true of
the polygon is also true of the circle. For this is only
to say that whatever is true of the variable polygon in
all its forms is true of it in its last form a truism
which may surely be seen without the aid of any
newly-invented axiom. According to his view, in-
deed, there was no chasm to be bridged over or spanned,
and consequently there was no need of any very great
labor to bridge it over or to span it. His axiom was,
at best, only a means devised for the purpose of pass-
ing over nothing, which might have been done just as
well by standing still and doing nothing. The truth

* Art. 201. f Preface. J Doctrine of Limits, Book II., Art. 4,

THE PHILOSOPHY OF MATHEMATICS.

3?

is, however, that although he said the two figures
would ultimately " coincide," leaving no chasm be-
tween them to be crossed, he felt that there would be
one, and hence the new axiom for the purpose of
bridging it over. But the man who can adopt such a
solution of the difficulty, and, by the authority of his
name, induce others to follow his example, only inter-
poses an obstacle to the progress of true light and
knowledge. Indeed, the attempts of Dr. Whewell to
solve the enigmas of the calculus are, as we shall have
occasion to see, singularly awkward and unfortunate;
showing that the depth and accuracy of his knowledge
are not always as wonderful as its vast extent and
variety.

CHAPTER II.

DEFINITION OF THE FIRST PRINCIPLES OF THE
INFINITESIMAL METHOD.

[N the preceding chapter it has been shown that
s an error to consider a circle as a polygon. It is
tainly a false step to assume this identity, in any
e, as a first principle or postulate, since so many
thematicians of the highest rank regard it, as evi-
itly untrue. Thus Carnot, for example, says, " It
absolutely impossible that a circle can ever be con-
ered as a true polygon, whatever may be the num-
of its sides." * The same position is, with equal
phasis, assumed by Berkeley, Maclaurin, Euler,
AJembert, Lagrange, and a host of other eminent
thematicians, as might easily be shown, if neces-
sary, by an articulate reference to their writings.
But, indeed, no authority is necessary either to estab-
lish or to refute a first principle or postulate in geome-
try. This is simply a demand upon our reason which
is only supported by assertion, and put forth either to
be affirmed or denied. If the reason of mathematicians
does not affirm it, then is there an end of its exist-
ence as a first principle or postulate. As no effort is
made to prove it, so none need be made to refute it.
For no one has a right to be heard in geometry who
makes the science start from unknown or contradicted

* Beflexions, etc., chapter I., p. II.
38

THE PHILOSOPHY OF MATHEMATICS.

39

principles, especially from such principles as have, in
all ages, been rejected by the mathematicians of the
very highest order. Yet has there been, in modern
times, an eager multitude of geometers who rush in
where a Euclid and an Archimedes feared to tread.
Let us see, then, if we may not find a safer and more
satisfactory road to the same result.

The problem to be solved is, as we have seen, how
to pass from the properties of rectilinear figures to those
of curvilinear ones. Or, in particular, how to pass
from the known properties of the polygon to a know-
ledge of the properties of the circle. Since no poly-
gon can, ex hypothesi, be found which exactly coincides
with the circle, we are not at liberty to transfer its
properties to the circle, as if it were a polygon with a
great number of sides. For, having inscribed a regu-
lar polygon in a circle, and bisected the arcs sub-
tended by its sides, we may double the number of its
sides, and continue to repeat the process ad libitum;
and yet, according to hypothesis, it will never exactly
coincide with the circumscribed circle. There will,
after all, remain a chasm between the two figures
between the known and the unknown. Now the ques-
tion is, how to bridge over this chasm with a perfectly
rigorous logic in order that we may clearly, directly,
and expeditiously pass from the one side to the other,
or from the known to the unknown? The method
of limits affords a perfect solution of this question.
Nor is this all. For, in the clear and satisfactory solu-
tion of this problem, the very first relating to the in-
finitesimal analysis, it opens, as we shall be enabled
to see, a vista into one of the most beautiful regions
of science ever discovered by the genius of man. Let

fh

us, then, proceed to Jay down trie first principles ot
this method^ and produce the solution of the above
problem.

The limit of a variable. When one magnitude takes
successively values which approach more and more
that of a constant magnitude, and in such manner that
its difference from this last may become less than any
assigned magnitude of the same species, we say that
the first approaches indefinitely the second, and that the
second is its limit.

Thus, the limit of a variable is the constant quantity
which the variable indefinitely approaches } but never
reaches.*

" The importance of the notion of a limit" says Mr.
Todhunter, "cannot be over-estimated; in fact, the
whole of the differential calculus consists in tracing
the consequences which follow from that notion." f
Now this is perfectly true. Duhamel says precisely
the same thing. But, then, the consequences of this
notion or idea may be traced clearly, and every step
exhibited as in the open light of day ; or they may be
traced obscurely, and almost the whole process con-
cealed from the mind of the student behind an im-
penetrable veil of symbols and formulae. They may
be shown to flow, by a perfectly clear and rigorous
course of reasoning, from the fundamental definition
or idea of the infinitesimal method, or they may be
deduced from it by a process which looks more like
legerdemain than logic. In this respect there ap-
pears to be a vast difference between the above-named

* Elements de Calcul Infinitesimal, par M. Duhamel, Vol. I,
Book I., chap. I., p. 9.
f Dif. and Int. Calculus, p. 4.

TEE PEILOSOPS7 OF MATHEMATICS. 41

mathematicians. The student who follows the guid-
ance of the one sees everything about him, and is at
every step refreshed and invigorated by the pleasing
prospects presented to his mind. On the contrary,
the student who pursues the analysis of the other re-
sembles, for the most part, the condition of a man who
feels his way in the dark, or consents to be led blind-
fold by a string in the hand of his guide.

The very first point of divergence in these two very
different modes of development is to be found in the
definition of the all-important term limit. In the
definition of M. Duhamel, the variable is said not to
reach its limit, while in that of Mr. Todhunter this
element of the "notion of a limit " is rejected. " The
following may," says he, "be given as a definition:
The limit of a function (or dependent variable) for
an assigned value of the independent variable, is that
value from wkieh the function can be made to differ as
little as we please by making the independent variable
approach its assigned value." * There is, in this defini-
tion, not a word as to whether the variable is supposed
to reach its limit or otherwise. But the author adds,
" Sometimes in the definition of a limit the words
' that value which the function never actually attains 5
have been introduced. But it is more convenient to
omit them." Now this difference in the definition of
a limit may, at first view, appear very trifling, yet in
reality it is one of vast importance. If, at the outset
of such inquiries, we diverge but ever so little from the
strict line of truth, we may ultimately find ourselves
involved in darkness and confusion. Hence, it is
necessary to examine this difference of definition, and,

* Chapter I., p. 6.

follow.

Is the definition of a limit, then, of the one all-im-
portant idea of the infinitesimal calculus, a mere mat-
ter of convenience, or should it be conformed to the
nature of things ? The variables in the calculus are
always subjected to certain conditions or laws of change,
and in changing according to those conditions or laws
they either reach their limits or they do not. If they
do reach them, then let this fact be stated in the defi-
nition and rigidly adhered to without wavering or
vacillation. Especially let this be done if, as in the
work before us, the same fact is everywhere assumed
as unquestionably true. Thus, the limit of a variable
is supposed to be its "limiting value/'* or the last
value of that variable itself. Again, he still more
explicitly says, " any actual value of a function may
be considered as a limiting value/ 5 f Having assumed
that the variable actually reaches its limit, it would,
indeed, have been most inconvenient to assert, in his
definition, that it never reaches it; for this would
have been to make one of his hypotheses contradict
the other. But if it be a fact that the variable does
reach its limit, and if this fact be assumed as true,
then why not state it in the definition of a limit ?

The reason is plain. This, also, would have been
very inconvenient, since the author would have found
it very difficult to verify the correctness of his defini-
tion by producing any variables belonging to the in-
finitesimal analysis that actually reach their limits.
He might easily find lawless variables, or such as
occur to the imagination while viewing things in the

* Chapter I., p. 6. f Ibid.

THE PHILOSOPHY OF MATHEMATICS. 43

abstract, which may reach their limits. But such
variables are not used as auxiliary quantities in the
infinitesimal analysis. They would be worse tiian
useless in all the investigations of that analysis. Hence,
if he would verify his assumption, he must produce
variables of some use in the calculus which are seen
and known to reach their limits. Can he produce
any such variables ? He has certainly failed to pro-
duce even one.

In order to illustrate his "notion of a limit," he
adduces the geometrical progression 1 + J + J -f J +?
etc. Now, as he truly says, "the limit of the sum
of this series, when the number of terms is indefinitely
increased, is 2." But does this sum actually reach its
limit 2 ? Or, in other words, if we continue to make
each term equal to one-half of the preceding term,
shall we ever reach a term equal to nothing? Or,
in other words again, is the half of something ever
nothing? If so, then two nothings may be equal to
something, and, after all, the indivisibles of Cavalieri
was no mathematical or metaphysical dream. If we
may divide a quantity until it ceases to have halves, or
until one-half becomes absolutely nothing, then have
the mathematical world greatly erred in rejecting these
indivisibles as absurd, and we may still say that a line
is equal to the sum of an indefinite number of points,
a surface to an indefinite number of lines, and a volume
to an indefinite number of surfaces. But is not the
mathematical world right? Is it not a little difficult
to believe that the half of something is nothing? Or
that a line which has length may be so short that its
half will be a point or no length at all ? Be this as it
niay ; the infinite divisibility of magnitude, as well as

44 THE PHIL OSOPHY OF MA THEM A TICS.

the opposite doctrine, may be a metaphysical puzzle ;
but it has no right to a place in mathematics, much
less to the rank of a fundamental assumption or postu-
late. But it must be regarded as such if we may assert
that the sum of the progression l + i + i + i+j etc.,
actually reaches its limit 2 by being sufficiently far
produced. We shall certainly escape such dark and
darkening assumptions if we can only find a method
for passing, in the order of our knowledge, from the
variable to its limit without supposing the variable
itself to pass to its limit. Precisely such a method we
have in the work of Duhamel, and nothing approxi-
mating to it in the differential calculus of the English
mathematician.*

Our author gives another illustration of the idea of

a limit. "Although approaches as nearly as we

please to the limit, it never actually attains that limit" f
Both the words and the italics are his own. Here it
is said that the variable " never actually attains its
limit," arid this, I apprehend, will be found to be the
case in relation to every variable really used in the
infinitesimal method. It will, at least, be time enough
to depart from the definition of Duhamel when vari-
ables are produced from the calculus which are seen to
reach their limits without violating the law of their

* It has often been a subject of amazement to my mind that the
English mathematicians derive so little benefit from the improve-
ments introduced by their French neighbors. "Why, in the republic
of letters and science, should there not be a free interchange of ideas
and improvements ? The French were not slow to borrow the methods
of Newton; but the English seem exceedingly slow, if not disin-
clined, to borrow from a Carnot, a Cauchy, or a Duhamel the im-
provements which they have made in these methods.

f Chapter I., p. 6.

TSE PSIL080PHY OF MATHEMATICS.

45

increase or decrease. If such variables should be
found, then, since some are admitted to exist which
never reach their limits, such quantities should be
divided into two classes and discussed separately.
That is to say, the analyst should then treat of those
variables which reach their limits and of those which
never reach their limits. But it is to be hoped that
he will cease to take any further notice of the first
class of variables until some such can be found that
are capable of being used in the calculus.

and the polygon, because this will make the idea per-
fectly plain. Duhamel knows, as Euclid demonstrated,
that such a variable polygon may be made to approach
the dimensions of the circle as nearly as one pleases.
He knows this, indeed, just as well as he knows any
property of the polygon itself, or of any other figure
in geometry. He takes his stand, then, upon the
demonstrated truth that the difference between the
dimensions, or the areas, of the two figures may be
made less than any "grandeur dcsign^e," than any
assigned magnitude of the same species. This know-
ledge, this clearly perceived, this demonstrated truth,
is the point from which he sets out to bridge the
chasm between the one figure and the other. He
never supposes the two figures to coincide or to be-
come equal, because he has the means of spanning the
chasm which separates them without either denying
its existence or filling it up with doubtful propositions
about what may be supposed to take place at the end
of an infinite process. He has no use for any such
assumptions or assertions even if true, because he has
a much clearer and better method to obtain the same

1:1

46 THE PHILOSOPHY OF MATHEMATICS.

result.. But before we can unfold that method in a
clear and perspicuous manner it will be necessary to
consider his next definition.

" We call" says he, " an infinitely small quantity., or
simply an infinitesimal, every variable magnitude of which
the limit is zero.

"For example, the difference between any vari-
able whatever, and its limit, is said to be infinitely
small, since it tends towards zero. Thus the difference
of the area of a circle from that of the regular inscribed
polygon of which the number of sides is indefinitely
multiplied, is infinitely small. It is the same with
the difference between a cylinder and an inscribed
prism, or a cone and the inscribed pyramid, etc., etc.

" We cite these particular cases in order to indicate
some examples, but. infinitely small quantities may pre-
sent themselves in a multitude of circumstances where
they are not differences between variables and their
limits."

It is to be regretted, perhaps, that Duhamel did
not use the term "infinitesimal" instead of the more
ambiguous words " infinitely small" in order to ex-
press the idea which he has so clearly defined. There
is, however, nothing obscure in his meaning. An in-
finitely small quantity is, as he defines it, not a fixed
or constant quantity at all, much less one abso-
lutely small, or one beyond which there can be no
smaller quantity. It is, on the contrary, always a
variable quantity, and one which has zero for its
limit. Or, according to his .definition of a limit, an
infinitesimal is a variable which may be made to ap-
proach as near to zero as one pleases, or so near as to
reduce its difference from zero to less than any assigned

THE PHILOSOPHY OF MATHEMATICS. 47

quantity. Thus, it never becomes infinitely small, in
the literal sense of the terms, or so small that it cannot
be made still smaller. It is, on the contrary, its dis-
tinguishing characteristic that it may become smaller
and smaller without ever acquiring any fixed value,
and without actually reaching its limit, zero. It is
from these two ideas of a limit and an infinitesimal,
says he, that the whole system of truths contained in
the calculus flows in the form of logical consequences.
But in order to develop these ideas, or apply them to
the investigation of truth, he found it necessary to
establish

THE FUNDAMENTAL PRINCIPLE OF LIMITS.

It is precisely for the want of this principle, and a
knowledge of its applications, that so many mathema-
ticians, both in England and America, have discussed
the processes of the differential calculus in so obscure
and unsatisfactory a manner. This principle is indis-
pensable to render the lamp of the infinitesimal ana-
lysis a sufficient light for our eyes, as well as guide for
our feet. This principle is as follows :

" If two variable quantities are constantly equal and
tend each toward a limit, these two limits are necessarily
equal. In fact, two quantities always equal present
only one value, and it seems useless to demonstrate
that one variable value cannot tend at the same time . ||

towards two unequal limits, that is, towards two con- 'i

stant quantities different from one another. It is,
moreover, very easy to add some illustrations which
render still clearer, if possible, this important propo-
sition. Let us suppose, indeed, that two variables Ji
always equal have different limits, A and B ; A being,

[8 TEE PHIL OSOPHT OF MA THEM A TICS.

or example, the greatest, and surpassing B b j a deter-
nlnate quantity ^. The first variable having A for
L limit will end by remaining constantly comprised
>etween two values, one greater the other less than A,
et us suppose, for instance, this difference less than
; //. Likewise the second variable will end by re-
aaining at a distance from B less than J A. Now it
3 evident that then the two values could no longer
>e equal, which they ought to be, according to the
lata of the question. These data are, then, incom-
patible with the existence of any difference whatever
>etween the limits of the variables. Then these limits
re equal.

The following principle is more general and more
iseful than that laid down by Duhamel, and, besides,
b admits of a rigorous demonstration :

If, while tending toward their respective limits, two
ariable quantities are always in the same ratio to
ach other, their limits will be to one another in the same
atio as the variables.

Let the lines A B and A C represent the limits of
ny two variable magnitudes which are always in the
arne ratio to one another, and let Ab, Ac represent

A b c B' b' B C' c f C C"

wo corresponding values of the variables themselves ;
lien Ab : Ac : : A B : A C.

If not, then Ab : Ac : : A B : some line greater or
2ss than A C. Suppose, in the first place, that Ab :
Lc : : AB : A C'; A C' being less .than A C. By
ypothesis, the variable Ac continually approaches
L C, and may be made to differ from it by less than

THE PHILOSOPHY OF MATHEMATICS. 49

any given quantity. Let Ab and Ac, then, continue
to increase, always remaining in the same ratio to one
another till Ac differs from A C by less than the quan-
tity C' C ; or, in other words, till the point c passes
the point C', and reaches some point, as c', between C'
and C, and b reaches the corresponding point b'.
Then, since the ratio of the two variables is always the
same, we have

Ab : Ac : : Ab' : Ac'.

By hypothesis, Ab : Ac : : A B : A C';
hence Ab ; : Ac':: AB : AC',

or AC'XAb'=Ac'XAB;

which is impossible, since each factor of the first
member is less than the corresponding factor of the
second member. Hence the supposition that Ab :
Ac : : A B : A C',.or to any quantity less than A C, is
absurd.

Suppose, then, in the second place, that Ab : Ac : :
A B : A C", or to some term greater than A C. Now
there is some line, as A B', less than A B, which is to
AC as A B is to A C". If, then, we conceive this
ratio to be substituted for that of A B to A C", we
have

Ab:Ac::AB':AC;

which, by a process of reasoning similar to the above, | *

may be shown to be absurd. Hence, if the fourth ft

term of the proportion can be neither greater nor less f
than AC, it mustJbe equal to AC; or we must have

Ab : Ac : : A B : A C. Q. E. B,

',*

50 TEE PHILOSOPHY OF MATHEMATICS.

Cor. If two variables are always equal, their limits
are equal.

The above truth is, as has already been said, the
great fundamental principle of the infinitesimal ana-
lysis, which, being demonstrated once for all by the
rigorous method of the reductio ad absurdum, will
easily help us over a hundred chasms lying between
rectilinear and curvilinear figures, as well as between
volumes bounded by plane surfaces and those bounded
by curved surfaces, and introduce us into the beauti-
ful world of ideas beyond those chasms. But before
we can apply this prolific principle to the solution of
problems or to the demonstration of theorems, it will
be necessary to establish one or two preliminary pro-
positions. These are demonstrated by Duhamel as
follows :

1. The limit of tlie sum of the variables x, y,z . . .u,
of any finite number whatever which have respectively for
their limits a, 6, c . . . l y . positive or negative) is the
algebraic sum of those limits. In fact, the variables
x, y, z . . . u can be represented by a + a, b + &
1 + %, the differences a, /?,... A having each zero for
its limit. We have then x + y + z + . . . + u = (a +
b + c+...l) + ( + +... + *). But a + /9 +
... A tends towards the limit zero, since it is thus
with each of the terms in any finite number which
composes that quantity. Then the limit of the second
member, and consequently of the first, which is always
equal to it, is a + b + c + . . . + 1, which was to be
demonstrated.

2. The limit of the product of several variables is the
product of their limits. In fact, if we employ the same
denomination as in the preceding case, we shall have

THE PHILOSOPHY OF MATHEMATICS.

51

x y z . . . u = ( a + ) (b + /3 +) (c + r ) . . (1 + A) =
a b c . . . 1 + to, co designating the sum of a finite num-
ber of terms, each .having zero for its limit, since they
contain as factors at least one of the quantities a, ft, ?
. . . ^, each of which has zero for its limit. We see,
then, that the second member, or the first x 7 z . . . u,
has for its limit a b c . . . 1, which was to be demon-
strated.

3. The limit of the quotient of two variables is the
quotient of their limits. In fact,

x _ a+# _ a, bet a/?

~~ '

But the denominator of the last fraction can be made
as nearly as we please equal to b 2 , which is a constant
quantity different from zero; its numerator tends
towards zero; then the fraction has zero for its limit.

The limit of - is then - , the proposition to be demon-

7 b

strated.

4. The limit of a power of a variable is the same
power of its limit. For, supposing the degree of its
power to be the entire number, m, then x m is the pro-
duct of m factors equal to x, and, according to the
case 2, the limit of x m will be a m .

Let m = 2, p and q being any entire numbers what-

P i

ever x<i' is the power p of x^ ; then, according to the

P
preceding case, the limit of x^ is the p power of the

i _

limit x* or of i/x ; it remains to find this last. But x,

being the product of q factors equal to i^x, has for its
limit the q power of the limit of iXx, and as x has for

52 THE PHIL OSOPHY OF MA THEM A TICS.

its limit a, it follows that a is the q power of the limit

__ p

of ]Xx, or that iXx has for a limit i/a. Then x q has

P
for a limit a<i, as we have enunciated.

These principles will be found exceedingly easy m
practice, as well as clear and rapid in arriving at the
most beautiful results. I shall begin with cases the
most simple, and proceed with equal ease and clear-
ness to solve problems and prove theorems which are
usually esteemed more difficult.

APPLICATION TO SIMPLE QUESTIONS IN THE ELEMENTS
OF GEOMETRY.

1. TJie surfaces of any two circles are to each other
as the squares of their radii.

Let S, S ; be the surfaces of any two circles, and E,
E ; their radii. These surfaces, we know, are the
limits of two regular inscribed polygons, whose sides,
always equal in number, are supposed to be doubled
an indefinite number of times. But these polygons
are always to each other as the squares of the radii of
the circumscribed circles. Hence their limits, the
circles themselves, are to each other in the same ratio.
That is,

S : S ; : : E 2 : E' 2 ,

which is the proposition to be demonstrated.

2. TJie circumferences of any two circles are to each

Let the inscribed auxiliary polygons be as in the
last case. The circumferences of the circles are then
the limits of the peripheries of the polygons. But
these peripheries are to each other as the radii of the

THE PHILOSOPHY OF MATHEMATICS

53

circumscribed circles. Hence their limits, the circum-
ferences, are in the same ratio to each other. That is,
If C, C' be circumferences, we shall have

C : C' : : E : R',

the proposition to be demonstrated.

3. The area of a circle is equal to half its circum-

Let P denote the inscribed auxiliary polygon, a its
apothem, and p its periphery. Then we shall always
have

P = | a, p.

But if two variables are always equal, their limits will
be equal. Hence

S = |E C,

since the limit of P is S, and the limit of the product
J a, p is the product of the limits J E C. Q. E. D.

4. The volume of a cone is equal to the product of its
base by one-third of its altitude.

The cone is the limit of a pyramid having the same
vertex, and for its base a polygon inscribed in the base
of the cone, of which the number of sides may be in-
definitely increased. Let V be the volume of the
cone, B its base, and H its height, and let V, B' be
the volume and the base of the inscribed pyramid,
whose height is also H; since every pyramid is mea-
sured by one-third of its base into its height, we have

V = i B' H.

But If two variables are always equal, their limits are
equal. Hence

5*

inciated.

aoove principle, that if
nave an invariable ratio to each

v/v~~, T ._c5ir umits will necessarily be in the same

ratio to each other, the student may easily demonstrate
other theorems in the elements of geometry. He may
easily prove, for example, that the convex surface of
the cone is equal to the circumference of its base into
half its slant height ; that the volume of a cylinder
is equal to its base into its height, and that" its convex
surface is equal to the circumference of its base into
its height; that the volume of a sphere is equal to its
surface into one-third of its radius, and that its sur-
face is equal to four great circles. In like manner, he
may easily find the measure for the volume and the con-
vex surface of the frustum of a cone, by considering
them as the limits of the volume and of the convex
surface of the inscribed frustum of a pyramid. Nay,
he may go back and by the use of the same method
easily find the area of any triangle and the volume of
any pyramid.

Nor is this all. For, after having demonstrated in
a clear and easy way the theorems in the elements of
geometry, the fundamental principle of limits, as above
conceived, carries its light into analytical geometry
and into the transcendental analysis. It is, indeed,
a stream of light which comes down from that ana-
lysis, properly understood, and irradiates the lower
branches of mathematical science, somewhat as the
sun illuminates the planets. If the student will only
familiarize his mind with that principle and its appli-

THE PHILOSOPHY OF MATHEMATICS.

55

cations, he will find it one of the most fruitful and
comprehensive conceptions that ever emanated from
the brain of man. At the end of the next chapter
but one, we shall see some of its most beautiful appli-
cations to the quadrature of surfaces And to the cuba-
ture of volumes.

CHAPTER III.

THE METHOD OF INDIVISIBLES.

KEPLER introduced the consideration of infinitely
great and infinitely small quantities into the science
of mathematics. Cutting loose from the cautious and
humble method of the ancients, which seemed to feel
its way along the shores of truth, this enterprising and
sublime genius boldly launched into the boundless
ocean of the infinite. His example was contagious.
Others entered on the same dark and perilous voyage
of discovery, and that, too, without chart or compass.
Cavalieri was the first to use such quantities systemati-
cally, or to lay down rules for the guidance of the
mind in dealing with them. The manner in which
he employed them is known as " The Method of In-
divisibles," which, it is well known, opened a new
and successful career to geometry. He has invariably,
and with perfect justice, been regarded as the precur-
sor of those great men to whom we owe the infinitesi-
mal analysis.* The study of his method is, indeed, a
necessary prerequisite to a knowledge of the rise, the
aiature, the difficulties, and the fundamental principles
of that analysis.

In the method of indivisibles lines are considered
as composed of points, surfaces as composed of lines,
and volumes as composed of surfaces. " These hypo-

* Curnot on UK; InfinilcHimal Analysis, chap. III., p. 141.
56

THE PHILOSOPHY OF MATHEMATICS. 57

theses," says Carnot, " are certainly absurd, and they
ought to be employed with circumspection." * Now
here the question very naturally arises, in every re-
flecting mind, If these hypotheses or postulates are
absurd, why employ them at all ? The only answer
that has ever been returned to this question is, that
such hypotheses should be employed because they lead
to true results. Thus, says Carnot, " It is necessary
to regard them as means of abbreviation, by means of
which we obtain promptly and easily, in many cases,
what could be discovered only by long and painful
processes according to the method of exhaustion."
This method is, then, recommended solely on the
ground of its results. We do not and cannot see the
justness of its first principles; but still we must accept
them as true, because they lead to correct conclusions.
That is to say, we must invert the logical order of our
ideas and judge of our principles by the conclusions,
not of the conclusions by our principles. Nay, how-
ever absurd they may appear in the eye of reason, we
must, in the grand march of discovery, ask no ques-
tions, but just shut our eyes and swallow them down !
All honor to Cavalieri, and to every man that makes
discoveries ! But as there is a time for the making
of discoveries, so is there also a time for seeing how

We are told, for example, that a line is made up of
points, and, at the same time, that a point has abso-
lutely no length whatever. How many nothings, then,
does it take to make something? Who can tell us?
The demand is too much for the human mind. The
hypothesis is admitted to be absurd, and yet its harsh-

* Carnot on the Infinitesimal Analysis, chap. III., p. 141,
0*

58 THE PHILOSOPHY OF MATHEMATICS.

ness is sought to be softened by the assurance that It
should be regarded merely as an abbreviation. An
. abbreviation of what ? If it is the abbreviation of
any true principle, then it is not absurd at all, since it
should evidently be understood to mean the principle
of which it is the abridged form or expression. But if
it is not an abbreviation of any such principle, then
we do not see how our condition is bettered by the
use of a big word. This apology for the so-called first
principles of the method of infinites has, indeed, been
made and kept up from Carnot to Todhunter; but
we have not been informed, nor are we able to dis-
cover, of what these hypotheses are the abbreviations.
If they are abridgments at all, they may be, for aught
we can see, abridgments of conceptions as " certainly
absurd" as themselves.

After giving one or two beautiful applications of
the method of indivisibles, Carnot says : " Cavalieri
well asserted that his method is nothing but a corol-
lary from the method of exhaustion ; but he acknow-
ledged that he knew not how to give a rigorous de-
monstration of it." This is true. Cavalieri did not
know how to demonstrate his own method, because he
did not understand it. He understood it practically,
but not theoretically. That is to say, he knew how
to apply it so as to make discoveries. But how or
why his method happened to turn out true results he
did not know, and consequently he could not explain
to others. His disciples had to walk by faith and not
by science ; but if the road was dark, the goal was
beautiful. Some of his disciples even eclipsed the
master in the beauty and the value of their discoveries.
But, after all, their knowledge of the method was

THE PHIL OSOPSY OF MA THEMA TICS. 59

only practical, and consequently they wisely abstained,
as a general thing, from attempts to elucidate the
principles and the working of its interior mechanism.
u The great geometers who followed this method/'
as Carnot well says, "soon seized its spirit; it was in
great vogue with them until the discovery of the new
calculus, and they paid no more attention to the ob-
jections which were then raised against it than the
Bernouillis paid to those which were afterwards raised
against the infinitesimal analysis. It was to this
method of indivisibles that Pascal and Roberval owed
their profound researches concerning the cycloid."*
Thus, while appealing to the practical judgment of
mankind, they treated the demands of our rational
nature with disdain, and the more so, perhaps, because
these demands were not altogether silent in their own
breasts. A man may, indeed, be well satisfied with
his watch, because it truly points to the hour of the
day. But when, as a rational being, he seeks to know
how this admirable result is brought to pass, is it not
simply a grand imposition to turn him off with the
assurance that his watch keeps the time ? Does this
advance his knowledge? Does this enable him to
make or to improve watches? Nay, docs this even
give him the idea of a watch, by showing him the in-
ternal mechanism and arrangement of the parts which
serve to indicate on its surface as it passes each flying
moment of time ? No one, says Bishop Butler, can
have " the idea of a watch " without such a knowledge of
its internal mechanism, or the adaptation of its several
parts to one another and to the end which it accom-
plishes. May we not, then, with equal truth, say that

* Chapter III., p. 144.

60 THE PHILOSOPHY OX' MATHEMATICS.

no one lias "the idea" of the method of indivisibles,
or of the infinitesimal calculus, unless he can tell by
what means and how it achieves its beautiful results ?
Without such knowledge the mathematician may, it
is true, be able to name his tools and to work with
them ; but does he understand them ? Does he com-
prehend the method he employs?

Blaise Pascal himself, though universally recog-
nized as one of the greatest geniuses that ever lived,
could not comprehend the hypotheses or postulates of
the method of indivisibles as laid down by Cavalieri.
Hence, while he continued to use the language of
Cavalieri, he attached a different meaning to it a
change which is supposed by writers on the his-
tory of mathematics to have improved the rational
basis of the method. By "an indefinite number of
lines," said he, " he always meant an indefinite number
of small rectangles," of which " the sum is certainly
a plane." In like manner, by the term "surfaces,"
he meant " indefinitely small solids," the sum of which
would surely make a solid. Thus, he concludes, if
we understand in this sense the expressions " the sum
of the lines, the sum of the planes, etc., they have
nothing in them but what is perfectly conformed to
pure geometry." This is true. The sum of little
planes is certainly a plane, and the sum of little solids
is as clearly a solid. But, from this point of view, it
seems improper to call it " the method of indivisibles,"
since every plane, as well as every solid, may easily
be conceived to be divided. The improved postulates
of Pascal deliver us, indeed, from the chief difficulty
of the method of indivisibles, properly so called, only
to plunge us into another into the very one, in fact,

THE PHILOSOPHY OF MATHEMATICS.

61

from which Cavalieri sought to effect an escape by the
invention of his method.

Let me explain. If we divide any curvilinear
figure into rectangles, no matter
how small, the sum of these rect-
angles will not be exactly equal
to the area of the figure. On the
contrary, this sum will differ from
that area by a surface equal to the
sum of all the little mixtilinear
figures at the ends of the rectangles. It is evident,
however, that the smaller the rectangles are made, or
the greater their number becomes, the less will be the
difference in question. But how could Cavalieri
imagine that this difference would ever become abso-
lutely nothing so long as the inscribed rectangles con-
tinue to be surfaces ? Hence, in order to get rid of
this difference altogether, and to arrive at the exact
area of the proposed figure, he conceived the small
rectangles to increase in number until they dwindled
into veritable lines. The sum of these lines he sup-
posed would be equal to the area of the figure in ques-
tion ; and he was confirmed in this hypo-thesis, because
it was found to conduct to perfectly exact results.
Thus, his hypothesis was adopted by him, not because
it had appeared at first, or in itself considered, as intui-
tively certain, but because it appeared to be the only
means of escape from a false hypothesis, and because
it led to so many exactly true results. But when this
hypothesis, abstractly considered, was found to shock
the reason of mankind, which, in the words of Carnot,
pronounced it " certainly absurd/ 7 the advocates of the
method of indivisibles were obliged to assume new

62 THE PHILOSOPHY OF MATHEMATICS.

ground. Accordingly, they discovered that indivisi*
bles might be divided, and that by "the sum of right
lines" was only meant " the sum of indefinitely small
rectangles/' Pascal seems to believe, in fact, that such
was the meaning of Cavalieri himself. It is certain
that history has decided other wise, and delivered the
verdict that by indivisibles Cavalieri really meant
indivisibles.

Now, it seems just as evident that a curvilinear
figure is not composed of rectangles, as that it is not
composed of right lines. Yet Pascal, the great dis-
oiple, adopted this supposition as the only apparent
means of escape from the absurdity imputed to that
of the master, and he pointed to the perfect accuracy
of his conclusions as a proof of the truth of his hypo-
thesis. For, strange to say, the sum of the rectangles,
as well as the sum of the lines, was found to be exactly
equal to the area of the curvilinear figure. What,
then, became of the little mixtilinear figures at the
extremities of the rectangles? How, since they were
omitted or thrown out, could the remaining portion
of the surface or the sum of the rectangles alone be
equal to the whole? Pascal just cut the Gordian
knot of this difficulty by declaring that if two finite
quantities " differ from each other by an indefinitely
small quantity/' then " the one may be taken for the
other without making the slightest difference in the
result." Or, in other words, that an infinitely small
quantity may be added to or subtracted from a finite
quantity without making the least change in its mag-
nitude. It was on this principle " that he neglected
without scruple," as Carnot says, " these little quai ti-
ties as compared with finite quantities ; for we see that

THE PHILOSOPHY OF MATHEMATICS.

Pascal regarded as simple ivctangk'S the trapeziums or
little porticos of the area of the curve comprise*! be-
tween two consecutive co-ordinates, neglecting conse-
quently the little mixtilinear triangles which have for
their bases the differences of those ordinates."*

Carnot adds, as if he intended to justify this pro-
cedure, that " no person, however, has been tempted to
reproach Pascal with a want of severity." This seems
the more unaccountable, because Carnot himself has
repeatedly said that it is an error to throw out such
quantities as nothing. Nor is this all. No one can
look the principle fairly and fully in the face, that, an
infinitely small quantity may be substraeted from a
finite quantity without making even an infinitely small
difference in its value, and yet regard it as otherwise
than absurd. It is when such a principle is recom-
mended to the mathematician by the desperate exi-
gencies of a system which strains his .reason, warps his
judgment, and clouds his imagination, that; it is ad-
mitted to a resting-place in his mind. It was thus, as
we have seen, that Pascal was led to adopt the prin-
ciple in question; and it was thus, as wo shall see,
that Leibnitz was induced to assume the same absurd
principle - as an unquestionable axiom in geometry.

Now iff with Cavalicri, we suppose a surface to be
composed of lines, or a line of points, then we shall
have to add points or no-magnitudes together until we
make magnitudes. Nay, if lines are composed of
points, surfaces of lines, and solids of surfaces, then is
it perfectly evident that solids are made up of points,
and the very largest magnitude is composed of that
which has no magnitude ! Or, in other words, every

* Carnot, chapter III., p. 140.

64 THE PHILOSOPHY OF MATHEMATICS.

magnitude is only the sum of nothings ! On the othes
hand, if we agree with Pascal that a curvilinear space
is, strictly speaking, composed of rectangles alone,
then we shall have to conclude that one quantity may
be taken from another without diminishing its value !
Which term of the alternative shall we adopt ? On
which horn of the dilemma shall we choose to be im-
paled? Any one is at liberty to select that which is
the most agreeable to his reason or imagination. But
is it, indeed, absolutely necessary to be swamped amid
the zeros of Cavalieri or else to wear the yoke of Pas-
cal's axiom ? May we not, on the contrary, guided
by the careful insight of some new Spallanzani, safely
sail between this Scylla and Charybdis of the infinitesi-
mal method? The reader will soon be enabled to

Many persons have embraced the axiom in question
without seeming to know anything of the motives
which induced a Pascal and a Roberval to invent and
use it. Thus, for example, in a " Mathematical Dic-
tionary and Cyclopedia of Mathematical Sciences/' it
is said, " When several quantities, either finite or in-
finitesimal, are connected together by the signs plus or
minus, all except those of the lowest order may be
neglected without affecting the value of the expres-
sion. Thus, a+ dx + dx 2 = a."* Is it possible
a + dx + dx 2 is exactly equal to a, and yet dx + dx 2
are really quantities? But, then, they are so very
small that they may be added to a, without affecting
its value in the least possible degree !

There is, it is true, high authority in favor of this

* Dictionary of Mathematics, etc., by Davies and Peck. Art
Infinitesimal.

THE PHILOSOPHY OF MATHEMATICS. 65

strange axiom. Roberval, Pascal, Leibnitz, the Mar-
quis de L'HSpital, and others, have all lent the sanc-
tion of their great names to support this axiom and
give it currency in the mathematical world. But does
a real axiom ever need the support of authority ? On
the other hand, there is against this pretended axiom,
as intrinsically and evidently false, the high authority
of Berkeley, Maclaurin, Carnot, Euler, D'Alembert,
Lagrange, and Newton, whose names preclude the
mention of any other. But "where doctors disagree?"
Doctors never disagree about the axioms of geometry.
The very fact of a disagreement among them proves
thing else which is set up as an axiom. It is, indeed,
of the very essence of geometrical axioms that they are
necessary and universal truths, absolutely commanding
the assent of all, and shining, like stars, above all the
dust and darkness of human controversy. But waiv-
ing this, I shall in the next chapter explode this pre-
tended axiom, this principle of darkness assuming the
form of light, which has so long cast its shadow on
some of the fairest portions of demonstrative truth.

I shall conclude the present chapter with the exam-
ples which Carnot has given from Cavalieri and Pascal
to illustrate and recommend the method of indivisibles.
" Let A B," says he, " be the diameter of a semicircle, f

A G B: let A B F D be the circumscribed rectangle; ;

C G the radius perpendicular to D F; let the two
diagonals C D, C F also be drawn ; and finally through
any point m of the line A D, let the right line mnpg
be drawn perpendicular to C G, cutting the circum-
ference of the circle at the point n, and the diagonal r
C D at the point _p. ^

66

THE PHILOSOPHY OF MATHEMATICS.

771

D G F

" Conceive the whole figure to turn around C G, as
an axis; the quadrant of the circle A C G will gener-
ate the volume of a semi-sphere whose diameter is A
B ; the rectangle A D C G will generate the circum-
scribed right cylinder; the isosceles right-angled tri-
angle C G D will generate a right cone, having the
equal lines C G, D G for its height and for the radius
of its base; and finally the three right lines or segments
of a right line m g, ng, p g will each generate a circle,
of which the point g will be the centre.

"But the first of these circles is an element of the
cylinder, the second is an element of the semi-sphere,
and the third that of the cone.

" Moreover, since the areas of these circles are as the
dently form the hypothenuse and the two sides of a
right-angled triangle, it is clear that the first of these
circles is equal to the sum of the other two; that is to
say,, that the element of the cylinder is equal to the sum
of the corresponding elements of the semi-sphere and
of the cone ; and as it is the same with all the other ele-
ments, it follows that the total volume of the cylinder is
equal to the sum of the total volume of the semi-sphere
and of the total volume of the cone.

THE PHILOSOPHY OF MATHEMATICS. 67

"But we know that the volume of the cone is one-
third that of the cylinder; then that of the semi-sphere
is two-thirds of it; then the volume of the entire sphere
is two-thirds of the volume of the circumscribed cylin-
der, as Archimedes discovered."

Again, says Carnot, " the ordinary algebra teaches
how to find the sum of a progression of terms taken
in the series of natural numbers, the sum of their
squares, that of their cubes, etc,; and this knowledge
furnishes to the geometry of indivisibles the means
of valuing the area of a great number of rectilinear
and curvilinear figures, and the volumes of a great
number of bodies.

Let there be a triangle, for example; let fall from
its vertex upon its base a perpendicular, and divide
this perpendicular into an infinity of equal parts, and
lead through each of the points of division a right
line parallel to the base, and which may be terminated
by the two sides of the triangle.

According to the principles of the geometry of in-
divisibles, we can consider the area of the triangle as
the sum of all the parallels which are regarded as its
elements ; but, by the property of the triangle, these
right lines are proportioned to their distances from the
vertex ; then the height being supposed divided into
equal parts, these parallels will increase in an arith-
metical progression, of which the first term is zero.

But in every progression by differences of which
the first term is zero, the sum of all the terms is equal,
to the last, multiplied by half the number of terms.
But here the sum of the terms is represented by the
area of the triangle, the last term by the base, and the
number of terms by the height. Then the area of

68 THE PHILOSOPHY OF MATHEMATICS.

every triangle is equal to the product of its base by
the half of its height.

Let there be a pyramid; let fall a perpendicular
from the vertex to the base; let us divide this perpen-
dicular into an infinity of equal parts, and through
each point of division pass a plane parallel to the base
of this pyramid.

According to the principles of the geometry of indi-
visibles, the intersections of each of these planes by
the volume of the pyramid will be one of the elements
of this volume, and this latter will be only the sum
of all these elements.

But by the properties of the pyramid these elements
are to each other as the squares of their distances from
the vertex. Calling, then, B the base of the pyramid,
H its height, b one of the elements of which we have
just spoken, h its distance from the vertex, and V the
volume of the pyramid, we will have

B:b::H 2 :h 2 ;

therefore,

Then V, which is the sum of all these elements, is

T>

equal to the constant multiplied by the sum of the

squares of Ji 2 ; and since the distances h increase in a
progression by differences of which the first term is
zero and the last H that is, as the natural numbers
from o to H the quantities K 2 will represent their
squares from o to H 2 .

Now common algebra teaches us that the sum of

THE PHILOSOPHY OF MATHEMATICS. 69

the squares of the natural numbers from o to H, in-
clusively, is

2H 3 +3H 2 + H

But here the number H being infinite all the terms
which follow the first in the numerator disappear in
comparison with this first term, then this sum of the
squares is reduced to -J H 3 .

T>

Multiplying, then, this value by the constant
found above, we will have for the volume sought

that is, the volume of the pyramid is the third of the
product of its base by its height."

Now here we see, in all their naked harshness, the
assumption of Cavalieri on the one hand and that of
Pascal on the other. An area is supposed to be made
up of lines, of that which, compared with the unit of
superficial measure, has absolutely no area at all !
This hypothesis is, as we have seen, pronounced c; cer-
tainly absurd" by Carnot, and yet it leads by some
unknown process to true results. How this happened,
or could have happened, Carnot is at no pains to ex-
plain. This seems the more extraordinary because
the clue to the secret was more than once in his hands,
and only required to be seized with a firm grasp and
followed out to its consequences, in order to solve
the enigma of "the method of indivisibles." He is,
in fact, the apologist rather than the expounder of tiur
method.

No one was more sensible than Cavalieri liinise.i

70 TEE PHILOSOPHY OF MATHEMATICS.

of the grave objections to his own method. Accord-
ingly he strove, as he tells us, " to avoid the suppos-
ing of magnitude to consist of indivisible parts/' because
there remained some difficulties in the matter which
he was not able to resolve.* Instead of pretending
that he could explain, or even see through, these ob-
jections, he exclaimed : " Here are difficulties which
the arms of Achilles could not conquer." He speaks,
indeed, as if he foresaw that his method would be, at
some future day, delivered in an unexceptionable form,
so as to satisfy the most scrupulous geometrician. But
free from the miserable sham of pretending to under-
stand it himself, he simply leaves, with a beautiful
candor worthy of his genius, this Qordian knot, as he
calls it, to some future Alexander. If that Alexander
appeared in the person of Carnot, it must be admitted
that, like the original, he was content to cut rather
than to untie the Gordian knot of the method of
indivisibles, f

Again, we are gravely told that infinity, plus 3
times infinity square, may be neglected, or thrown out
as nothing, by the side of infinity cube. Now such
propositions (I speak from experimental knowledge)
tend to disgust some of the best students of science
with the teachings of the calculus, and to inspire nearly
all with the conviction that it is merely a method of
approximation. How could it be otherwise ? How
oan reflecting minds, or such as have been trained and
encouraged to think, be told, as we are habitually told
in the study of the differential calculus, that certain

* Cavalieri, Georn. Indivis., lib. 7.

f I speak in this way, because in my laborious search after light
respecting the enigma of the method of Cavalieri, I applied to Oar-
not in vain.

THE PHILOSOPHY OF MATHEMATICS. 71

f*

quantities are thrown out or neglected on one side of
a perfect equation, without feeling that its perfection
has been impaired, and that the result will, therefore,
be only an approximation to the truth ? This is the
conclusion of nearly all students of the calculus, until
they are better informed by their instructors. Every
teacher of the calculus is often called upon to encounter
this difficulty; but, unfortunately, few are prepared
to solve it either to their own satisfaction or to that
of their pupils.

Thus, for example, in one of the latest and best
treatises on the "Differential Calculus" which has
been issued from the University of Cambridge, we
find these words : "A difficulty of a more serious kind,
which is connected with the notion of a limit, appears
to embarrass many students of this subject namely, a
suspicion that the methods employed are only approxi-
mative, and therefore a doubt as to whether the results
are absolutely true. This objection is certainly very
natural, but at the same time by no means easy to
meet, on account of the inability of the reader to point
out any definite place at which his uncertainty com-
mences. In such a case all he can do is to fix his
attention very carefully on some part of the subject,
as the theory of expansions for example, where specific
important formulas are obtained. He must examine
the demonstrations, and if he can find no flaw in
them, he must allow that results absolutely true and
free from all approximation can be legitimately derived
by the doctrine of limits." *

Alas! that such teaching should, in the year of

* Tod hunter's Differential Calculus, etc. Cambridge : Macmillau
& Co. 1855.

7 2 TEE PHIL OSOPHY OF MA THE MA TICS.

grace 1866, issue from the most learned mathematical
University in the world, and that, too, nearly two
centuries after its greatest intellect, Newton, had
created the calculus ! What ! the reader, the student
not able to point out the place at which his difficulty
begins ! Does not every student know perfectly well,
in fact, that when he sees small quantities neglected,
or thrown out on one side of an equation, and nothing
done with them on the other, he then and there begins
to suspect that the calculus is merely an approxima-
tive method? In view of the rejection of such quan-
tities his " objection is," as the author says, " certainly
very natural." Nay, his "suspicion" is not only
natural; it is necessary and inevitable. But if any
student should be unable to tell where his " difficulty,"
his "suspicion," his "uncertainty" commences, why
should not this be pointed out to him by his teacher ?
Surely, after the labors of a Berkeley, a Carnot, a
D'Alembert, and of a hundred more, the teacher of
mathematics in the most learned University in the
world should be at no loss either to explain the origin
of such a difficulty, or to give a rational solution of it.
Is the philosophy, the theory, the rationale of the in-
finitesimal calculus not at all studied at Cambridge ?
The truth is, that the teacher in question, like many
others, found it "by no means easy to meet" the diffi-
culty which haunts the mind of every student of the
calculus, just because he himself had studied the won-
derful creation of Newton merely as a practical art to
be used, and not as a glorious science to be under-
stood.

CHAPTER IV.

SOLUTION OF THE MYSTERY OF CAVALIERl's METHOD,
AND THE TRUE METHOD SUBSTITUTED IN ITS PLACE.

IN the preceding chapter, the difficulty, the enigma,
the mystery of Cavalieri's method was fully exhibited.
It is my object, in this chapter, to clear up the mys-
tery of that method, and to set the truth in a trans-
parent and convincing point of view. Or to untie,
as he calls it, " the Gordian knot" of his method, and
to replace it by a perfectly clear train of reasoning,
which shows the necessary connection between unde-
nied and undeniable principles, and the conclusion at
which he arrived, as well as conclusions lying beyond
the reach of his obscure and imperfectly developed
system.

I shall begin with the first of the examples or illus-
trations produced from the work of Carnot. By con-
sulting the last chapter the reader will perceive that
Cavalieri finds the area of any triangle by obtaining,
as he supposes, the sum of its elements or of all right
lines parallel with its base, and included between its
two sides. Now, although this hypothesis is "cer-
tainly absurd," yet is there at the bottom of it a pro-
found truth which was most obscurely seen, and there-
fore most inadequately expressed, by the great Italian.
Nor from that day to this has the truth in question
been any better seen or more adequately expressed a

7 I> 73

74 TEE P&IL OSOPSY OF MA THEM A TICS.

fact which will in due time be demonstrated in the
following pages. As often as the mathematician has
by his reasoning been brought face to face with this
great truth he has failed to see it, because he has mis-
conceived and misinterpreted his symbols. But we
are not, as yet, quite prepared to set this singular and
instructive fact in a perfectly satisfactory and convin-
cing light.

is represented by the figure A 1> (J.
Let its altitude A D be divided
into any number of equal parks,
as seen in the figure, and through
each point of division let a rig-lit
line be drawn parallel to its base
and terminating in its two sides.
Let there be, as in the figure, n
system of rectangles constructed,
each having in succession one of the parallel linos for
its base. Now the question is, what course should
the geometer pursue in order to obtain by a clear and
unexceptionable logical process the area of the triangles
ABC?

Cavalieri, as we have already seen, would not proeeed
on the assumption that the sum of the rectangles, how
great soever their number, is equal to the area of the
triangle, because he believed that it would always he
greater than that area. Hence, in order to arrive at
the exact area, he conceived the triangle to be coin-
posed, not of rectangles however small, but of ri<r] lt
lines. Pascal, on the other hand, acknowledging the
absurdly of such an hypothesis, supposed the tri<7 n ^
to be composed of the rectangles when their munber

THE PHILOSOPHY OF MATHEMATICS. 75

was indefinitely increased. Thus, by a slight diverg-
ence between the courses of the two great geometers,
the one was landed in Scylla and the other in Cha-
ry bdis.

The method of Pascal is founded in error. Its
basis, its fundamental conception, is demonstrably
false. It is evident that the sum of the rectangles
can never be exactly equal to the area of the triangle
unless the broken line AlmnopqrstuvC can be
made to coincide with the line A C. But this can
never be, since, however great the number of rect-
angles may be conceived to be, still the sum of all the
little lines, such as A I, m n, o p, and so forth, parallel
to the base of the triangle will always continue equal
to D C, and the sum of all the little lines, such as n o,
p g, r s, and so forth, parallel to the altitude A D of
the triangle, will always continue equal to A D. Hence
the broken line A.lmn op qrstuvG will always re-
main equal to D C + A D; and if it should ever coin-
cide with A C, then one side of the triangle ADC
would be equal to the sum of the other two, or the
hypothenuse of a right-angled triangle would be equal
to the sum of its two sides, which is impossible. In-
deed, the broken line in question is a constant quantity;
the number of parallels may be increased ad libitum, and
yet the length of the broken line will remain invariably
the same. Hence the difference between this constant
length and the length of A C is itself a constant quan-
tity, and the length of the one line never even approxi-
mates to that of the other, much less can the one ever
coincide with the other. A C is not even the limit of
the broken line A I m n, etc., since the value of the
latter does not tend toward that of the former as the

76 THE PHILOSOPHY OF MATHEMATICS.

number of its parts is increased. But the area of the
triangle A D C is the limit of the area of the figure
CD Almnopr stuvC, since the last area continu-
ally tends toward an equality with the first area, with-
out ever becoming absolutely equal to it. The same
things are, it is obvious, equally true in regard to the
right line A B, and the broken line on the other sid<*
of the triangle ABC.

We should, then, discard the fundamental concep-
tion of Pascal and Eoberval as false; which we may
do at the present day without falling into the hypo-
thesis of Cavalieri or any of its manifold obscurities.
If, instead of seeking the sum of the rectangles, whose
number is supposed to be indefinitely increased, we
seek the limit of that sum, we shall find the exact area
of the triangle by a logical process as clear in itself as
it is true in its conclusion.

For this purpose let B represent the base of the
triangle A B C, b the base b e of any triangle, A b e
formed by one of the lines parallel to B C, H and h
the respective heights of these two triangles, and k
one of the equal parts into which the line A D has
been divided. Then, by similar triangles, we have

b : B::7 i: H,

, B _
or >

in which b Jc is the area of the little rectangle, whose base
is b and altitude k. Now, the limit of the sum of all
such rectangles being the exact area of the triangle

THE PHILOSOPHY OF MATHEMATICS.

77

ABC, we have only to find the limit of that sum in
order to obtain an expression for the area sought.
That is to say, we have only to find the limit of the sum

T>

of k .h for all the values of h. But the value of Ji
H

varies from A to A D or from zero to H, and since
the heights of the little rectangles are all equal to each
other, we shall have for the successive values of 7i,

Jc 9 2k, 3 Jc . . . n k,

in which n denotes the whole number of rectangles,
or of equal parts into which A D is divided. Let it
be observed that

Then the sum of

H

H

or

Jti

But since the sum of the series 1 + 2 + 3 . . . + n is,
according to a well-known algebraic formula, equal to

n (n + 1) ,
_A ! L^ we have

- (1 + 2 + 3. .. ft ) = A

H H

k(nlc

H*

_B H (H + ^)^B
~H 2 H

7*

78 THE PHIL OSOJPSY OF MA TSEMA TICS.

Now, if S be the sum of the rectangles, we shall have

B EP + Hfe
S = -X

However small k may be made, or however great, in
other words, the number of rectangles may be con-
ceived to be, the two variables S and its value will be
equal to each other. Hence, as has been demonstrated,
their limits are equal. But the limit of S is the area

13 TT2 I TT 7.

of the triangle ABC, and the limit of - X :

H 2

B H 2 B TT

is X , or . That is, the area of the triangle

A B C is one-half the product of its base by its alti-
tude.

Now, it may be clearly shown how it was that Pas-
cal, as well as Roberval and others, started from a
false hypothesis or first principle, and yet arrived at a
perfectly correct conclusion. He committed an error
first in supposing that the sum of the rectangles would
ultimately be equal to the area of the triangle; he
committed another error, in the second place, in sup-
posing that he could reject indefinitely small quanti-
ties without making any difference in the result; and
these two errors, being opposite and equal, just exactly
neutralized each other. Thus, the quantities which
he rejected did make a most important difference in
false. It is, in the natural world, experimentally
proved that two rays of light may cross each other so
as to produce darkness. But this is nothing to the
wonder of the infinitesimal method as used by Rober-

THE PHILOSOPHY OF MATHEMATICS. 79

val and Pascal. For here two rays of darkness are

Thus, in the logic of Pascal, there was an unsus-
pected compensation of unsuspected errors. This
might, indeed, have been conjectured from the nature
of his procedure. For, if we look at the figure, we
shall perceive that the sum of the rectangles is made
up of the triangle ABC, which is always constant,
and of all the little variable triangles which serve to
complete that sum. In like manner, if we examine
the expression for the sum of the rectangles, we shall
find that it is composed of a constant term and of a
variable term. For that expression is, as we have

B H 2 + H BH,B . ,.,

seen, X , or 1 , an expression winch,

BH

literally understood, has no meaning. For is a

T>

surface, and is a line, and it is impossible to add a

2

line to a surface. Hence, according to the well-known
principle of homogeneity, we must in all such cases
restore the understood unit of measure, which is, in
the present case, the variable quantity h. The above

T> TT T> L

expression then becomes \~ . The constant

2 2

o TT

term is the measure of the constant triangle A B C.

2

Is not the variable term, then, , the expression for

2

the sum of all the little variable triangles ? That is
to say, have not all these little triangles been elded
to the area ABC, and then thrown away as if they
were nothing in their last stage of littleness ? Such a

80 THE PHILOSOPHY OF MATHEMATICS.

suspicion, it seems to me, ought to have arisen in the
mind of any one who had looked closely and narrowly
into the mysteries of this method.

But this charge of a compensation of errors is some-
thing more than a shrewd suspicion or conjecture. It
is a demonstrative certainty. The opposite errors may
be easily seen and computed, so as to show that they
exactly neutralize each other. Thus, when it is asserted
that the triangle A B C is equal to the sum of all the
rectangles set forth in the figure, it is clear that the
measure is too great, and exceeds the area of A B C by
the sum of all the aforesaid little triangles. But the

rejected term is exactly equal to that excess, or to

2

the sum of all the little triangles. For the triangle

o
, o p q =

v //x/v , T /->,

T 8 * - s t u = - , and u v C =

2 ' 2 '

Hence their sum is equal to

st-j- uv)k _ D C X

2 2 '

In like manner it may be shown that the sum of the
triangles on the other side of the triangle A B G is

equal - . Hence the sum of all the triangles

2

on both sides of A B C is equal to (P + B P) k =

H 2

T) 7,

. But this is precisely the quantity which has
2

been thrown away, as so very small as to make abso-
lutely no difference in the result ! It is first added

THE PHILOSOPHY OF MATHEMATICS. 81

by a false hypothesis, and then rejected by virtue of a
false axiom ; and the exact truth is reached, both to
the astonishment of the logician not to say magician
and of all the world beside.

If we may, openly and above-board, indulge in such
a compensation of errors, then we need not go down
into the darkness of the infinite at all. For the above
reasoning if reasoning it may be called is just as
applicable to a finite as it is to an infinite number of
terms. Let us suppose, for example, that the number
of rectangles constructed, as above, are finite and
fixed instead of variable and indefinite. Let this
finite fixed number be denoted by n and the sum of
the rectangles by S.

rni . Q B H . B Jo
Inen fo= .

22

Now, if we may be permitted to assert, in the first
place, that this sum is equal to the area of the triangle,

TR Z*
and, in the second, throw away as unworthy of

2i

"B "FT
notice, then we shall obtain , or one-half the pro-

Zi

duct of the base by the height as an expression for the
area of the triangle. The result is exactly correct.
But, then, in asserting that the sum of the rectangles,
say of ten for example, is equal to the triangle, we
make its area too great by the sum of twenty very
respectable triangles. We correct this error, however,

by throwing away , or rather , which is
2 2i

exactly equal to the sum of these twenty triangles.
D*

82 THE PHIL OSOPHT OF MA THEM A TICS.

Precisely such, in nature and in kind, is the reason-
ing of the more approved form of the method of indi-
visibles. It is, indeed, only under the darkness of
the infinite that such assertions may be made and such
illicit processes carried on without being detected, and
they expire under the scrutiny of a microscopic in-
spection.

How different the method of limits ! If properly
understood, .this proceeds on no false assertion and
perpetrates no illicit process. No magnified view can
be given to this method which will show its propor-
tions to be otherwise than just or its reasonings to be
otherwise than perfect. Having found the above
expression for the sum of its auxiliary rectangles,

T> TT2 I TT

which is S = ~ X - , this method does not
JnL 2*

throw away H in the numerator of the last term, be-
cause H, though infinite, may therefore be treated as
nothing by the side of H 2 . On the contrary, it simply
makes that term homogeneous by restoring the sup-
pressed or understood unit of measure &, so that it
becomes

a BH .

b = -- h - ;

22'

and then proceeds on the demonstrated truth thai If
two variables are always equal, their limits must also
be equal. But the area of the triangle is the limit of

T> TT

S (the sum of the rectangles), and - is the limit of

2
the second member of the above equation. Hence, if

T> TT

A be the area of the triangle, we have A = - .

2

THE PHIL OSOPHY OF MA THEM A TICS. 83

T rl , ,, . 2H 3 +3H 2 4 H

In like manner, from the expression -

* 6

found by Carnot in the last chapter, the method of
limits does not reject infinity, plus 3 times infinity
square, as nothing by the side of twice infinity cube,
i)i order to reach the conclusion that the whole ex-
pression is exactly equal to the needed result H 3 .
In the sublime philosophy of Pascal, " the number H
being infinite, all the terms which follow + 2 H 3 in the
numerator disappear by the side of that first term ;
then that sum of the squares reduces itself to J HV
But the method of limits, more humble and cautious
in its spirit, takes its departure from the demonstrated
proposition that if two variable quantities are always
equal, then their limits must be equal, and arrives at

. , ,, u _, 2H 3 +3H 2 +H

precisely the same result. For - ,

i *n A i . iu

when fully expressed, is -- 1 --- h - , and by

making Jc = o, we find its limit |- H 3 .

From the above example it will be seen that Pascal,
instead of taking the sum of his auxiliary rectangles
and the sum of his auxiliary prisms, as he supposed
he did, in finding the area of a triangle and the
volume of a pyramid, really took the limits of those
sums, and that, too, without even having had the idea
of a limit, or comprehending the nature of the process
he performed. Nor is this all. For he arrived at p|

this result only by a one-sided and partial application I j

of his own principle. In order to explain, let us re- i

sume the above expression for the sum of the auxiliary /f '

T> TT2 I TT

rectangles, which is X - . Now if II be
Jo. 2

84 THE PHIL OSOPHY OF MA THEMA TICS.

infinite, and may be omitted as nothing compared

TT 2
with H 2 , reducing the last factor to , it should be

2t
remembered that, according to the same supposition,

T>

the first factor becomes = o. Hence the expression

CO

for the sum of the rectangles is reduced to o X , or
an infinite number of zeros. Precisely the symbol of
the great truth which lies at the bottom of Cavalieri's
hypothesis, and which, as we shall hereafter see ; still
remains to be correctly interpreted by the mathemati-
cal world. In like manner the sum of the auxiliary
prisms used in finding the volume of the pyramid, or

8 = 5- X 2H3 + 3H2 + H i s reduced by the same
H 2 6

suppositions to S = X = o X o , a symbol which

never could have been understood or correctly inter-
preted without a knowledge of the method of limits.
But ever since that knowledge has been possessed and
more clearly developed, the meaning of the symbol
o X co has been, as it were, looking the mathematician
in the face and waiting to be discovered. No attempt
can, however, be made to construe it, until the methods
of Leibnitz and Newton be passed under review.

Before leaving this branch of the subject, it may be
well to show how, by the method of limits, the volume
of the pyramid is determined. Let V, then, be the
volume of any pyramid, B its base, and H the perpen-
dicular fro in its vertex on the plane of its base. Let H
be divided into any number of equal parts, each repre-
sented by Jc, and planes passed through the several
points of division parallel to the base. On the base

THE PHILOSOPHY OF MATHEMATICS.

85

of the pyramid, and on every similar section of the
pyramid cut out by the parallel planes, conceive right
prisms to be constructed, each equal in height to /c,
the distance between any two adjacent parallel planes.
Let S represent the sum of these prisms, b the base of
any one of them except the lowest, and li its distance
from the vertex of the pyramid.

Then, by a well-known property of the pyramid,
we shall have

or

and

.,

H" '

/ 7 B/C 72

for the volume of the prism whose base is 6. Now
S, the sum of all the prisms, is evidently equal to

B Jc '

multiplied into the several values of Ji 2 . But, if

n be the whole number of prisms, then the several
values of li will be

Jo, 2 k, 3 Jc, 4 7c . . . + n L nJc = H.
Hence, S = ? (# + 2 2 P + 3 2 F + 4?1P... a #),

or

But, according to a well-known algebraic formula^

1 . 2i . o

86 TSE PHILOSOPHY OF MA THEMA TICS.

B

Hence, S = .

__
or S-

___

~~

1.2.3

Now, if we conceive k to become smaller and smaller,
or the number of prisms to become greater and greater,
their sum will continually tend more and more to an
equality with the volume of the pyramid, without ever
becoming exactly equal to that volume. Hence V is
the limit of S. In like manner, as k becomes smaller
and smaller, the expression

l_ x

_ /\

H 2 \ 1.2.3

or

tends continually more and more toward an equality

T>

with X J H 3 , without ever reaching that value,
while k remains a real quantity, or the prisms have

T>

the least possible thickness. Hence X H 3 is the

limit of the variable in question.

But as these two variables are always equal, then
are their limits also equal. That is to say,

# Here, as the unit of measure k is not dropped or suppressed,' the
expression is homogeneous, as it should always be understood to Ib
even when not expressed.

THE PHILOSOPHY OF MATHEMATICS. 87

limit of S = limit of ~ X (J H 8 -f J H 2 + | H ),

the well-known measure for the volume of a py-
ramid.

In the above example I have used a good many
words, because the beginner, for whom it is written,
is not supposed to be familiar with the method of
limits. But the process is in itself so direct, simple,
and luminous, that a little familiarity with the method
of limits will enable the student to repeat it or any
similar process almost at a glance. He will only
have to conceive the pyramid with its system of aux-
iliary prisms, form the expression for their sum, pass
to its limits, and the problem is solved, or the volume
of the pyramid found. And he may do this, too,
with little or no aid from the use of diagrams or sym-
bols. He may, in fact, bring his mind into direct
contact with geometrical phenomena, and reason out
his results in full view of the nature of things, or of
their relations, rather than in the blind handling of
mere formula), and thus beget a habit of meditation
and of close discriminating attention, which are among
the very best effects of any system of mental edu-
cation.

AKEA OF THE PAKABOIJC SEGMENT.

This question will offer us examples of very various
procedures which may be employed in the search of
quadratures, and will give an idea of the variety of
resources which the infinitesimal method presents.

88

TEE PHILOSOPHY OF MATHEMATICS.

First Solution. Beferring the parabola to the diame-
ter A B and to the tangent A Y parallel to the base
of the segment ADC, the squares of the ordinates

are proportional to the abscissas, and the equation of
the curve is

y 2 = 2p x.

Let us cut the surface by parallels to A Y; the area,
ADC may be considered as the limit of the sum of
interior parallelograms,, which will be divided into two
equal parts by the diameter A B, so that the two areas
A B C, A B D, being limits of equal sums, are equiva-
lent It suffices, then, to calculate the half ABC.

We shall know the area A B C if we know its ratio
to the complement AEG of this area in the parallelo-
gram A B C E, and to find this we will compare two
corresponding parallelograms P M H P', Q, M K Q/
of the sums which have these two areas for their limits*

I

TEE PHILOSOPHY OF MATHEMATICS. 89

Designating by x, y the co-ordinates of any point M,
and by Ji and k the increments that they acquire In
passing from M to M ; , we will have

PMHP 7 = yA
QMKQ'~~o?i'

and we have, from the equation of the curve,

hence 2yk + k 2 = 2 ph.

Hence,

h = 2y + &. or y h = y

I 2_p ' * k x

xk

This ratio then tends toward the limit 2, when k
tends toward zero ; the two areas A C B ; AEG are
then the limits of sums of such infinitely small
quantities that the ratio of any two corresponding
ones tends toward the same limit 2 ; then, according
to the principle already demonstrated, the ratio of
the areas A C B, A E C is exactly 2. Thus, the
area of A C B is two-thirds of the parallelogram
A E G B, and the proposed area A G D is four-thirds
of this same parallelogram, or two-thirds of the whole
circumscribed parallelogram.

Second Solution. It is easy to calculate directly
the area AEG, which is the limit of the sum of the
parallelograms Q M K Q' or Q II M 7 Q', of which
the general expression is x k sin A, A designating the

90 THE PHIL OS PHY OF MA THE MA TICS.

angle Y A X, k designating the increment of y. It Is
necessary to express x in terms of y, which will give
for the expression of any one of the parallelograms

2 7 * A

y sm . Now, if we suppose in this case that the
2p

altitudes of the parallelograms are all equal, which
was useless in the preceding solution, the successive
values of y will be

k, 2 k, 3k...nk,
and we shall have

n k = A E, or (n + 1) k = A E,

according as we take the parallelograms Q, M ; or Q, K,
which is indifferent.

It is required, then, to find the limit of the sum

when n increases indefinitely, and k decreases at the
same time, so that we always have n k = A E.

Now, Archimedes has given for the summation of
the squares of the natural numbers a formula which,
written with the signs used by the moderns, gives

1.2.3

It is necessary, then, to find the limit of the follow-
ing expression

k* sin A n (n + 1) (2 n + 1)

1.2.3

or

k) sin A

TEE PHILOSOPHY OF MATHEMATICS.

91

when Je tends towards zero. That limit is evidently

AJE 3 . sin A A B . A E sin A
or

2.3jp 3 '

observing that = A B.

2p

The area A E C is, then, the third of the parallelo-
gram A B C E, and the area A B C is two-thirds of it,
as we found by the first solution.

Third Solution. This solution will have the advan-
tage of giving an example of a mode of decomposition
very different from- the preceding ones. We shall in
this consider an area as the limit of a sum of areas,
indefinitely small, determined by tangents to the same
curve.

Let A C C ; be the parabolic segment, A B the

diameter, C D the tangent at C ; from, which results
A D = A B ; let us compare the two areas A C B P
I) AC.

92 THE PHILOSOPHY OF MATHEMATICS.

"We may consider A C B as the limit of a sum of in-
scribed trapeziums P M M 7 P 7 , whose sides P P 7 lying
upon A B tend all in any way whatever towards zero.

As to the area D A C, we will draw at M and M 7
the two tangents M T, M' T 7 , from which will result
AT-=AP, AT 7 = AP 7 , TT 7 P P 7 . If, through
the point of meeting R of these tangents, we draw a
parallel to A B, we shall have the diameter of the
chords M M 7 , which will pass in consequence through
the middle of M M 7 , so that the area of the triangle
T E T 7 will be half of that of the trapezium P P 7 M M 7 .
Now it is easy to see that the area of D A C is the
limit of the sum of the triangles T 7 T E. In fact, this
area is exactly the sum of the areas comprised be-
tween each of the arcs M M 7 , the tangent M 7 T 7 , the
base of T 7 T and the tangent T M terminating in. M.
But each of these areas differs from the corresponding
triangle T 7 T E by a quantity infinitely small in com-
parison with it, when P P 7 tends towards zero ; for this
difference is less than the rectilinear triangle M E M 7
whose ratio to the triangle T 7 E T is that of the rect-
angles of the sides which include their angles at R,
which are supplementary ; a ratio which is evidently
infinitely small. Then the area D A C is the limit of
the sum of the triangles T 7 E T.

This being established, the two areas A C B, D A C
being limits of sums of infinitely small quantities
which are in the ratio of 2:1, will be themselves in
this ratio. Then A C B is two-thirds of the triangle
DBG, or of the parallelogram constructed upon A B
and B C, which leads us back to the result obtained
before..

CHAPTER V.

THE METHOD OF DESCARTES, OR ANALYTICAL
GEOMETRY.

DESCARTES is the great connecting link between
the ancient and the modern geometry. For two thou-
sand years, or a little less, the science of geometry had
remained nearly stationary when this extraordinary
man appeared to give it a new and prodigious im-
pulse. During that long and dreary period not one
original mind dared to assert its own existence. " It
is not surprising," says the Marquis de L'HSpital,
" that the ancients did not go farther ; but we know
not how to be sufficiently astonished that the great
men without doubt as great men as the ancients
should so long have stopped there, and that, by an
admiration almost superstitious for their works, they
should have been content to read them and to com-
ment upon them without allowing themselves any
other use of their lights than such as was necessary
to follow them, without daring to commit the crime
of sometimes thinking for themselves, and of carrying
their mind beyond what the ancients had discovered.
In this manner many worked, wrote, and books mul-
tiplied, but yet nothing advanced ; all the productions
of many centuries only sufficed to fill the world with
respectable commentaries and repeated translations of
originals often sufficiently contemptible." Thus, there

93

9 4 THE PHILOSOPHY OF MA THEMA TICS.

Was, in tlie mathematical world, no little activity ; but
it moved on hinges, not on wheels. It repeated, for
tlie most part, the same everlasting gyrations, but

" Such was the state of mathematics," continues the '
Marquis, "and above all philosophy, up to the time
of J Descartes. That great man, impelled by his
genius, and by the superiority which he felt in him-
self, quitted the ancients to follow the same reason
which the ancients had followed ; and that happy
boldness in him, though treated as a revolt, was
crowned with an infinity of new and useful views con-
cerning Physics and Geometry."

The Marquis knew, of course, that there were some
exceptions to the above general statement. The time
was sufficiently gloomy, it must be conceded, both
with respect to mathematics and philosophy; but it
was, nevertheless, relieved by the auspicious dawn
that ushered in the brilliant era of Descartes. Alge-
bra had been created, and Vieta, himself a man of
great original genius, had effected that happy alliance
between algebra and geometry which has been the
prolific source of so many important results. But
tliis detracts nothing from the glory of Descartes. For
it is still true of him, as de L'HSpital says, that " he
commenced where the ancients had finished, and began
by a solution of the problem at which Pappus said
they had all been arrested. Nor is this all. It is
merely the first step in his great career. He not only
solved the problem which had, according to Pappus,
proved too much for all the ancients, but he also in-
vented a method which constitutes the foundation of
t.he modern analysis, and which renders the most diffi*

TITJS PHILOSOPHY OF MATHEMATICS. 95

cult questions considered by tlie ancients quite too
easy and simple to tax even the powers of the merest
tyro of the pi^esent day. The method which he dis-
covered for tangents, the one great and all-compre-
hending question of the modern analysis, appeared to
him so beautiful that he did not hesitate to say, " That
that problem was the most useful and the most general
not only that he knew, but even that he ever desired to
know in geometry." *

But although Descartes, like every true king of
thinkers, extended the boundaries of science, he could
not set limits to them. Hence, it was only a little
while after the publication of his method for tangents,
that Fermat invented one which Descartes himself ad-
mitted to be more simple and felicitous than his own.f
It was the invention or discovery of his method of
tangents which led Lagrange, in opposition to the
common opinion, to regard Fermat as the first author
of the differential calculus. But the method of Barrow
was more direct and simple, if not more accurate, than
that of Fermat. He assumed that a curve is made up
of an infinite number of infinitely small right lines,
or, in other words, to be a polygon, the prolongation
of whose infinitely small side is the tangent to the
curve at the point of contact. On this supposition the
" differential triangle" formed by the infinitely small
side of the polygon, the difference between the two
ordinates to the extremities of that side, and the differ-
ence between the two corresponding abcissas, is evi-
dently similar to the triangle formed by the tangent,
the ordinate, and the subtangent to the point of con-
tact. Hence the subtangent is found simply by means

* Geometric, Liv. 2. t Lettre 71, Tom. 8.

96 THE PHILOSOPHY OF MATHEMATICS.

of these two similar triangles, a method which dis-
penses with the calculations demanded by the method
of Fermat, as well as by that of Descartes.

Barrow did not stop, however, at his " differential
triangle;" he invented a species of calculus for his
method. But it was necessary for him, as well as for
Descartes , to cause fractions and all radical signs to
disappear in order to apply or use his calculus. This
was, says the Marquis de L'HSpital, " the defect of
that calculus which has brought in that of the celebrated
M. Leibnitz, and that learned geometer has commenced
where M. Barrow and the others had terminated. His
calculus has led into regions hitherto unknown, and
made those discoveries which are the astonishment of
the most skillful mathematicians of Europe. The
Messrs. Bernouilli (and the Marquis might have added
himself) were the first to perceive the beauty of that
calculus ; they have carried it to a point which has
put it in a condition to surmount difficulties which no
one had ever previously dared to attempt.

" The extent of that calculus is immense ; it applies
to mechanical curves as well as to geometrical ; radical
signs are indifferent to it, and even frequently con-
venient; it extends to as many indeterminates as one
pleases ; the comparison of infinitely small quantities
of all kinds is equally easy. And hence arises an in-
finity of surprising discoveries with respect to tangents,
whether curvilinear or rectilinear ones, to questions
of maxima and minima, to points of inflexion and of
rebrousement of curves, to develop^es, to caustics by
reflexion and by refraction," etc.*

Thus, by his method of tangents, Descartes opened

* Preface to Analyse des Infinites Petites.

THE PHIL OSOPH Y OF MA THEM A TICS. 97

the direct route to the differential calculus. Nor is
this all. For, by the creation of his co-ordinate
geometry, he enabled Fermat, Barrow, Newton, and
Leibnitz to travel that route with success. A more
happy or a more fruitful conception had never, up to
that time, emanated from the mind of man, than Des-
cartes' application of indeterminate analysis to the
method of co-ordinate geometry a method which was
due exclusively to his own genius.

We shall, then, proceed to give, as far as possible,
an accurate and complete idea of Aanalytical Geome-
try the wonderful method created by Descartes.
This branch of mathematics has one thing in common
with the application of algebra to geometry namely,
the use of algebraic symbols and processes in the treat-
ing of geometrical questions. Hence, if we would
obtain clear views respecting its first principles or its
philosophy, we must, in the first place, form a just
idea of the precise relation which these symbols sus-
tain to geometrical magnitudes. We proceed, then,
to illustrate and define this relation.

THE KELATION OF ALGEBRAIC SYMBOLS TO GEOMETRI-
CAL MAGNITUDES.

On this subject geometers have advanced at least
three distinct opinions. The first is, that in order to
represent the length of a line (to begin with the most
simple case) by a letter, we must apply to it some
assumed unit of lineal measure, as a foot or a yard,
and see the number of times it contains this unit.
Then this number may, as in ordinary algebra, be re-
presented by a letter. According to this view, the
number represents the line and the letter the number.

9 E

98 THE PHIL OSOPHY OF MA THEM A TICS.

Such process of comparison, it is supposed, must either
be executed or conceived in order to establish the
possibility of expressing geometrical magnitudes by
the characters of algebra.

The second opinion is, that " geometrical magni-
tudes may be represented algebraically in two ways :
first, the magnitudes may be directly represented by

letters, as the line A B, given

j^ a -g absolutely, may be represented

by the symbol a; secondly, in-
stead of representing the magnitudes directly, the alge-
braic symbols may represent the number of times that
a given or assumed unit of measure is contained in the
magnitudes; as for the line A B, a may represent the
number of times that a known unit is contained in it."
In this case, as it is said, " the algebraic symbol repre-
sents an abstract number," which, in its turn, is sup-
posed to represent the line.

The third opinion is, that the letter represents not
the number of units contained in the line, but the
length of the line itself. Thus, we are told, "the
numerical measure of the line may, when known, be
substituted at pleasure for the letter which stands for
the line ; but it must always be remembered that what the
letter denotes is not the number which measures the length,
but the length itself. Thus, if AB (denoted by a) is
A a B two inches long, and an inch is the
unit of length, we shall have a = 2; but if half an
inch is the unit, a = 4. Here a has two different
numerical values, while that which a really represents,
the actual length of the line, is in both cases the same."

Now, it there be no real conflict of views in such
diverse teaching, there must certainly be some want

THE PHILOSOPHY OF MATHEMATICS. 99

of precision and clearness in the nse of language. If
the student should confine his attention exclusively to
any one of these opinions, he might consider the authors
who teach it as quite clear and satisfactory ; but if he
should extend his researches into other writers on the
same subject, he might, perhaps, begin to find that he
had something to unlearn as well as something further
to learn. He might be made to believe, as thousands
have believed, that algebraic symbols can only repre-
sent numbers and that, therefore, the only way to
bring geometrical magnitudes within the domain of
algebraic analysis is to reduce them to numbers by
comparing them with their respective units of measure.
But, then, if he should happen to see in the work of
some celebrated author the still more obvious position
laid down that algebraic symbols may be taken to re-
present magnitudes directly, as well as numbers, it is
highly probable he would be disturbed in his former
belief. It is likely that he would vacillate between
his old conviction and the new idea, and be perplexed.
Nor would he be delivered from this unpleasant
dilemma on being assured that in Analytical Geometry
symbols never denote numbers, but always the un-
divided magnitudes themselves. Suppose, then, that
each of these opinions contains the truth, it is evident JJ

that it cannot contain the whole truth, and nothing jj\

but the truth, clearly and adequately expressed. On tjj

the contrary, the rays of truth they contain are so irn-
perfectly adjusted that, in crossing each other, they ^

produce darkness, perplexity, and confusion in the
mind of the student. It is necessary, if possible, so to
eliminate and readjust the truths exhibited in these
opinions as to avoid all such interference, and all

100 THE PHILOSOPHY OF MATHEMATICS.

such darkening of the very first principles of the
science.

When It is said that a line is measured by a num
ber, it is evident that an abstract number, such as 2
or 4, cannot be intended. Such numbers represent or
measure, not the length of a line, but only the ratio
of one line to another. If a line two inches long, for
example, be compared with an inch as the unit of
measure, the abstract number 2 will be the ratio of
this unit to the line, and not "the measure of the
line" or its " numerical value." Supposing the line
of two inches to be denoted by a, then we shall have,

~ i , a o 2 inches rt T ,.,

not a = 2, but = 2, or : = 2. In like

1 inch 1 inch

manner, if half an inch be the unit of measure, we

shall have, not a = 4, but ; = 4. In the first

|- inch

case, a = 2 inches, and in the second, a = 4 half
inches, so that, in both cases, we shall have the same
value for the same thing, since 2 inches and 4 half
inches are not " different numerical values."

It should always be remembered that it is only a
denominate number which truly " measures the length
of a line," and that abstract numbers merely represent
the ratios of lines. Thus, for example, if a line one
yard in length be compared with a foot as the unit of
measure, the abstract number 3 will be the ratio of
this unit to the line, and if an inch be the unit, then
36 will express this ratio, or the number of times the
unit is contained in the line measured. In neither
case, however, is 3 or 36 " the numerical measure of
the line" or the yard. This is measured, not by the
abstract number 3 or 36, but by the denominate nuoi-

THE PHIL OSOPHY OF MA THEM A TICS. 101

ber 3 feet or 36 inches. Thus, for one and the same
thing we have not " two different numerical values,"
but only one and the same value.

The third opinion, then, appears to have arisen from
the supposition that an abstract number, such as 2 or
4, can measure the length of a line, whereas this is
always measured by a denominate number. And
this being the case, it makes no difference whether the
letter be taken to denote the number which measures
the length of the line or the length itself. For whether
a be taken to represent the length itself, as one yard,
or the number which measures it, as 3 feet or 36
inches, it will stand for precisely the same magnitude.
In one case it will stand for the whole, and in the
other for the undivided sum of the parts ! Hence, we
reject the third opinion as founded on a wrong notion
respecting the nature of the number which serves to
measure the length of a line, and as being a distinc-
tion without a difference.

The second opinion is involved in a similar fallacy.
For it proceeds 011 the assumption that a linear mag-
nitude may be " represented" by an " abstract num-
ber ;" whereas this can only represent the ratio of one
line to another. Indeed, an abstract number bears no
relation to the length of a line, and can be brought
into relation with it only by means of the unit of
measure, either expressed or understood. If, for ex-
ample, any one were asked how long a particular line
is, or how it should be represented, and he were to
answer it is three long, or should be represented by 3 ;
he would talk unintelligible nonsense. But if he were
to reply it is 3 feet, or 3 miles in' length, and should
be represented accordingly, he would be understood.

102 THE PHIL QSOPHY OF MA THEMA TICS.

Hen^e, as abstract numbers do not represent lines, j
the letters which stand for such numbers do not repr<
sent them.

There is, then, only one way of representing a lir
by a letter, and that is by taking the letter to denoi
the line itself, or, what amounts to the same thing, 1
j denote the denominate number which measures tfc

line. This may be done, no doubt, if we please ; bi
is this way of representing lines admissible in Analyt:
cal Geometry ? It is certainly embarrassed with diflE
culties which the authors of the second opinion do nc
seem to have contemplated. If, for example, one lin
6 feet long is denoted by a, and another 3 feet long i
denoted by 6, it is easy to see that a + b = 9 feei

a b = 3 feet, and - = 2 ; but what shall we say o:
b

the product abf Or, in other words, of 6 feet by I
feet ? Almost any student, after having gone througl
with elementary works on pure Geometry or Analyti
square feet. Yet there is no rule in mathematics fo;
the multiplication of one denominate number by an-
other. The product of feet by feet is just as unintel-
ligible as the product of cents by cents; an absurc
operation with which some people perplex themselves
a great deal to no purpose. The multiplier musi
always be an abstract number. The present writei
has often been asked by letter, " What is the prodticl
of 25 cents by 25 cents ?" an inquiry as unintelligi-
ble as if it were what is the product of 25 cents by 25
apples, or the product of 25 apples by 25 sheep ? Such
an absurdity would be less frequently committed if
elementary works on arithmetic had thrown sufficient

THE PHILOSOPHY OF MATHEMATICS. 103

light on the nature of multiplication. But, however
obvious this error, it is precisely similar to that com-
mitted by geometers when they seek the product of
any one concrete magnitude, such as a line or a sur-
face, by another.*

If we would avoid all such errors and difficulties,
we must lay aside the notion that magnitudes are
represented either directly or indirectly by letters.
There is no such representation in the case. Indeed,
the rationale or analysis of the whole process of sym-
bolical reasoning lies, as we shall see, beneath this
notion of representation, and is something deeper than
is usually supposed. Certainly, the abstract number
obtained by comparing a line with an assumed unit
of length cannot properly be called " the numerical
value of the line," as it is by so many authors. For,
if it could, then one and the same line might have an
infinity of numerical values, since the abstract number
would vary with every change in the assumed unit of
measure. But surely, if an infinity of numerical
values for one and the same thing be not an absurdity
in mathematics, it is far too vague and indefinite a
notion to find a place in the domain of the most precise
and exact of all the sciences.

The precise truth is, that in establishing the theo-
rems of geometry we do not aim to determine the
length of lines, but the relations they sustain to
each other, as well as to surfaces and solids. In
trigonometry, for example, we are concerned, not
with the absolute value of the magnitudes considered,
but with the relations existing between them ; so that
when a sufficient number of these magnitudes are

* Note A.

1 04 THE PHIL OSOPHY OF MA THEM A TICS.

known, or may be measured, the others may be de-
duced from them by means of the relations they bear
to each other. The same is true of all other parts of
geometry. Hence, what we need is not a representa-
tion of the magnitudes themselves, but of the relations
existing between them. We start from certain given
relations, we pass on to other relations by means of
reasoning; and having found those which are most
convenient for our purpose, the theorems of geometry
are established and ready for use. The precise man-
ner in which this is accomplished we shall now pro-
ceed to explain.

In all our reasoning we deal with abstract numbers
alone, or the symbols of abstract numbers. Tjiese, it
is true, do not, strictly speaking, represent lines or

g;i other magnitudes, but the relations between these may,

and always should, represent the relations between
the magnitudes under consideration. This representa-
tion of relations, and not magnitudes, is all that is
necessary in symbolical reasoning, and if this be borne
in mind the rationale of the whole process may be

81 H made as clear as noonday.

The unit of linear measure is altogether arbitrary.

|/, r It may be an inch, a foot, a yard, a mile, or a thou-

sand miles. But this unit once chosen, the square de-
scribed on it should be the unit of measure for sur-
faces, and the cube described on it the unit of measure
for solids. Each magnitude, whether a line, surface,
or solid, might be compared directly with its own
unit of measure, and the abstract number thence re-
sulting might be represented by a single letter. But
this course would be attended with much confusion
and perplexity. Hence it is far more convenient, and

THE PHILOSOPHY OF MATHEMATICS.

105

consequently far more common, to represent only the
abstract number obtained from a line by a single
letter.

Then will the product of two letters represent the
abstract number answering to a surface. Suppose, for
example, that the line A B contains 6 units, and the
line CDS units. Let a denote the abstract number

[B

C>

-iD

6, and b the abstract number 3, then ab = 18. Now
this product a b is not a surface, nor the representative
of a surface. It is merely the abstract number 18.
But this number is exactly the same as the number
of square units contained in the rectangle whose sides
are A B and CD, as may be seen, if necessary, by
constructing the rectangle. Hence the surface of the
rectangle is represented or measured by 18 squares
described on the unit of length. This relation is uni-
versal, and we may always pass from the abstract unit
thus obtained by the product of any two letters to the
measure of the corresponding rectangle, by simply
considering the abstract units as so many concrete or
denominate units. This is what is intended, or at
least should always be intended, when it is asserted
that the product of two lines represents a surface ; a
proposition which in its literal sense is wholly unin-
telligible.

In like manner the product of three letters, a b c, is
not a solid obtained by multiplying lines together,
which is an impossible operation. It is merely the
product of the three abstract numbers denoted by the

S# E*

106 THE PHILOSOPHY OF MATHEMATICS,

letters a, b, and c } and is consequently an abstract
number. But this number contains precisely as many
units as there are solid units in the parallelopipedon
whose three edges are the lines answering to the num-
bers denoted by a, b, and c ; and hence we may easily
pass from this abstract number to the measure of the
parallelopipedon. "We have merely to consider the
abstract number as so many concrete units of volume,
or cubes described on the linear unit. It is in this
sense, and in this sense alone, that the product of three
lines, as it is called, represents a solid. Bearing this
in mind, as we always should do, we may, for the sake
of brevity, continue to speak of one letter as repre-
senting a line, the product of two letters as represent-
ing a surface, and the product of three letters as repre-
senting a solid.

THE COMMONLY EECEIVED DEFINITION OF ANALYTICAL
GEOMETRY.

In most definitions this branch of mathematics is
exhibited as merely the application of algebra to
geometry. Thus, says M. De Fourcy in his treatise
on the subject, "Analytical Geometry, or in other
terms, the Application of Algebra to Geometry, is that
important branch of mathematics which teaches the
use of algebra in geometrical researches." This defi-
nition, like most others of the same science, can impart
to the beginner no adequate idea of the thing defined.
It fails in this respect, partly because the geometrical
method used in this branch of mathematics is different
from any with which his previous studies have made
him acquainted, and partly because algebra itself un-
dergoes an important modification in its application to

THE PHILOSOPHY OF MATHEMATICS. 107

this new geometrical method. These points must be
cleared up and the student furnished with new ideas
before he can form a correct notion of Analytical
Geometry,

According to the above definition, no new method,
no new principle is introduced by Analytical Geome-
try ; it is simply the use of algebra in geometrical in-
vestigations. Precisely the same idea underlies nearly
all definitions of this branch of mathematics. Thus,
in one of these definitions, we are told that " in the
application of algebra to geometry, usually called Ana-
lytic Geometry, the magnitudes of lines, angles, sur-
faces, and solids are expressed by means of the letters
of the alphabet, and each problem being put into
equations by the exercise of ingenuity, is solved by
the ordinary processes of algebra." In another it is
said that " Analytical Geometry" is that " branch of
mathematics in which the magnitudes considered are
represented by letters, and the properties and relations
of these magnitudes are made known by the applica-
tion of the various rules of algebra." Now these defi-
nitions, and others which might be produced, convey
no other idea of the science in question than that it is
the use of algebraic symbols and methods in geometri-
cal researches. They contain not the most distant
allusion to that new and profoundly conceived geo-
metrical method, nor to that peculiar modification of
algebra, by the combination of which Analytical Geo-
metry is constituted.

This beautiful science, it is universally conceded,
was created by Descartes. But if the above definition
be correct, then Vieta, and not Descartes, was the
creator of Analytical Geometry; for he made precisely

108 THE PHIL OSOPHY OF MA THEM A TICS.

such use of algebra in his geometrical researches. In-
deed, we could not better describe the method of Vieta
than by adopting some one of the current definitions
of Analytical Geometry. It is most accurately ex-
hibited in the following -words of a recent author :
" There are three kinds of geometrical magnitudes, viz.,
lines, surfaces, and solids. In geometry the properties
of these magnitudes are established by *a course of
reasoning in which the magnitudes themselves are
constantly presented to the mind. Instead, however,
of reasoning directly upon the magnitudes, we may,
if we please, represent them by algebraic symbols.
Having done this, we may operate on these symbols
by the known methods of algebra, and all the results
which are obtained will be as true for the geometrical
quantities as for the algebraic symbols by which they
are represented. This method of treating the subject
is called Analytical Geometry." Now every word of
this description is accurately and fully realized in the
labors of Vieta. Hence, if, it had been given as a
definition of the " Application of Algebra to Geome-
try," as left by him, it would have been free from
objection. But it cannot be accepted as a definition
of Analytical Geometry. For such method, however
valuable in itself, is not Analytical Geometry, nor
even one of its characteristic properties. It is not that
grand era of light by which the modern geometry is
separated from the ancient. For that era, or the
creation of Analytical Geometry, is, according to the
very author of the above definition himself, due to
Descartes. Yet his definition of Analytical Geometry,
like most others, includes the method of Vieta, and ex-
cludes the method of its acknowledged author, Descartes*

THE PHILOSOPHY OF MATHEMATICS. 109

Even before the time of Vieta, Kegiomontanus,
Tartaglia, and Bombelli solved problems in geometry
by means of algebra. But in each case they used
numbers to express the known lines and letters to
represent the unknown ones. Hence their method
was confined within narrow limits when compared
with the method of Vieta. He was the first who em-
ployed letters to represent known as well as unknown
quantities ; a change, says Montucla, " to which alge-
bra is indebted for a great part of its progress." It
enabled Vieta and his successors to make great ad-
vances in Geometry as well as in algebra. But it did
not enable him to reveal or to foresee the new method
which was destined to give so mighty an impulse to
the human mind, and produce so wonderful a revolu-
tion in the entire science of mathematics, whether
pure or mixed. This was reserved for the tra.nscend-
ent genius of Descartes.

It seems to me that a definition of Analytical Geo-
metry should include the method of Descartes (its
acknowledged author) and exclude that of Vieta. It
is certain that if we should adopt the above definition,
we should be compelled to include Trigonometry, as
well as " the solution of determinate problems," in
Analytical Geometry. Indeed, M. De Fourcy, after
having defined Analytical Geometry as above stated,
expressly acids, " Under this point of view it ought to
comprehend trigonometry, which forms the first part
of this treatise." In like manner, Biot, Bourdon,
Lacroix, ar-d other French writers, embrace trigo-
nemetry, as well as " the solution of determinate pro-
blems," in their works on Analytical Geometry. This
seems to be demanded by logical consistency, or a
10

110 TEE PHILOSOPHY OF MATHEMATICS.

strict adherence to fundamental conceptions, since
analytical trigonometry, no less than determinate
problems of geometry, is clearly included in iheir
definitions. We shall exclude both, because neither
conies within the definition which we intend to adopt.
Trigonometry and determinate problems of geometry
were, indeed, both treated by means of algebra long
before Analytical Geometry, properly so called, had
an existence or had been conceived by its great author.

Those American writers, however, who have adopted
the above definition, exclude trigonometry, though
not the solution of determinate problems, from their
works on Analytical Geometry. Hence, in excluding
both, the additional omission will be very slight, inas-
much as the solutions of determinate problems in the
works referred to constitute only a few pages. These
few pages, too, being little more than a mere exten-
sion of ordinary algebra, should, it seems to me, form
a sequel to that branch of mathematics, rather than a
heterogeneous prefix to Analytical Geometry. It is
certain that by such a disposition of parts we should
restore an entire unity and harmony of conception to
the beautiful method of Descartes, by which a new
face has been put on the whole science of mathematics

This method and that of Vieta are, as M. Biot says,
" totally separated in their object." Hence he was
right in determining, as he did, " to fix precisely and
cause to be comprehended this division of the Appli-
cation of Algebra to Geometry into two distinct
branches," or methods of investigation. Since these
two branches, then, are so " totally separated in their
objects," as well as in their methods, we shall separate
them in our definitions. We certainly shall not, in

THE PHIL OSOPH Y OF MA THEM A TICS. Ill

our definition, cause the method of Vieta to cover the
whole ground of Analytical Geometry, to the entire
exclusion of the method o Descartes. The method
of Vieta is, indeed, nowhere regarded as constituting
Analytical Geometry, except in the usual definitions
of this branch of mathematics. The authors of these
definitions themselves entertain no such opinion. On
the contrary, they unanimously regard the method of
Descartes as constituting Analytical Geometry, though
this view is expressed elsewhere than in their defini-
tions. Thus, after having disposed of " determinate
problems," and come to those investigations which
belong to the Cartesian method, one of these authors
adds in a parenthesis, " and such investigations consti-
tute the science of Analytical Geometry" If so, then
surely the nature of such investigations should not
have been excluded, as it has been, from his definition
of this branch of mathematics. In like manner, an-
other of these authors, after having discussed the sub-
ject of determinate problems, enters on the method
of Descartes with the declaration that this philosopher
by his great discovery " really created the science of
covery excluded from -his definition of the science?
In spite of their definitions, we have, indeed, the
authority of these writers themselves that Analyti-
cal Geometry, properly so called, is constituted by
something different and higher and better than
the algebraic solutions of determinate problems of
geometry.

112 THE PHILOSOPHY OF MATHEMATICS.
THE OBJECT OF ANALYTICAL GEOMETRY.

In order to unfold in as clear and precise a manner
as possible the great fundamental conceptions of Ana-
lytical Geometry, we shall consider first, the object of
the science; and, secondly, the means by which this
object is attained.

" Geometrical magnitudes, viz., lines, surfaces, and
solids," are, it is frequently said, the objects of Ana-
lytical Geometry. But this statement can hardly be
accepted as true. For lines, surfaces, and solids, con-
sidered as magnitudes, are not, properly speaking, the
objects of this science at all. Lines and surfaces, it is
true, as well as points, are considered in Analytical
Geometry; but then they are discussed with reference
to their form and position, and not to their magnitude.
Questions of form and position are those with which
Analytical Geometry, as such, is chiefly and pre-emi-
nently conversant. So long, indeed, as our attention
is confined to questions of magnitude, whether pertain-
ing to lines, surfaces, or solids, we are in the domain
of the old geometry. It is the peculiar province and
the distinctive glory of the new that it deals with the
higher and more beautiful questions of form.

In relation to the discoveries of Descartes in mixed
analysis, Montucla says, " That which holds the first
rank, and which is the foundation of all the others, is
the application to be made of algebra to the geometry
of curves. We say to the geometry of curves, because
we have seen that the application of algebra to ordi-
nary problems is much more ancient." If we would
obtain a correct idea of his method, then, we must lay
aside as unsuited to our purpose the division of geo-

THE PHILOSOPHY OF MATHEMATICS.

113

metrical magnitudes into lines, surfaces, and solids.
For however important this division or familiar to the
. mind of the beginner, it is not adapted to throw light
on the nature of Analytical Geometry. If we would
comprehend this, we must divide all our geometrical
ideas into three classes namely, into ideas of magni-
tude, position, and form. Of these the most easily
dealt with are ideas of magnitudes, because magni-
tudes, whether lines, surfaces, or solids, may be readily
represented by algebraic symbols.

Indeed, to find a " geometrical locus," or, in other
words, to determine the form of a line, was the unsolved
problem bequeathed by antiquity to Descartes, and
with the solution of which he bequeathed his great
method to posterity. Thus the new geometry had its
beginning in a question of form, and, from that day
to this, all its most brilliant triumphs and beautiful
discoveries have related to questions of form. These
high questions, it is true, his method brings down to
simple considerations of magnitude, or, more properly
speaking, the relations of linear magnitudes. The
objects it considers are not magnitudes; they are forms
and the properties of form. The magnitudes it uses
and represents by letters are only auxiliary quantities
introduced to aid the mind in its higher work on
forms. They are the scaffolding merely, not the edi-
fice. In what manner this edifice, this beautiful
theory of the ideal forms of space, has been reared by
Analytical Geometry, we shall now proceed to ex-
plain.

1 14 THE PHILOSOPHY OF MA THEMA TICS.

THE GEOMETRIC METHOD OF ANALYTICAL GEOMETRY.

Tke new geometry consists, as we have said, of a
geometric method and a modified form of algebra.
Both of these should, therefore, be embraced in its
definition. We begin with an explanation of its geo-
metric method.

"By co-ordinate geometry," says Mr. O'Brien, "we
mean that method or system invented by Descartes, in
which the position of points are determined and the
forms of lines and surfaces defined and classified by
means of what are called co-ordinates." This appears
to be a correct definition of the system of Descartes ;
at least in so far as its geometric method is concerned.
But that method, as we shall see, however important
as an integral portion of the system, is barren in itself,
and becomes fruitful only by a union with the analytic
method of the same system. Adopting, for the pre-
sent, the above definition as applicable to the geometric
method of Descartes, it remains for us to unfold and
illustrate its meaning.

It is easy to see that every question of form depends
on one of position, since the form of any line or surface
is constituted by the position of its various points. If,
then, the position of every point of a line (to begin
with the more simple case) be determined, it is clear
that the form of the line will be fixed. Hence the
first step in the system of Descartes, or in the modern
doctrine of form, is the method by which it determines
the position of a point in a plane.

TEE PHILOSOPHY OF MATHEMATICS.

115

THE METHOD OF DETERMINING THE POSITION OF A
POINT.

From time immemorial the position of a point on
the surface of the earth has been determined by its dis-
tance from two fixed lines namely, an assumed meri-
dian and the equator. These two distances, kncwn
as the longitude and latitude of the point or place, are
among the most natural and easy means by which its
position can be fixed. Yet this method, although so
natural, so simple, and so familiar in practice, lay
upon the very surface of things for many centuries
before its immense scientific value began to be appre-
hended. Descartes, in the seventeenth century, was
the first philosopher by whom it was adopted into
geometry, generalized, and made to impart incalcula-
ble new resources to the science.

In order to fix the position of a point on a plane,
we trace in the plane, in conformity with the method
of Descartes, two right lines X X 7 and Y Y', which

Y

p/

M

P

M

X

X

P"

p///

Y'

lib THE PHILOSOPHY OF MA THEM A TICS.

make a given angle (usually a right angle) with each
other, and we draw through the point P, whose posi-
tion is to be determined, parallel to these lines, the
two right lines P M and P N, cutting them in the
points M and N. Now it is evident that the point P
will be determined when we know the points M and
N, for we can draw through these points M P parallel
to O Y, and N P parallel to O X 7 , and these parallels
will intersect in the point P. But the point M is
determined when we know the distance O M, and
the point N when we know the distance O N. Hence
the point P is determined or fixed by means of the
distance O M or its equal N P, and the distance O N"
or its equal M P. That is, on the supposition that P
lies in the angle Y O X'; otherwise its position could
not be fixed by these magnitudes alone.

For were these magnitudes O M and O N given,
this would not serve to determine the point to which
they answer, since there are four points P, P', P",
and P ;// , all of which answer to precisely the same
magnitudes or distances. To avoid the confusion
which must have resulted from such uncertainty of
position, Descartes adopted a very simple and efficient
artifice. Instead of employing a different set of letters
for each of the angles in which the required point
might be found, he effected his object and cleared
away every obscurity by the simple use of the signs
+ and . That is, he represented the magnitudes
O M and O N by the same letters, and they were
made to determine the point P, P', P", or P //; , ac-
cording to the signs attached to these letters.

Thus, for example, O M is represented by a and
O N by 6; when a is plus, it is laid off in the direction

THE PHIL OSOPH Y OF MA THEM A TICS. 117

from O toward X', and when it is minus it is> mea-
sured in the opposite direction, or from O toward X.
In like manner, when b is plus, it is counted from O
toward Y, or above the line X X', and when it is
minus, in the opposite direction, or from O toward
Y'. "What is thus said of the distances O M and
O N, or their representatives a and b, is applicable to
all similar distances. Thus, by the use of two letters
and two signs, the position of any point in any one of
the four infinite quarters of the plane is indicated
without the least uncertainty or confusion.

The distance O M, or its equal N P, is called the
abscissa, and the distance O N or its equal M P, is
called the ordinate of the point P to which they
answer. These distances wfren taken conjointly are
denominated the co-ordinates of the point. Instead
of saying the point whose abscissa is denoted by a and
ordinate by 6, we simply say, the point (a, 6). The
line X X', on which the abscissas are laid off, is called
the axis of abscissas, and the line Y Y ; the axis of
ordinates. Both together are denominated the co-or-
dinate axes. The point O, in which the co-ordinate
axes intersect, is known as the origin of co-ordinates ;
or more briefly, as the origin.

As the object is to determine, not the absolute but
only the relative position of points, so the co-ordinate
axes, or lines of reference, may be assumed at pleasure.
We may place the origin, or incline the axes, so as to
meet the exigencies of any particular case, or to an-
swer any special purpose. In general, however, it is
more convenient to refer points to axes which make
right angles with each other; in which case the sys-
tem of co-ordinates is rectangular. If they are in-

118

THE PHILOSOPHY OF MATHEMATICS.

clined to each other, then they form an oblique system
of co-ordinates. The former, or the rectangular sys-
tem of co-ordinates, should always be understood^
unless it be otherwise expressed.

In the foregoing remarks we have spoken of the
point P which is supposed to remain fixed, and whose
co-ordinates a and b are therefore constant. But sup-
pose this point, or any other, to move on the plane of
the co-ordinate axes, it is evident that its co-ordinates
will no longer remain constant or unchanged. On
the contrary, as the point moves, either one or both of
its co-ordinates must undergo corresponding changes
of value. These variable co-ordinates, answering to
all the positions of the movable point, or to all the
points of the line it describes, are in general denoted
by the letters x and y, and the line X X 7 on which
the abscissas are measured is sometimes called the axis
of x, and the line Y Y' ; on which ordinates are laid
off, the axis of y.

As x and y may assume all possible values, whether
positive or negative, so they may represent the co-or-

-X
-X

TX

X'

-Y

THE PHILOSOPHY OF MA THEMA TICS. 119

dinates of any point in the plane of the axes. The
angle to which the point belongs will depend, as we
have seen, on the algebraic signs of its co-ordinates x
and y. By means of the preceding diagram we may
perceive at a glance the angle to which any point be-
longs when the signs of its co-ordinates are known.
Thus we always have

x positive and y positive for the angle Y O X',
x negative and y positive for the angle Y O X,
x negative and y negative for the angle Y'O X,
x positive and y negative for the angle Y'O X 7 .
In Analytical Geometry, then, the letters x and y re-
present not unknown, determinate values or magni-
tudes as in algebra, but variable quantities. It is
this use of variable co-ordinates and symbols of inde-
termination to represent them which constitutes the
very essence of the Cartesian system of geometry a
system of whose analytic portion, however, we have
as yet caught only an exceedingly feeble glimpse. It
justly claims, in this place, a somewhat fuller expo-
sition, especially since its value is so completely over-
looked in the definitions of most writers on the sub-
ject. Even the definition of Mr. O'Brien contains,
as we have seen, only the geometrical method of Des-
cartes, and not the most distant allusion to its ana-
lytic method. Indeed, in his preface, this author
asserts that the subject of which he treats "is usually
styled Analytical Geometry, but its real nature seems
to be the better expressed by the title Co-ordinate
Geometry, since it consists entirely in the application of
the method of co-ordinates to the solution of geometrical
problems" Yet this method of co-ordinates, if sepa-
rated from the method of indeterminate analysis, can

120 n . ori[A

*

the

,

other . D ,l * Wlloll y <lependent

ihe

described by a poin . mn '. . , Ilne or curve thus

Joes not come XE ST? ^ 1&W r wder
metry. ^ Ae doma ^ of Analytical Geo-

according to S0 me d 1 "? " random ' are
> such cas es there7itl f rinVariab]eOTd cr.
ordinate of each pi O f .^ ^ the absci -a and
or unchanging mi ftu a] ^^ a c relation
one point of such line toTn T' ^ PaSSm ^ from
ordinate must chan^ bu t 7 f^ * he absdssa ^
^Y remain the sante' In I ^ bet <he m
t unif om re]ation " is ^ a ^*7rf instances

THE PHILOSOPHY OF MATHEMATICS.

121

ordinate of each point of a line is called the equation of
the line. The line, in its turn, is called the loeus of the
equation; but it is still more frequently called the locus
of the point by which it is described.

The equation of a line once formed will enable us,
by suitable operations upon it, to detect all the circum-
stances and to discover all the properties of the line
or locus to which it belongs. A few simple illustra-
tions will serve to put this great fundamental truth in
a clearer light. Suppose, then, that there is a right line,
such as B B', which divides the angle Y O X' into

two equal parts, then it is clear that its ordinate will
always be equal to its abscissa. The abscissa may
assume all possible values from zero to plus infinity;
yet through all its changes it will remain constantly
equal to the corresponding ordinate. This invariable
relation between the two variable co-ordinates is per-
fectly expressed or represented by the equation

* A '

which is therefore the equation of the line B B'. In

11 F

122

THE PHILOSOPHY OF MATHEMATICS.

like manner, the equation of any line institutes precisely
the same relation between the symbols x and y as that
which exists between the co-ordinates symbolized or re-
presented by them.

If In this equation we give positive values to ir, we
shall find positive values for y, and these values will
determine points of the right line in the angle Y (.) X'.
If we give negative values to x, we shall then have;
negative values for y, and these values will determine
points of the line in the angle Y 7 O X. But, leaving-
this most simple of all cases, let us pass on to still
more interesting examples.

Let it be required, then, to find the equation of the
circumference of a given circle, and show how thin
equation may be made to demonstrate some of the
properties of that curve. We place the origin of co-
ordinates at the centre of the given circle A C B D,

and denote its known radius by the letter r Then
for any point of the circumference, as P, the square of
the ordmate, plus the square of the abscissa, is equal
to the square of the radius, since the sum of the squares
on the sides of a right-angled triangle is equal to th*

THE PHILOSOPHY OF MATHEMATICS.

123

square of its hypothenuse. This relation is expressed
by the equation

f -f #2 = r\

and as this is true for every point of the circumference,
so this is the equation of the curve. Now from this
equation all the properties of the circumference of the
circle may be evolved by suitable transformations.

As our present purpose is merely illustrative, we
shall, in this place, evolve only a single property.
Then the equation gives

f ( r + x) (r x);
bu.tr + x= AM; r # = MB; and?/ =

; hence

that is, the perpendicular let fall from a point of the
circumference of a circle to its diameter is a mean pro-
portional between tJie segments into which it divides the
diameter a well-known property of the circle.

Or, if we choose, we may set out from this property,
and, putting it into an equation, deduce therefrom the
ordinary definition of the circumference of a circle.
Thus, let it be required to prove that the line whose
ordinate is always a mean proportional between the
segments into which it divides a given distance on the
axis of X, is the circumference of a circle ; or, in other
words, is everywhere equidistant from a certain point
in the plane of the axes. We suppose A B, the given
distance, = 2 r, and we place the origin of co-ordinates
O at its middle point, so that A O = O B = r. Then,

*

i

124

TEE PHILOSOPHY OF MATHEMATICS.
\Y

^y. = , + , ; and MB==r _,

hence, by substitution, f = ( r + x] (r __ x] = ^ _

bnt
therefore

THE PHILOSOPHY OF MATHEMATICS.

125

into an equation, deduce its other properties therefrom
by suitable transformations of its equation.

We shall add one more illustration. Suppose the
question, for example, be to determine the principal
circumstances of position and form of the line, the
square of whose ordinate varies as the corresponding
abscissa ; or, in other words, the square of whose ordi-
nate is always equal to the rectangle of the abscissa
into some constant line, as 2 p. The equation of this
line is simply the analytic statement of its definition^
and is

or

Now if, in the first place, we wish to find the point in
which the line cuts the axis of X, we must determine
the co-ordinates of that point, since every point is made
known by its co-ordinates. But as the required point
lies on the axes of x, it is evident that its ordinate is
o } and this, substituted for y in the equation, gives

o =

or x = o

for the corresponding value of the abscissa. Hence
the line cuts the axis of X at the origin of co-ordi-
nates, since that is the only point whose co-ordinates
are both o.

Again, we put the equation in this form :

from which we see that if x be minus, then y will be
imaginary; or, in other words, there will be no corres-
n*

l?l

lli >.

J%l

126 ZHE PSILOSOfSY

THE PHILOSOPHY OF MATEEMA TICS.

127

ous method, and would require immense calculations
to determine the curve with any degree of accuracy.
It is, indeed, the method of co-ordinates, and serves
to illustrate the imperfections of that method when
unaided by the higher powers of the analytic portion
of the Cartesian system.

By calling this to our aid, we may easily discover a
property of the line in question which \vill enable us
to describe it by a continuous motion without the
necessity of such tedious or operose calculations. Thus,

if we lay off O F = -, and O L = % and through the

point L erect an indefinite perpendicular D D f to the
axis X X', then each and every point of the curve in
question will enjoy this remarkable property namely,
it will be at an equal distance from the point F and the
line D D'.

That is, if from any point, as P, of the curve, a
right line P F be drawn to F, and another perpen-
dicular to D D', then P F = P D. Tor

P F 2 = P M 2 + F M 2 = y 2 -+ (a? - ^)

2

or P F 2 = f + x* p x + ;

4

but y 2 = 2jp&';

hence, P F 2 = a? + p x + & = (x +- P -)*

or

128 TSE PSIL OSOP3Y OF MA TJSEMA TICS.

but

tlierefore ;

as enunciated. This remarkable property enables us
to describe the curve in question by a continued
motion.

The above very simple illustrations, or instances of
discovery, are but mere scintillations of that great
analytic method which seems as inexhaustible as the
sun, and which has already poured floods of light on
every department of the exact sciences. The geomet-
rical method itself, however important, is chiefly valu-
able as a basis for this analytic method. The two
methods are, however, indispensable to each other,
and it was by the happy union of both that Analytical
Geometry was created. It was, moreover, by the
wonderful fecundity ancl power of this combination
that the way was opened for the discovery of the In-
finitesimal Calculus, and for the solution of the grand
problems of the material universe, as well as for the
renovation ancl reconstruction of all the physical
sciences.

The great beauty of this method consists in the
generality of its solutions a generality which is capa-
ble of being rendered far greater than is usual in works
on Analytical Geometry. To illustrate this point:
Ld ^ be required to find the equation of a curve suck
that Hie square of any ordinate shall be to tlie rectangle
of the distances between its foot and two fixed points on
the axis of x in a given ratio.

Let O ancl D be the two fixed points on the axis of

THE PHILOSOPHY OF MATHEMATICS.

129

x, and let the distance between them, O D, be denoted
by 2 a ; take the origin of co-ordinates at O, and sup-
pose the given ratio is 6 2 : a 2 , in which b represents

any line whatever. At the point A, the middle of the
line O D, erect the perpendicular A B = 6, and let P
be any point of the curve whose equation is required.
O M is the abscissa, and P M is the ordinate of that
point, and the two distances between its foot, M, and
the two fixed points O and D are M D and O M.
Hence, by the condition of the problem, we have

PM 2 :OMxMD::6 2 :a 2 ,

or y 2 : x (2 a x) : : 6 2 : a 2 ;

since P M, for any point of the curve, is the variable
ordinate y, and O M the variable abscissa x } and since
OD OM = 2a x.

Hence

is the equation of the curve required.

F*

130 THE PHILOSOPHY OF MATHEMATICS.

Now the whole folio of Apollonius, in which he dis-
cusses with such wonderful ability the conic sections, is
wrapped up and contained in this one equation. For,
by the discussion of this one equation, we may easily
ascertain the form and all the other properties of the
circle, of the ellipse, of the hyperbola, and of the para-
bola;* unfolding from one short analytic expression
the whole system of beautiful truths which caused
Apollonius to be regarded as the greatest of all the
geometers of the ancient world, except Archimedes.

The method of Descartes consisted in the happy use
of a system of auxiliary variables, such as x and y,
representing the variable co-ordinates of a series of
points. In addition to these variables, Newton and
Leibnitz employed another system composed of the
variable increments or decrements which x and y tmcler-
dergo in passing from one point to another ; or, in
other words, the variable differences of the variable
co-ordinates. Thus, the systems of these two illus-
trious geometers were both erected on the foundation
so wonderful a revolution into the whole science of
mathematics. It has been well said, then, that "Des-
cartes not only perfected the work of Vieta, but he
also invented methods at once simple and fruitful, in
order to bring the theory of curves within the grasp
of the algebraic analysis, and these methods are, in the
eyes of posterity, the most beautiful title to glory of
that celebrated philosopher." f

* Any one who is master of the fundamental idea of Descartes
may easily do this; it is done in my unpublished work on Analyti-
cal Geometry.

t See Note B.

TEE PHILOSOPHY OF MATHEMATICS. 131

THE METHOD OF INDETERMINATES.

Descartes approached the differential calculus in more
directions than one. " It seems to me," says Carnot,
" that Descartes, by his method of indeterminates, ap-
proached very near to the infinitesimal analysis ; or
rather, it seems to me, that the infinitesimal analysis
is no other than the happy application of the method
of indeterminates."

" The fundamental principle of the method of inde-
terminates, or of indeterminate co-efficients, consists in
this, that if we have an equation of the form

A + B a; + Cz 2 + J)x s + etc. = 0,

in which the co-efficients A, B, C, etc., are constant,
and x a small quantity, which can be supposed as small
as we please, it necessarily follows that each of these
co-efficients taken separately must be equal to zero;
that is to say, that we shall always have

A = 0, B = 0, C = 0, etc.,

whatever may be the number of the terms of the
equation.

" Indeed, since we can suppose x as small as we
sum of all the terms which has x for its factor; that is
to say, the sum of all the terms which follow the first.
Then that first term A differs as little as we please
from ; but A being a constant cannot differ as little
as we please from 0, since then it would be a variable,
then it can be only 0, then we have A = ; there re-
maining thus :

132 THE PHILOSOPHY OF MATHEMATICS.

Bo? + Cz 2 + r) 3 + etc. = 0.
I divide the whole by x, and I have

B + Ca? + Da? 8 + etc. = 0,

from which we deduce B = by the same reasoning
that we have given to prove A = ; the same reason-
ing would likewise prove = 0, D = 0, etc.

" That granted, let there be an equation with only
two terms

in which the first term is constant and the second sus-
ceptible of being rendered as small as we please ; that
equation cannot subsist after what has been said, un-
less A and Bx are each in particular equal to zero.
Then we may establish this as a general principle, and
as an immediate corollary from the method of inde-
terminates, that if the sum or the difference of two pre-
tended quantities is equal to zero, and if the one of the
two can be supposed as small as we please, while the
other contains nothing arbitrary, then the two pretended
quantities will be each in particular equal to zero"

"This principle alone suffices for the resolution
by ordinary algebra of all the questions which be-
long to the infinitesimal analysis. The respective
procedures of the one and of the other methods, sim-
plified as they ought to be, are absolutely the same ;
all the difference is in the manner of considering the
question ; the quantities which are neglected in the one
as infinitely small are unexpressed in the other, though
considered as finite, because it is demonstrated that
they ought to eliminate themselves by themselves, that

THE PHILOSOPHY OF MATHEMATICS. 133

is to say, to destroy one another in the result of the
calculus.

" Indeed, it is easy to perceive that the result can
be only an equation with two terms of which each in
particular is equal to zero; we can then suppress before-
hand, in the course of the calculus, all the terms of
these two equations of which we do not wish to make
use. Let us apply this theory of indeterminates to
some examples.

" For a second example/' * says the author, " let us
propose to prove that the area of a circle is equal to the
product of its circumference by the half of its radius ;
that is to say, denoting the radius by E, the ratio of
the circumference to the radius by TT, and consequently
that circumference by JT R, S the surface of the circle,
we ought to have

" In order to prove this I inscribe in the circle a regu-
lar polygon, then I successively double the number of
its sides until the area of the polygon differs as little as
we please from the area of the circle. At the same
time the perimeter of the polygon will differ as little
as we please from the circumference TT R, and the apo-
the area S will differ as little as we please from the
| TT E 2 ; then if we make

the quantity p, if it is not zero, can at least be sup-
posed as small as we please. That supposed, I put the
equation under the form

* The first example is quite too long for my purpose, and besides,
it would not be understood by the reader without a knowledge of
what had gone before in the work of Carnot.

i 34 THE PHIL OSOPHY OF MA THEM A TICS.

an equation of two terms, the first of which contains
nothing arbitrary, and the second of which, on the
contrary, can be supposed as small as we please ; then,
by the theory of indeterminates, each of these terms in
particular is equal to ; then we have

S i n E 2 == 0, or S = } TT B?;

which was to be demonstrated.

" Let it be proposed now to find the value which it
is necessary to give to x, in order that its function
a x x 2 may be a maximum.

" The case of a maximum ought evidently to have
place, when by adding to the indeterminate x an arbi-
trary increment x ; } the ratio of the corresponding aug-
mentation of the proposed function ax y? to x' can
be rendered as small as we please by diminishing x r
more and more.

"But if I add to x the quantity x', I shall have for
the augmentation of the proposed function

a (x + x f ) (x + xj (ax x*),
or by reducing

(a

it is then the ratio of this quantity to x f , or
a 2 x x',

which we ought to have the power to suppose as small
as we please. Let this quantity = p 9 we shall then have

a 2x x f = /?.

THE PHILOSOPHY OF MATHEMATICS. 135

or (a

an equation of two terms, the first of which contains
nothing arbitrary, and the second of which may be
supposed as small as we please ; then by the theory of
indeterrninates, each of these terms taken separately
is equal to 0. Then we have

a 2 x = 0, or x = ^ a,

which was required to be found.

" Let it be proposed to prove that two pyramids with
the same bases and the same heights are equal to each
other.

" Consider these pyramids divided into the same
number of frustums, all of the same height. Each
of these frustums may evidently be regarded as com-
posed of two parts,, the one of which will be a prism
having for its base the smaller of the two which termi-
nates the frustum, and the other will be a sort of
aglet which surrounds that prism.

" If, then, we call "V, "V, the volumes of the two pyra-
mids, P, P', the respective sums of the prisms, of
which we have just spoken, q } q', the respective sums
of the aglets, we shall have

But it is clear that P = P', then

V j = V g',or(V V 7 ) (?20 = 0.

But the first term of this equation contains nothing
arbitrary, and the second can evidently be supposed as

136 THE PHIL OSOPHY OF MA THEMA TICS.

small as we please. Then, by the theory of indetermi*
nates, each of these terms in particular is equal to zero,
Then we have

which was to be demonstrated." *

After proving, by the same method, that the volume
of a pyramid is equal to one-third of its base by its
altitude, and showing that the process is identical with
that of the calculus, Carnot adds : " We then see that
the method of indeterminates furnishes a rigorous
demonstration of the infinitesimal analysis, and that it
gives at the same time the means of supplying its
place, if we wish, by the ordinary analysis. It is
desirable, perhaps, that the differential and integral
calculus had been arrived at by this route, which
was as natural as the road that was actually taken,
and would have prevented all the difficulties." f
But however ingenious or striking such application
of the method of indeterminates, if Carnot had only
tried that method a little further, he would have found
that it is an exceedingly poor substitute for the differ-
ential and integral calculus. For these, in fact, grap-
ple successfully with an infinity of difficult questions
which the method of indeterminates is wholly unable
to solve.

* Reflexions, etc., Chapter III.

CHAPTER VI.

THE METHOD OF LEIBNITZ.

" IJEIBNITZ who was the first," says Carnot,* " to
give rules for the infinitesimal calculus, established it
upon the principle that we can take at pleasure, the
one for the other, two finite magnitudes which differ
from each other only by a quantity infinitely small.
plicity and of a very facile application. It was adopted
as a kind of axiom, and he contented himself with re-
garding these infinitely small quantities as quantities
less than those which can be appreciated or seized by
the imagination. Soon this principle operated prodi-
gies In the hands of Leibnitz himself, of the brothers
Bernouilli, of de L'H6pital, etc. Still it was not free
from objection; they reproached Leibnitz (1) with em-
ploying the expression infinitely small quantities with-
out having previously defined it ; (2) with leaving in
doubt, in some sort, whether he regarded his calculus
as absolutely rigorous, or as a simple method of ap-
proximation." t

* Reflexions, etc., Chapter I., p. 36,

f This objection to the calculus is two hundred years old ; it has
always arisen, naturally, if not necessarily, in view of the fact that
infinitely small quantities are thrown out as nothing. And yet a
Cambridge mathematician says, even at the present day, that we
cannot so easily answer this objection, because we cannot see how it
arises !

12* 137

138 THE PHILOSOPHY OF MATHEMATICS.

This principle was adopted as an axiom? or rather
as " a sort of axiom." Now is this really an axiom
or otherwise ? Is it true or false ? "Will it make no
possible difference in the result whether we throw
away as nothing or retain as something these infinitely
small quantities? If we subtract one quantity, how-
ever small, from another, shall we not at least dimin-
ish that other by an amount equal to the quantity sub-
tracted? It seems so to me. Yet Carnot, who has
looked so deeply into the " metaphysics" of the calculus,
appears, at least occasionally, to entertain a different
opinion. For having referred to the brilliant career
of the axiom, in question, and to the prodigies it had
performed in the hands of Leibnitz and others, he
adds : " The illustrious author and the celebrated men
who adopted his idea (i. e., the above axiom) contented
themselves with showing by the solution of problems
the most difficult, the fecundity of the principle, and
the constant agreement of its results with those of the
ordinary analysis, and the ascendency which it gives
to the new calculus. These multiplied successes vic-
toriously proved that all the objections were only
specious ; but these savans did not reply in a direct
manner, and the knot of the difficulty remained. There
are truths with which all just minds are struck at
first, of which, however, the rigorous demonstration
escapes for a long time the most skillful." We should
not be surprised, however, if we should hereafter find
Carnot himself urging the very objection he here pro-
nounces "only specious/' and branding the above
axiom as an error ; for it seems to be one of the estab-
lished penalties of nature that the man who begins by
denying the truth shall end by contradicting himself

THE PHILOSOPHY OF MATHEMATICS. 139

This truth,, if it be one, has certainly had to wait a
lono* time on "the skillful" for a demonstration. It

o

is now, indeed, more than two hundred years since it
is supposed to have " struck all just minds," and yet,
although it has always been objected to, it is just as
far as ever from having been demonstrated.

The calculus of Leibnitz, we are told, was " estab-
lished upon this principle" by its author. If, then,
the thing were possible, why did not Leibnitz himself
demonstrate this principle, and put the foundation of
his system beyond cavil and controversy? Or why
did not M. Carnot, or some other admirer of this great
fundamental truth, vouchsafe a demonstration of it to
the world ? Shall it wait for ever on the " most skill-
ful" for a demonstration, and wait in vain? Carnot
offers a graceful apology for Leibnitz. " It is not
astonishing," says he, " that Leibnitz should not have
attempted the rigorous demonstration of a principle
which was then generally regarded as an axiom."*
But he knew that this was an axiom only with the
initiated few, while, on all sides, there came up against
it objections from the common sense and reason of
mankind. He only replied, if we may believe M.
Carnot, by "the solution of the most difficult pro-
blems," and by showing "the ascendency which it
gave to the new calculus," and thus " victoriously
proved that every objection was only specious."
" But," adds our author, " he did not reply in a direct
manner, and the knot of the difficulty remained."
"Why, then, did he not reply in a direct manner, and
for ever dissipate the knot of the difficulty ? The truth
is, that the dark knot of the difficulty was in the mind

* Bo flexions, etc., Oluipter III., p. H.

1 40 THE PHIL OSQPHY OF MA THEMA TICS.

of Leibnitz himself, as well as in tlie minds of those
who objected against the logical basis of his method.

If, as Carnot says, Leibnitz failed to reply in a
direct manner to the objection in question, it cannot be
For it is well known that he did attempt such a reply,
and also that it was a failure.* " Leibnitz," says M.
tion entirely erroneous, saying that he treated infinitely
small quantities as incomparables, and that he neglected
them in comparison with finite quantities, 'like
grains of sand in comparison with the sea;' a view
which would have completely changed the nature of
his analysis by reducing it to a mere approximative
calculus," etc.f A greater than M. Comte had, many
years before him, said precisely the same thing. " M.
Leibnitz," says D'Alembert, " embarrassed by the ob-
jections which he " felt would be made to infinitely
small quantities, such as the differential calculus con-
sidered them, has preferred to reduce his infinitely
small quantities to be only incomparable^, which ruined
the geometrical exactitude of the calculus." Now, if
instead of all this embarrassment, vacillation, and
uncertainty, Leibnitz had only demonstrated his funda-
mental principle, then his reply would have been far
more satisfactory. Even the unskillful would have
been compelled to recognize its truth, and lay aside
their objections to his method. But, as it was, this
illustrious man bequeathed, with all its apparent un-
certainty and darkness, the fundamental principle of
his method to his followers.

* Montucla's Ilistoiro de Mathemutiques, Vol. I., Leibnitz,
f Philosophy of Mathematics, Chapter III., p. 99.

THE PHILOSOPHY OF MATHEMATICS. 141

"The Marquis de L'HSpital," one of the most cele-
brated of those followers, " was the first to write a
systematic treatise on the ' Analysis of Infinitely Small
Quantities," Hitherto its principles constituted, for
the most part, a sort of esoteric doctrine for the initi-
ated ; but now, by this most venerable man and accom-
plished mathematician, they were openly submitted to
the inspection of the world. The whole superstruc-
ture rests on two assumptions, which the author calls
" demands or suppositions." The first of these is thus
stated by the Marquis :

I. FIRST DEMAND OB SUPPOSITION.

" We demand that we can take indifferently the one
for the other, two quantities which differ from each
other only by an infinitely small quantity, or (which
is the same thing) that a quantity which is increased
or diminished only by another quantity infinitely less
than itself, can be considered as remaining the same." *
True, it may be considered as remaining the same, if
actually remain the same ? If we " increase or dimin-
ish" a quantity, ever so little, will it not be increased
or diminished ? And if we throw out any quantities,
however small, as nothing, will not this make some
difference in the result ? Thus, it seems to be written
over the very door of the mathematical school of Leib-
nitz and de L'Hopital, " let no man enter here who
cannot take his first principles upon trust." When
young Bossut, afterward the historian of mathematics,
ventured to hint his doubts respecting this first de-
mand, and ask for light: "Never mind," said his

* Analyse des Iiifiniment Petits, Art. 2, p. 3.

1 42 THE PHIL OSOPHY OF MA THEM A TICS.

teacher ; " go to work with the calculus, and you will
soon become a believer." * He took the advice what
else could he do ? and ceased to be a thinker in order
to become a worker.

II. SECOND DEMAND OK SUPPOSITION.

" "We demand," says the Marquis, " that a curve
line can be considered as the assemblage of an infinity
of right lines, each infinitely small; or (what is the
- same thing) as a polygon with an infinite number of
sides, each infinitely small, which determine by the
angles which they make with each other the curva-
ture of the line."

Now this is the principle which, in the preceding
pages, I have so earnestly combated. The truth is,
as I shall presently demonstrate, that these two &lse
principles or demands lead to errors, which, being
opposite and equal, exactly neutralize each other, so
that the great inventor of this intellectual machine, as
well as those who worked it the most successfully, were
blindly conducted to accurate conclusions.

In the preface to his work, the Marquis de L'Hfipital
says, " The two demands or suppositions which I have
made at the commencement of this treatise, and upon
which alone it is supported, appear to me so evident,
that I believe they can leave no doubt in the mind of
attentive readers. I could easily have demonstrated
them after the manner of the ancients, if I had not
proposed to myself to be short upon the things which
are already known, and devote myself principally to

* Most teachers of the present day are wiser : they avoid all such
difficulties; they do not state the first principles at all; they just set
their pupils to work with the calculus, and they become believers
rather than thinkers.

THE PHIL OSOPIIY OF MA THEMA TICS. 1 43

those which are new." I have been curious to know
what sort of demonstrations the Marquis had found
for this "sort of axioms." It is pretty certain, ii
seems to me, that they could hardly have been per-
fectly clear and satisfactory to his own mind, or else
he would have laid such demonstrations, like blocka
of transparent adamant, at the foundation of his
system. It is evident he should have done so, for
this would have removed a world of doubt from the
opponents of the new system, and a world of difficulty .
from its friends. Indeed, if there be principles by
which his postulates or demands could have been de-
monstrated, then those principles must have been more
evident and satisfactory than these postulates or de-
mands themselves, and should, therefore, have been
made to support them. A little time would not, most
assuredly, have been misemployed in giving such
additional firmness and durability to so vast and com-
plicated and costly a structure.

The third edition of the " Analyse," the one now
before me, is " followed by a commentary [nearly half
as large as the book itself], for the better understand-
ing of the most difficult places of the work." Now,
strange to say, one of these " most difficult places"
which a commentary is deemed necessary to clear up,
is the first " demand or supposition" which is laid
down by the author as self-evident. " The demand,
or rather the supposition of article 2, page 3," says the
commentary, " which beginners consider only with
pain, contains nothing at the bottom which is not very
reasonable." Then the commentator proceeds to show,
by illustrations drawn from the world of matter, that
this first axiom is not unreasonable. Not unreason-

144 TEE PHIL OSOPHY OF MA THEMA TICS.

able indeed I Should not the axioms of geometry be
reason itself, and so clear in the transcendent light of
their own evidence as to repudiate and reject all illus-
trations from the material world ? The very existence
of such a commentary is, indeed, a sad commentary on
the certainty of the axioms it strives to recommend.

" In fact," says the commentator, " we regard as in-
finitely exact the operations of geometers and astrono-
mers; they make, however, every day, omissions much
more considerable than those of the algebraists. "When
a geometer, for example, takes the height of a moun-
tain, does he pay attention to the grain of sand which the
wind has raised upon its summit? When the astrono-
mers reason about the fixed stars, do tliey not neglect
the diameter of the earth, whose value is about three
thousand leagues ? When they calculate the eclipse
of the moon, do they not regard the earth as a sphere,
and consequently pay no attention to the houses, the
towns, or the mountains which are found on its sur-
face ? But it is much less to neglect only d x, since it
takes an infinite number ofdx's to make one x- then
the differential calculus is the most exact of all calcu-
luses; then the demand of article 2 contains nothing
unreasonable. All these comparisons are drawn from
the Course of Mathematics of Wolff, torn. 1, p. 418."
Thus, from this curious commentary it appears that
the editor of the work of de L'H6pital in 1798, as
well as Wolff, the great disciple of Leibnitz, regarded
the differential calculus as merely a method of approxi-
mation. Leibnitz himself, as we have already seen,
was at times more than half inclined to adopt the same
view; plainly confessing that he neglected infinitely
small quantities in comparison with finite ones, " like

THE PHILOSOPHY OF MATHEMATICS. 145

grains of sand in comparison with the sea." Indeed,
he must have been forced to this conclusion and fixed
in this belief, if pure geometry had not saved him
from the error. He certainly expected that the rejec-
tion of his infinitesimals would tell on the perfect
accuracy of his results; but he found, in fact, that
these often coincided exactly with the conclusions of
pure geometry, not differing from the truth by even so
much as a grain of sand from the sea, or from the
solar system itself. But, not comprehending why there
should not have been at least an infinitely small error
in his conclusions, he simply stood amazed, as thou-
sands have since done, before the mystery of his
method, sometimes calling his " infinitely small quan-
tities zeros," sometimes "real quantities/' and some-
times " fictions." "When he considered these quantities
in their origin, and looked at the little lines which
their symbols represented, he thought they must be
real quantities; but since these quantities might be
infinitely less than the imagination of man could con-
ceive, and since the omission of them led to absolutely
exact results, he was inclined to believe that they must
be veritable zeros. But, not being able to reconcile
these opposite views, or to rest in either, he sometimes
effected a sort of compromise, and considered his infi-
nitely small quantities as merely analytical " fictions."
The great celebrities of the mathematical world since
the time of Leibnitz, the most illustrious names, in-
deed, in the history of the science, may be divided into
three classes, and ranged, as advocates of these three
several views of the differential calculus.

It was long before the true secret was discovered.
" After various attempts, more or less imperfect," says

13 G

146 .THE PHILOSOPHY OF MATHEMATICS.

M. Comte, "a distinguished geometer, Carnot, pre-
sented at last the true, direct, logical explanation of
the method of Leibnitz, by showing it to be founded
on the principle of the necessary compensation of errors,
this being, in fact, the precise and luminous manifes-
tation of what Leibnitz had vaguely and confusedly
perceived."* Now Carnot owed absolutely no part
of his discovery to Leibnitz. If Leibnitz, indeed,
obscurely perceived the existence of such a compensa-
tion of errors in the working of his method, he has
certainly nowhere given the most obscure intimation
of it in his writings. Such a hint from the master,
however unsupported by argument, would have served j

at least to put some of his followers on the true path [

of inquiry. But no such hint was given. Leibnitz, t

it is true, perceived several things very obscurely ; but
the real secret of his method was not one of them.
Hence he put his followers on the wrong scent only,
and never upon the true one. Indeed, if he had sus- !

pected his system of a secret compensation of errors,
then he must also have suspected that the two ee de- ;

mands or suppositions" on which it rested were both
false, and it would not have been honest in him to lay
them down as self-evident truths or axioms.

The explanation of Carnot is certainly, as far as it j

goes, perfectly satisfactory. In the second edition of [

his work he quotes with an evident and justifiable f

satisfaction, the approbation which the great author !

of the Theory of Functions had bestowed on his expla- i

nation. "In terminating," says he, "this exposition
of the doctrine of compensations, I believe I may
honor myself with the opinion of the great man whose

* Philosophy of Mathematics, Chap. III., p. 100.

THE PHIL OSOPH Y OF MA THE MA TICS. 1 47

recent loss the learned world deplores, Lagrange ! He
thus expresses himself on the subject in the last edition
of his ' The'orie des Fonctions Analytiqucs :'

" ' In regarding a curve/ says Lagrange, ' as a poly-
gon of an infinite number of sides, each infinitely small,
and of which the prolongation is the tangent of the
curve, it is clear that we make an erroneous supposi-
tion but this error finds itself corrected in the calcu-
lus by the omission which is made of infinitely small
quantities. This can be easily shown in examples,
but it would be, perhaps, difficult to give a general
demonstration of it/

" Behold," exclaims Carnot, " my whole theory re-
sumed with more clearness and precision than I could
put into it myself!" * Let us, then, see this theory, or
rather its demonstration. Carnot begins with a special
case.

" For example," says he, " let it be proposed to

* It is, then, Garnet's emphatic opinion that the two demands or
postulates of the method of Leibnitz are both " clearly erroneous sup-
positions" or false hypotheses. Yet, as we have seen, when the first
of these demands was assailed by others as untrue, he pronounces
the objection " only specious,-" he excuses Leibnitz for not having
demonstrated it, because it was "generally regarded as an axiom,"
and even places it among those truths which, at first, "strike all
just minds/' but of which the demonstrations long remain to be dis- j*

covered. H\$Tow, how could Leibnitz have demonstrated an error? J j|

Or how long will it be before such a thing is demonstrated? Or,

again, if any one objects to receiving as true a manifest error, how ' j

can it be said that his objection is "only specious?" The truth is, f|

that instead of that metaphysical Clearness and firmness of mind .

which never loses sight of a principle, but carries it as a steady light
into all the dark regions of speculation, there is some little wavering
and vacillation in the views of Carnot, and occasionally downright
contradictions, especially in what he says in regard to the method
of Leibnitz. f <

l/f

148

THE PHILOSOPHY OF MATHEMATICS.

draw a tangent to the circumference B M D at the
given point M,

" Let C be the centre of the circle, D C B the axis;

suppose the abscissa D P = x, the ordinate M P = y,
and let T P be the subtangent required.

" In order to find it, let us consider the circle as a
polygon with a very great number of sides ; let M N"
be one of these sides, prolonged even to the axis; that
will evidently be the tangent in question, since that
line does not penetrate to the interior of the polygon ;
let fall the perpendicular M O upon N Q, which is
parallel to M P, and name a the radius of the circle ;
this supposed, we shall evidently have

MO:NO::TP:MP,

MO = TP = TP

NO ~

or

MP

y

On the other hand, the equation of the curve for the
point M being y 1 = 2 a x x 2 , it will be for the point
N,(y + N O) 2 = 2 a (x + M O) (x + M O) 2 , taking
from this equation the first, found for the point M r
and i educing, we have

MO_ 27/ + NO
~~

THE PHIL OSOPHY OF MA THEM A TICS. 149

equaling; this value of to that which has been

^ b NO

found above, and multiplying by y, it becomes

TP== 2/(2y + NO)
2a2x MO'

" If then M O and N O were known, we should have
the required value of TP; but these quantities MO
N O are very small, since they are less than the side
M N, which, by hypothesis, is itself very small. Then
we can neglect without sensible error these quantities
in comparison with the quantities 2 y and 2 x 2 a to
which they are added. Then the equation reduces
itself to

a x

which it was necessary to find.

" If this result is not absolutely exact, it is at least
evident that in practice it can pass for such, since the
quantities M O, N O are extremely small ; but any one
who should have no idea of the doctrine of the infinite,
would perhaps be greatly astonished if we should say

?/ 2
to him that the equation T P = - , not only ap-

a x

proaches the truth very nearly, but is really most per-
fectly exact ; it is, however, a thing of which it is easy
to assure one's self by seeking T P, according to the
principle that the tangent is perpendicular to the ex-
tremity of the radius ; for it is obvious that the similar
triangles C P M, M P T, give

C P : M P : : M P : T P,

13 *

150 THE PHIL 080PHY OF MA THEM A TICS.

2

Lence, T P = ^ , as above.

a x

({ Let us see, then, how in the equation

found above, it has happened that in neglecting M O
and N O we have not altered the justness of the result,
or rather how that result has become exact by the
suppression of these quantities, and why it was not so
before.

" But we can render very simply the reason why
this has happened in the solution of the problem
above treated, in remarking that, the hypothesis
from which it set out being false, since it is absolutely
impossible that a circle can ever be considered as a true
polygon, whatever may be the number of its sides,
there ought to result from this hypothesis an error in
the equation

2a 2# MO'

if

and that the result T P = - being nevertheless

a x

certainly exact, as we prove by the comparison of
the two triangles C P M, M P T, we have been able
to neglect M O and N O in the first equation ; and
indeed we ought to have done so in order to rectify
the calculus, and to destroy the error which had arisen
from the false hypotheses from which we had set out.
To neglect the quantities of that nature is then not

THE PHILOSOPHY OF MATHEMATICS. 151

only permitted in sucli a case, but it is necessary : it is
the sole manner of expressing exactly the conditions
of the problem." *

" The exact result T P = - has then been ob-

a x

tained only by a compensation of errors; and that com-
pensation can be rendered still more sensible by treat- !
ing the above example in a little different manner,
that is to say, by considering the circle as a true curve,
and not as a polygon.

"For this purpose, from a point E, taken arbi-
trarily at any distance from the point M, let the line
E S be drawn parallel to M P, and through the points
E and M let the secant E T' be drawn ; we shall evi-
dently have

T'P:MP::MZ:EZ,

and dividing T ; P into its parts, we have
TP+TT' = MP .

This laid down, if we imagine that E S moves parallel
to itself in approaching continually to M P, it is ob-
vious that the point T ; will at the same time approach
more and more to the point T, and that we can conse-
quently render T' T as small as we please without
the established relations ceasing to exist. If then I
neglect the quantity T' T in the equation I have just
found, there will in truth result an error in the equa-
tion T P = M P . to which the first will then be
EZ

* This -last expression seems a little obscure, since it is difficult to
perceive how the neglect of such quantities is necessary "to express
the conditions of the problem."

l

152 TEE PHILOSOPHY OF MATHEMATICS.

reduced ; but that error can be attenuated as much a.n
we please by making E S approach M P as much as
will be necessary ; that is to say, that the two members
of that equation may be made to differ as little as we

T n i, MZ 2v+EZ

" In like manner we have - = - - - ,

EZ 2a 2x MZ'

and this equation is perfectly exact, whatever may be
the position of E ; that is to say, whatever may be the
values of M Z and E Z. But the more E S shall ap-
proach M P, the more small will the lines M Z and
R Z become, and if we neglect them in the second
member of the equation, the error that will result

M Z v

therefrom in the - = ^ to which it will then
E Z #

be reduced, would, as in the first, be rendered as small
as we might think proper.

" This being so, without having regard to the errors
which I may always render as small as I please, I
treat the two equations

rn-r> , r T> M Z , MZ V

T P == M P - and

EZ EZ ax

which I have just found, as if they were both perfectly
exact; substituting then in the first, the value of

- taken from the last, I have for the result
E Z

TP ==--, as above.
a x

This result is perfectly just, since it agrees with that
which we obtain by comparing the triangles C P M,

TEE PHIL OSOPHY OF MA THEM A TICS. 1 53

TV/T *7

MPT, and yet the equations T P = y and

R Z

# ? from which it is deduced, are both cer-

R Z a x

tainly false, since the distance of R S from M P has
not been supposed nothing, nor even very small, but
equal to any arbitrary line whatever. It follows, as
a necessary consequence, that the errors have been
mutually compensated by the combination of the two
erroneous equations."

" Behold, then," says Carnot, " the fact of the com-
pensation of errors clearly and conclusively proved."
He very justly concludes that there is a mutual
compensation of errors in the case considered by him,
because the combination of the two imperfect equations
resulted in an absolutely perfect one. If, however, he
had pointed out the error on the one side and on the
other, and then proved that they were exactly equal
and opposite, his exposition would, it seems to me,
have been rather more " precise and luminous." This
is precisely the course pursued by Bishop Berkeley in
his demonstration of the same fact. It may be well,
therefore, to give his illustrative proof of this compen-
sation of errors in the ordinary use of the calculus.
" Forasmuch," says he, " as it may perhaps seem an
deduce true propositions from false principles, be right
in the conclusion, and yet err in the premises, I shall
endeavor particularly to explain why this may come
to pass, and show how error may bring forth truth, though
ii cannot bring forth science"

" In order, therefore, to clear up this point, we will
suppose, for instance, that a tangent is to be drawn to
G*

154

THE PHILOSOPHY OF MATHEMATICS.

a parabola, and examine the progress of the affair as
it is performed by infinitesimal differences. Let A B
be a curve, the abscissa A P = cc, the ordinate P B = y ;
the difference of the abscissa P M = d x, the difference

M

of the ordinate E N = d y. Now, by supposing the
curve to be a polygon, and consequently B 1ST, the in-
crement or difference of the curve to be a straight line
coincident with the tangent, and the differential tri-
angle B E N to be similar to the triangle T P B, the
subtangent is found a fourth proportional to K N :
E B : P B ; that is, to dy :dx :y. Hence the sub-

f\i r\ nf

tangent will be . But then there is an error

dy

arising from the forementioned false supposition (I. e. }
that the curve is a polygon with a great number of
sides), whence the value of P T comes out greater than
the truth ; for in reality it is not the triangle E N B,
but E L B, which is similar to P B T, and therefore
(instead of E N") E L should have been the fourth term,
of the proportion; i. e. y E N + N L, i. e., dy -f- 2;
whence the true expression for the subtangent should

i T V dx
have been

dy

There was, therefore, an error of

THE PHILOSOPHY OF MATHEMATICS. 155

defect in making d y the divisor, which error is equal
to z ; i. e., N L the line comprehended between the
curve and the tangent. Now, by the nature of the
curve y 2 = p x } supposing p to be the parameter,
whence by the rule of differences, 2 y dy =p dx and

dy=-B- . But if you multiply y + dy by itself,

and retain the whole product without rejecting the
square of the difference, it will then come out, by sub-*
stituting the augmented quantities in the equation of

T) (L X d 7/^

the curve, that dy=- truly. There was,

2y 2y

therefore, an error of excess in making d y = ,

dy

which followed from the erroneous rule of differences.

dif-
And the measure of this error is = z. Therefore

2y

the two errors, being equal and contrary, destroy each
other, the first error of defect being corrected by a
second of excess.

" If you had committed only one error, you would
not then come at a true solution of the problem. But
by virtue of a twofold mistake you arrive, though not
at science, yet at truth. For science it cannot be
called when you proceed blindfold and arrive at truth
not knowing how or by what means. To demonstrate

d if *
that z is equal to , let B R or dx be m, and E N

^ y
or d y be n. By the thirty-third proposition * of the

first book of the Conies of Apollonius, and from similar
triangles,

* Which is, that the subtangent T P is equal to 2 x.

156 THE PHIL OSOPHY OF MA THEM A TICS.

. my

2x : i/ : : m : n-\- z = - .
y 2x

Likewise from the nature of the parabola y 2 + 2 y n +
n 2 = p x + p my and 2y n + n* =p m\ wherefore

2 v n -f- n 2 T i o 'V 2 n i i

__^ ! r= m and because y* = p x, *_ will be equal

to a;. Therefore, substituting these values instead of
m and x, we shall have

my 2 y 2 ft. + -y n 2
n + 2 = -H *

, .
that is.

which being reduced gives

* = 5l = ^. Q.B.D.*

2y 2y

Thus it is shown that when we seek the value of
the subtangent on the supposition that the curve is a
polygon, we make d y too small by the line N L. On
the other hand, it is shown that when in seeking the
value of dy from the differential equation of the curve,

dif
we throw out the minus quantity ~ as infinitely

2y
small in comparison with dy, we make dy too great

d'v 2
by this quantity . But if we first make dy too

'
small by N L, and then too great by ^ it only re-

%y t .

mains to be shown that these two quantities are ex-

The Analyst, XXI. and XXII. Berkeley's Works, Vol. II., p.

422.

THE PHILOSOPHY OF MATHEMATICS.

157

actly equal in order to establish the fact of a compen-
sation of errors. Accordingly, this is done by the
Bishop of Cloyne, with the addition of the " Q. E. D."
That is to say, the error resulting from one " demand
or supposition" of the Leibnitzian method is corrected
by the error arising from its other demand or sup-
position.

It is not true, then, as M. Comte alleges, that Car-
not was the first to present the true " explanation of the
method of Leibnitz." This honor is due to the Bishop
of Cloyne, not to the great French minister of war ; to
the philosopher, not to the mathematician ; for the ex-
planation of Berkeley preceded that of Carnot by more
than half a century. Both explanations rest, as we
have seen, on particular examples instead of general
demonstrations. Hence Lagrange, after approving
the explanation or adopting it as his own view of the
subject, adds : " This may be easily shown in exam-
ples, but it would be, perhaps, difficult to give a gene-
ral demonstration." Carnot dissents. "I believe,"
says he, that "in the demonstration which I have
given of it" there " is wanting nothing either of ex-
actitude or of generality." Plis general demonstration,
however, is metaphysical rather than mathematical
a sort of demonstration which does not always carry
irresistible conviction to the mind.

Having exhibited his examples, Carnot proceeds to
ascertain " the sign by which it is known that the
compensation has taken place in operations similar to
the preceding, and the means of proving it in each
particular case." This is done only by a process of
'*" general reasoning," as it is very properly called by
M. Comte, and it is fairly exhibited in his Philosophy

14

158 TEE PHIL OSOPITT OF MA THEM A TICS.

of Mathematics. fi In establishing," says M. Comte,
*' the differential equation of phenomena, we substi-
tute for the immediate elements of the different quan-
tities considered other simpler infinitesimals, which
differ from them infinitely little in comparison with
them, and this substitution constitutes the principal
artifice of the method of Leibnitz, which without* it
would possess no real facility for the formation of
equations. Carnot regards such an hypothesis as
really producing an error in the equation thus obtained,
and which for this reason he calls imperfect ; only it
is clear that this error must be infinitely small. Now,
on the other hand, all the analytical operations,
whether of differentiation or of integration, which are
performed upon these differential equations in order to
raise them to finite equations by eliminating all the
infinitesimals which have been introduced as auxili-
aries, produce as constantly by their nature, as is easily
seen, other analogous errors, so that an exact compen-
sation takes place, and the finite equations, in the
words of Carnot, become perfect. Carnot views, as a
certain and invariable indication of the actual estab-
lishme'nt of this necessary compensation, the complete
elimination of the various infinitely small quantities,
which is always, in fact, the final object of all the
operations of the transcendental analysis; for if we
have committed no other infractions of the general
rules of reasoning than those thus exacted by the very
nature of the infinitesimal method, the infinitely small
errors thus produced cannot have engendered other
than infinitely small ones in all the equations, and the
relations are necessarily of a rigorous exactitude as soon
as they exist between finite quantities alone, since the

THE PHILOSOPHY OF MATHEMATICS.

159

only errors then possible must be finite ones, while
none such can have entered. All this general reason-
ing is founded on the conception of infinitesimal quan-
tities, regarded as indefinitely decreasing, while those
from which they are derived are regarded as fixed/' *

Lagrange had, perhaps, no objection to offer to this
"general reasoning;" it appears certain that he did
not regard it as a "general demonstration." "It
would, perhaps," says he, "be difficult to give a
general demonstration" of the fact of a compensation
of errors in the use of the calculus. It is easy to give
a far more general demonstration than that proposed
by either Carnot or Berkeley. For the one of these
relates, as we have seen, merely to the question of
finding a tangent to the circumference of a circle at
a given point, f and the other to the same problem
with reference to the parabola. Now, this compensa-
tion of errors may be demonstrated to take place in
the process for finding the tangent to curve lines in
general.

Let y = Fx be the equation of such a curve, in

Fig. 1.
Y

y

E

7'

A D D / X

* Philosophy of Mathematics, Book L, Chap. III., p. 101.
j- Carnot, I am aware, has furnished a second example; but this
does not make the proof general.

100 THE PHILOSOPHY OF MATHEMATICS.

which y is equal, not to one particular function of x 9
as in the case of the circle or parabola, but to any
algebraic function whatever. Then, if the curve be
convex toward the axis of x, as in Fig. 1, and if an
increment h be given to the abscissa AD = #, the
ordinate y will take an increment E P', whose value
may thus be found :

dxl da? 1.2

, *

. -- K etc..
dx* 1.2.3 ' ?

by Taylor's Theorem,
or P'D'

.

do? 1 da; 2 1.2

and the ordinate of the tangent line P T will take the
corresponding increment E T, whose value, found in
the same way, gives

^ -
dx I

Hence

- .--,.

dx* 1.2 dx* 1.2.3 '

Now, the subtangent s is the fourth term of the
exactly true proportion,

but E T, being unknown, cannot be used for the pur-

THE PHILOSOPHY OF MATHEMATICS.

161

pose of finding the value of s. Hence E P ; is in the
method of Leibnitz substituted for E T, and this sub-
stitution is justified on the ground that the difference
between the two quantities is so very small. But still
this difference is, as we have seen,

h* . d s v h 3

s/ etc.

.2 dx* 1.2.3'

When the operator comes, however, to find the value
of P ; E from the equation of the curve, this value of
P' T is precisely the quantity thrown away as nothing
by the side of an infinitely small one of the first order.
Thus, by the one step, the true value is made too
great by the quantity P ; T, and by the other the sub-
stituted value is reduced by precisely the same amount
P' T. That is to say, the same quantity was first
difference in the result.

If the curve be concave toward the axis of x, as

Fig. 2.

D'

seen in Fig. 2, then the true value, T E, or the line
which gives the exact value of s by the proportion,

T E : h : : y : s y

162 THE PHIL OSOPHY OF MA THEM A TICS.

will be made too small by the substitution of P' E in
its place. But, in this case, the value of P' T, or

^__
.2 da; 3 1.2. 3

which is rejected in finding the value of P ; E, is a
negative quantity; and, consequently, in throwing it
away from the value of P' E as nothing, that value
was increased by the same amount it had been dimin-
ished. That value of P 7 T is negative, because the
curve being concave toward the axis of x, its first term

^ *

- . - is negative, and, since it is supposed very
d y? 1.2

small, this term is greater than the sum of all the
others.* In this case, then, the same quantity was
subtracted and added, which, of course, did not aifect
the result. Behold the very simple process which, by
means of signs and symbols and false hypotheses, has
been transformed into the sublime mystery of the trans-
cendental analysis.

In spite of its logical defects, however, the method
of Leibnitz has generally been adopted in practice;
because of the facility with which it reduces questions
of the infinitesimal analysis to equations, and arrives at
their solutions. Suppose, for example, the question
be to find the tangent line to the point M of any curve
A P, which is given by its equation. The method of
Leibnitz identifies the infinitely small chord M D with
the corresponding arc of the curve, and, consequently,
regards the figure M D E, composed of the small arc
M D and the increments of x and y, as a rectilinear

* See any work on the Differential Calculus.

THE PHIL OSOPH Y OF MA THEM A TICS. . 163

triangle. (This is, in fact, the differential triangle of
Barrow.) Hence, according to this method,

D E dy _EN.

B ?

or the tangent of the angle which the tangent line re-
quired makes with the axis of x. To find the value
of this tangent, then, it is only necessary to ascertain

the value of for the point M from the equation of
dx

the curve.

If the curve, for example, be the common parabola,
whose equation is y 1 = %p x, the value in question
may be easily ascertained. Thus, give to A B, or to
x for the point M, the infinitely small increment dx,
and B M, or y for the same point, will take the in-
finitely small increment d y. Then,

or if + 2 y dy + d if = 2p x + 2p dx.

Hence 2 y dy + d f = 2 p dx.

But df, being an infinitely small quantity of the second
order, may be rejected as nothing by the side of 2 y dy,
and hence we have

164

TEE P&ILOSOPSY OF MATHEMATICS-

dy = p
dx y'

for the value of the tangent required, which, by the
rigorous method of geometry, is known to be perfectly
exact.

Now, as we have already seen, there are in this pro-
cess two errors which correct each other; the one
arising from the false hypothesis that the curve A P
is made up of infinitely small right lines, such as M D;
and the other from the equally false postulate or de-
mand that the infinitely small quantity dy* may be
rigorously regarded as nothing by the side of 2ydy.
These false hypotheses are, however, wholly unneces-
sary, and only serve to darken science by words with-
out knowledge. That is to say, we may, in the per-
fectly clear light of correct principles, do precisely the
same thing that is done in the method of Leibnitz by
means of his false hypotheses and false logic. To
show this, let us resume the question of finding the
tangent to point M of the common parabola A P. The

tangent line T D' has, according to the definition, only
the point M in common with the curve, and

D' E = B M
ME T B'

THE PHILOSOPHY OF MATHEMATICS. 165

But D r E is unknown, since the equation of the tan-
gent line, the very thing to be determined, is not given.
Hence we adopt or apply the method of Leibnitz,
without adopting his view of that method. That is

D E

TV TT

, or we take the small quantity D E as the same

with D f E, just as he does ; but not because D E is the
same as D' E, or because M D coincides with M D'. On

-TT\ -TJ\

the contrary, we set out with because its value

ME
may be found from the equation of its curve, and be-

D' E
cause its limit is , the thing to be determined.

ME

Thus, give any increment B C or A to x, and y will
take a corresponding increment D E or k. Then

Now, if we conceive A to become less and less, then will

k

k also decrease; but the ratio j will continually in-
crease, and approach more and more nearly to an

D' E
equality with ^, which is a constant quantity. It

M E

is obvious that by making li continue to decrease, the

T) "P
fraction may be made to differ as little as we

ME

166 THE PHIL OS OPHY OF MA THE MA TICS.

, , D'E ,, D'E. . r ., ,DE
please Iroin . and hence is the limit 01

r ME ME ME

Jc

or -. But since the two members of the above equa-
li

tion are always equal, their limits are equal. That is
to say, the limit of

k . D f E dy p
_ 1S or __ = .

A M E' dx V 9

as above found. Thus, the two processes not only
lead to the same result, but they are, from beginning
to end, precisely the same. The steps in both methods
are precisely the same, and the only difference is in the
rationale or explanation of these steps. In the one
method the steps are only so many false hypotheses or
assertions, while, in the other, they are carried on in
the light of clearly correct principles. In the method

T) ~W
of limits we begin with in order to find the value

ME

D' E
of , not because the two lines D E and D' E are

M E

equal to each other, or because the difference between
them is so small that it is no difference at all, but

T) "FT

simply because the limit of , which may be easily

* J ME y y

found from the equation of the curve, is exactly equal

D' E

to the constant quantity , which is the quantity

M E
required.

The principle of this case is universal in its appli-
cation. That is to say, in the method of limits we
may always put one set of variables for another, pro-

I

j

THE PHILOSOPHY OF MATHEMATICS.

167

vicled that in passing to the limit the result will be the
same. We may not only do this, but in many cases
we must do it, in order to arrive at the desired result.
It is, in fact, the sum and substance of the infinitesimal
analysis to put one set of quantities for another ; i. e.,
of auxiliary quantities for the quantities proposed to
be found, in order to arrive indirectly at the result or
value, or relation, which cannot be directly obtained.
In the method of Leibnitz, it is supposed that one set
of quantities may be put for another, because they
differ so little from each other that they may be re-
garded as rigorously equal, and that an infinitely small
quantity may be rejected as "absolutely nothing."
On the other hand, the method of limits proceeds on
the principle that any one quantity may be put for
another, provided that in passing to their limits, the
limit of their difference is zero.

In order to illustrate his first " demand or supposi-
tion," the Marquis de L'Hopital says : ' ' We demand,
for example, that we can take Ap for A P, pm for

P M, the space Apm for the space A P M, the little
space MPpw for the little rectangle MPjpB, the
little sector A M m for the little triangle A M S, the
angle p A m for the angle P A M." Now he sup-

168 THE PHILOSOPHY OF MA THEM A TICS.

posed that we can with impunity take the one of these
several quantities for the other, because they are equal,
and also that an infinitely small quantity may be re-
jected as nothing with equal impunity. But, in fact, I
these quantities can, in the infinitesimal analysis, be ^f
respectively taken for one another, because their limits
are precisely the same, and because, by throwing out
the indefinitely small increments as nothing, or by
making them zeros, we pass to their limits, which are
the same. The space MP_pm, for example, is not
equal to the space M P p R, but always differs from it
by the little space M R m. But yet M Pp K. may be
put for MPpm, because their limit is precisely the
same line M P, and because when P p is made
equal to zero, or treated as nothing, the limit M P is
reached.

"We demand," says the Marquis de L'Hfipital,
" that we can take indifferently the one of two quan-
tities for the other which differ from each other only
by a quantity infinitely small ; or (what is the same
thing) that a quantity which is augmented or dimin-
ished by another quantity infinitely small can be con-
sidered as remaining the same." This demand is
refused. The two quantities are not equal; they differ
by an indefinitely small quantity, but their limits are
the same; and when the indefinitely small difference is
reduced to nothing, the same limit or value is obtained.
Leibnitz put the one of two quantities for another, be-
cause they were the same, whereas he should have
done so because their limits were the same. Again,
in throwing out indefinitely small quantities as zeros,
he supposed that, instead of affecting the result by this
step, everything would "remain the same;" whereas,

THE PHILOSOPHY OF MATHEMATICS.

169

in fact, this step perfected the operation and reached
the true result by passing to the limit. Thus, the
true route of the infinitesimal analysis is an indirect
one, and Leibnitz, by seeking to make it direct, only
caused it to appear absurd.

15 EC

:

CHAPTER VII.

THE METHOD OF NEWTON.

THE method of Newton, as delivered by himself,
has never been free from difficulties and objections.
Indeed, even among learned mathematicians and his
greatest admirers there have been obstinate disputes
respecting his explanation or view of his own method
of " prime and ultimate ratios." The very first de-
monstration, in fact, of the first book of his Prindpia,
in which he lays the corner-stone of his whole method,
has long been the subject of controversy among the
friends and admirers of the system ; each party show-
ing its veneration for the great author by imputing its
own views to him, and complaining of the misunder-
standing and wrong interpretation of the other. This
controversy has, no doubt, been of real service to the
cause of science, since it enables the studious disciple
of Newton to obtain a clearer insight into the princi-
ples and mechanism of his method than he himself
ever possessed. It has, indeed, been chiefly by the
means of this controversy that time and the progress
of ideas have cleared away the obscurities which origi-
nally hung around the great invention of Newton.
But if we would profit by the labors of time in this
respect, as well as by those of Sir Isaac, we must lay
aside the spirit in which the controversy has been car-
ried on, and view all sides and all pretended demou-
nt)

THE PHILOSOPHY OF MATHEMATICS. 171

strations with an equal eye, not even excepting those
of the Prinoipia itself.

The corner-stone or foundation of Newton's method
is thus laid in the Principia: "Quantities, and the
ratios of quantities, that during any finite time constantly
approach each other, and before the end of that time
approach nearer than any given difference, are ultimately
equal."

" If you deny it, suppose them to be ultimately un-
equal, and let D be their ultimate difference. There-
fore they cannot approach nearer to equality than by
that given difference D ; which is against the suppo-
sition." *

The above demonstration is thus given by Dr.
"Whewell, " Prop. I. (Newton, Lemma I.) :

" Two quantities which constantly tend towards
equality while the hypothesis approaches its ultimate
form, and of which the difference, in the course of
approach, becomes less than any finite magnitude, are
ultimately equal."

"The two quantities must either be ultimately equal,
or else ultimately differ by a finite magnitude. If
they are not ultimately equal, let them ultimately have
for their difference the finite magnitude D. But by
supposition, as the hypothesis approaches its ultimate
form, the differences of the two magnitudes become
less than any finite magnitude, and therefore less than
the finite magnitude D. Therefore D is not the ulti-
mate difference of the quantities. Therefore they are
not ultimately unequal. Therefore they are ultimately
equal." f

* Principia, Book I., Section I., Lemma I.
j- Doctrine of Limits, Book II.

172

THE PHILOSOPHY OF MATHEMATICS.

K

In the two following lemmas Newton proceeds to
give particular instances or illustrations of the import
of the above general proposition. As these are neces-
sary to render his meaning plain, they are here added :

LEMMA II.

"If in any figure A a c E, terminated by the right lines
A a, A E, and the curve

a; z a o E, there be inscribed any

number of parallelograms,
A by B c, C d, etc., compre-
hended under equal bases
n A B, B C, C D, etc., and

the sides B b, C c, D d, etc.,
parallel to one side A. a of
the figure, and the parallelo-
grams a K b I, b L c m,
c M d n, etc., are completed.
Then if the breadth of those

parallelograms be supposed to be diminished, and their
number to be augmented in infinitum / I say that the
ultimate ratios which the inscribed figure AK6LcMc?D,
the circumscribed figure A.albmcndo'E, and curvi-
linear figure A a b c d E, will have to one another, are
ratios of equality.

" For the difference of the inscribed and circumscribed
%ures is the sum of the parallelograms K /, L m, M n,
D o, that is (from the equality of all their bases), the
rectangle under one of their bases K b, and the sum
of their altitudes A a, that is, the rectangle A B I a.
But this rectangle, because its breadth A B is sup-
posed diminished in infinitum, becomes less than any
given space. And therefore (by Lem. I.) the figure

THE PHIL OSOPHY OF MA THEM A TICS. 173

Inscribed and circumscribed become ultimately equal
one to the other, and much more will the intermediate
curvilinear figure be ultimately equal to either. Q. E. B."

LEMMA III.

" The same ultimate ratios are also ratios of equality,
when the breadths A B, B C, D C, etc., of the parallelo-
grams are unequal, and are all diminished in infinitum

" For suppose A F equal to the greatest breadth, and
complete the parallelogram
FA af. This parallelo-
gram will be greater than
the difference of the in-
scribed and circumscribed
figures ; but, because its
in infinitum, it will become
less than any given rect-
angle. Q. E. D.

"Con. 1. Hence the ul-
timate sum of those evanes-
cent parallelograms will in all parts coincide with the
curvilinear figure.

" COR. 2. Much more will the rectilinear figure com-
prehended under the chords of the evanescent arcs
a by b c, c d y etc., ultimately coincide with the curvi-
linear figure.

"CoR. 3. And also the circumscribed rectilinear
figure comprehended under the tangents of the same
arcs.

"CoR. 4. And therefore these ultimate figures (as to
their perimeters a c E), are not rectilinear, but curvi-
linear limits of rectilinear figures."

15 *

I
f l

If

174 THE PHILOSOPHY OF MATHEMATICS.

In these celebrated demonstrations; as well as in
those which follow, there are very great obscurities
and difficulties. The objections to them appear abso-
lutely insuperable. How, for example, can the cir-
cumscribed figures in lemmas two and three ever be-
come equal to the curvilinear space A a E? If these
spaces should ever become equal, then the line Alb m
c n d o E would necessarily coincide with the curve
A b c d E, which seems utterly impossible, since a
broken line whose sides always make right angles with
each other cannot coincide with a curve line. I should
not, indeed, believe that the author of the Principia
contemplated such a coincidence, if his express words,
as well as the validity of his demonstration, did not
require me to believe it; that is, if he had not expressly
said in the first corollary, that "the ultimate sum
of these evanescent parallelograms will in all parts
coincide with the curvilinear figure." The supposi-
tion of such a coincidence, even if it were conceivable,
leads to an absurdity. For the sum of the horizontal
lines alj b m, c n, d o, or however far their number
may be augmented, will always be equal to the line
A E, and the sum of the corresponding vertical lines
I by m c, n d, o E, etc., will always be equal to the line
A. a. Hence, if the two figures should ultimately co-
incide, then the line A I b m c n d o E, or its equal
A.bcd E, would be equal to the sum of the two lines,
AE and A a. Or, if the curvilinear space AaE
were the quadrant of a circle, then one-fourth of its
circumference would be equal to the sum of the two
radii A E and A a, or to the diameter, which is im-
possible. Or, again, if the line a b c d E were a straight
line, it might be proved by the same reasoning that

THE PHILOSOPHY OF MATHEMATICS 175

the hypothenuse of the right-angled triangle A a E
is equal to the sum. of its other two sides, which is a
manifest absurdity.

The truth is, that the sum of the circumscribed or
of the inscribed parallelograms will never become
equal to the curvilinear figure. No possible increase
of the number of parallelograms can ever reduce their
sum to an equality with the curvilinear space. What,
then, shall we say to the above demonstrations ? Or
rather to the demonstration of the first lemma, on
which all the others depend ? I do not know that any
one has ever directly assailed this demonstration ; but,
unless I am very grievously mistaken, its inherent
fallacy may be rendered perfectly obvious. It may
be refuted, not only by the reductio ad absurdum, or
by showing the false conclusions to which it necessarily
leads, but by pointing out the inherent defect of its
logic.

The attempt is made to prove that the sum of the
circumscribed parallelograms will ultimately become
equal to the sum of the inscribed parallelograms.
Now it is evident that the difference of these sums is
a variable quantity which may be made as small as
we please. This is, indeed, one of the suppositions
of the case the circumscribed and the inscribed figures
are supposed to vary continually, and to " approach
nearer the one to the other than by any given differ-
ence." Of course, then, they can by this hypothesis
be made to " approach nearer to equality than by the
given difference D." If you. deny the two variable
figures to be ultimately equal, says the demonstrator,
" suppose them to be ultimately unequal, or let I) be
their ultimate difference. Therefore they cannot ap-

176 TEE PHILOSOPHY OF MATHEMATICS.

proach nearer to equality than by that difference D^
which is against the supposition." True. If the
difference be supposed to be variable, and then sup-
posed to be constant, the one supposition will, of
course, be against the other. If the difference in
question be a variable, which may be rendered less
than any given difference, then, of course, it may be
rendered less than the constant quantity D. Hence,
to suppose its ultimate value equal to D, is to contradict
the first supposition or hypothesis. Indeed, according
to that hypothesis, the difference in question has no
ultimate or fixed value whatever. It is, on the con-
trary, always a variable, and its limit is not D, nor
any other magnitude but zero. To say, then, that Its
ultimate value is equal to the constant quantity D, is
clearly to contradict the supposition that it is always
a variable which may be made to approach as near as
we please to zero. But is not that a very precarious
and unsatisfactory sort of demonstration which sets
out with two contradictory suppositions, and then con-
cludes by showing that the one supposition contradicts
the other ?

Let us apply this sort of demonstration to another
case. If a quantity be reduced, by repeated opera-
tions, to one-half of its former value, its successive
values may be represented by 1, J, J, J, and so on,
ad infinitum. By repeating the process sufficiently far,
it may be made less than any given quantity, or it
may be made to approach as near as we please to zero.
But will it ever become zero or nothing ? Is the half
of something, no matter how small, ever exactly equal
to nothing ? No one will answer this question in the
affirmative. And yet, if the above reasoning be cor-

TSJS PHILOSOPHY OF MATHEMATICS. 177

rect, it may be demonstrated that a quantity may be
divided until its half becomes equal to nothing. For,
by repeating the process ad libitum, it may be supposed
to " approach nearer to zero than by a given differ-
ence." Hence it will ultimately become equal to zero.
" If you deny it, suppose it be ultimately unequal
[to zero], and let D be its ultimate difference [from
zero]. Therefore it cannot approach nearer to equality
[with zero] than by the given difference D, which is
against the supposition." But if it be not unequal, it
must be equal to zero or nothing. That is, the ulti-
mate half of something is exactly equal to nothing ;
Q. E. D.

In his first attack on the reasoning of Sir Isaac
Newton, contained in " The Analyst," Bishop Berkeley
did not notice the above demonstration of the first
lemma of the first book of the Principia. Jurin, his
antagonist, complained of this neglect, and Berkeley
replied : " As for the above-mentioned lemma, which
you refer to, and which you wish I had consulted
sooner, both for my own sake and for yours, I tell you
I had long since consulted and considered it. But I
very much doubt whether you have sufficiently con-
sidered that lemma, its demonstration, and its conse-
quences." He then proceeds to point out one of these
consequences, which seems absolutely fatal to Sir
Isaac Newton's view of his own method. " For a
fluxionist," says he, " writing about nioxnenturns, to
argue that quantities must be equal because they have no
assignable difference, seems the most injudicious step
that could be taken ; it is directly demolishing the very
doctrine you would defend. For it will thence follow
that all homogeneous momentums are equal, and con-
's*

178 THE PHIL OSOPHY OF MA THEM A TIGS.

sequently the velocities, mutations ; or fluxions, propor-
tional thereto, are likewise equal. There is, therefore,
only ane proportion of equality throughout ; which at
once overthrows the whole system you undertake to
defend," ^ This objection appears absolutely unan-
swerable. For if all quantities, which "during any
finite time constantly approach each other, and before
the end of thiat time approach nearer than any given
difference ^ are ultimately equal," then are all indefi-
nitely small quantities ultimately equal, since they all
approach each other in value according to the hypothe-
sis. Tint is to say, as zero is the common limit toward
which tliej all continually converge, so they continu-
ally converge toward each other, and may be made to
" approach, nearer the one to the other than by any
given difference." If, then, it follows from this that
they are all c ' ultimately equal," " there is only one
proportion of equality throughout," and the whole
fabric of the infinitesimal analysis tumbles to the
ground. IFor this fabric results from the fact that, in-
stead of one uniform proportion, there is an infinite
variety of ratios among indefinitely small quantities.
If these veie ultimately equal, then their ultimate ratio
would arrays be equal to unity. But instead of
always tending toward unity, the ratio of two indefi-
nitely small quantities may, as every mathematician
knows, tend toward any value between the extreme
limits zero and infinity.

The objections of Berkeley, not to the method of
Newton y bmt to Newton's view or exposition of his
method, have never been satisfactorily answered.
" The A.nalijst was answered by Jurin," says Play fair,

* A. Defence cf Free Thinking in Mathematics, XXXII.

THE PHILOSOPHY OF MATHEMATICS. 179

" under the signature of Pliilalethes Cantabrigiensis,
and to this Berkeley replied in a tract entitled A De-
fence of Free TJilnldng in Mathematics. Replies were
again made to this, so that the argument assumed the
form of a regular controversy ; in which, though the
be acknowledged that they did not always argue the
matter quite fairly, nor exactly meet the reasoning of
their adversary." * This is the judgment of the ma-
thematician, not of the historian or the philosopher.
ISTo one, it seems to me, ever argued any question of
science more intemperately or more unfairly than
Jurin did in his reply to Berkeley. But it is not my
design to enter, at present, into the merits of this con-
troversy. I merely wish to quote Berkeley's experi-
ence among men, which so nearly coincides with my
own among books. " Believe me, sir," said he to
Philalethes, " I had long and maturely considered the
principles of the modern analysis before I ventured to
publish my thoughts thereupon in the Analyst. And,
since the publication thereof, I have myself freely con-
versed with mathematicians of all ranks, and some of
to be informed of the opinions of others, being very
desirous to hear what could be said towards clearing
my difficulties or answering my objections. But
though you are not afraid or ashamed to represent the
analysts as very clear and uniform in their conception
of these matters, yet I do solemnly affirm (and several
of themselves know it to be true) that I found no har-
mony or agreement among them, but the reverse thereof,
the greatest dissonance and even contrariety of opinions,

* Progress of Mathematical and Physical Science, Part II., Sec. 1.

180 THE PHILOSOPHY OF MATHEMATICS.

employed to explain what after all seemed inexplicable.
Some fly to proportions between nothings. Some reject
quantities because infinitesimal. Others allow only
finite quantities, and reject them because inconsider-
able. Others place the method of fluxions on a footing
with that of exhaustion, and admit nothing new therein.
Some maintain the clear conception of fluxions. Others
hold they can demonstrate about things incomprehen-
sible. Some would prove the algorithm of fluxions by
reductio ad absurdwn, others d priori. Some hold the
evanescent increments to be real quantities, some to be
nothings, some to be limits. As many men, as many
minds ; each differing from one another, and all from
Sir Isaac Newton. Some plead inaccurate expressions
in the great author, whereby they would draw him to
speak their sense; not considering that if he meant as
they do, he could not want words to express his mean-
ing. Others are magisterial and positive, say they are
satisfied, and that is all; not considering that we, who
deny Sir Isaac Newton's authority, shall not submit
to that of his disciples. Some insist that the conclu-
sions are true, and therefore the principles, not con-
sidering what hath been largely said in the Analyst
on that head. Lastly, several (and these none of the
meanest) frankly owned the objections to be unan-
swerable. All wliich I mention by way of antidote
to your false colors, and that the unprejudiced inquirer
after truth may see it is not without foundation that
I call on the celebrated mathematicians of the present
age to clear up these obscure analytics, and concur in
giving to the public some consistent and intelligible
account of their great master, which, if they do not, I

THE PHILOSOPHY OF MATHEMATICS. 181

believe the world will take it for granted that they
cannot." *

More than one champion entered the lists against
Berkeley. Besides Philalethes Cantabrigiensis, or Jurin,
another eminent mathematician, Mr. Robins, pub-
lished replies to both of the papers of the celebrated
Bishop of Cloyne. But, unfortunately, in attempting
to re-demonstrate the demonstrations of Newton, and
clear away every obscurity from his method, the two
disciples, instead of demolishing Berkeley, got into an
animated controversy about the meaning of the great
master. Newton, as understood by Jurin, was utterly
unintelligible or false in the estimation of Robins, and,
as interpreted by Robins, he was vehemently repudi-
ated by Jurin. Now this disagreement respecting the
true interpretation of Newton's interpretation of his
own method is well stated by Mr. Robins.

" It was urged," says he, " that the quantities or
ratios, asserted in this method to be ultimately equal, ^

were frequently such as could never absolutely coin- yi

cide. As, for instance, the parallelograms inscribed
within the curve, in the second lemma of the first book
of Sir Isaac Newton's Prineipia, cannot by any divi-
sion be made equal to the curvilinear space they are
inscribed in, whereas in that lemma it is asserted that
they are ultimately equal to that space."

" Here," says he, " two different methods of expla-
nation have been given. The first, supposing that by
ultimate equality a real assignable coincidence is in-
tended, asserts that these parallelograms and the curvi-
linear space do become actually, perfectly, and abso-
lutely equal to each other." This was the view of

~ A Defence of Free Thinking in Mathematics, XLIII. and XLIV.
10

182 THE PHIL OSOPHY OF MA THEM A TICS.

, and it seems difficult to understand how any
could arrive at any other conclusion. Newton
iumself, as we have seen, expressly asserts that the
** parallelograms will in oil parts coincide with the
Curvilinear figure." But Mr. Bobins, in his explana-
tion., understands Newton to mean that they will not
Coincide. Newton asserts, apparently as plainly as
language could enable him to do so, " the coincidence
of the variable quantity and its limits/' and yet the
clisciple denies, in the name of the master, the reality
of any such coincidence. Newton declares that the
variable becomes " ultimately equal" to its limit, and
yet Mr. Robins insists that he must have seen they
"Would always remain unequal. Now is this to inter-
pret, or simply to contradict, Sir Isaac Newton's ex-
planation of his own method? No one could possibly
entertain a doubt respecting the meaning of Mr.
IRoblns. If Newton had meant unequal, could he not
liave said so just as well as Mr. Robins, instead of
saying equal ? Or, if he did not believe in " the coin-
old enee of the variable and its limit," could he not
lia,ve denied that coincidence just as clearly as he has
asserted it? It is certain that from Jurin to Whe-
\vell, and from Whewell to the present mathematicians
of Cambridge, Newton has generally been understood
to contend for an ultimate equality between the vari-
able quantity and its limit. Thus, in expounding the
cLoctrine of Newton, which he adopts as his own, Dr.
"\Vhewell says: "A magnitude is said to be ultimately
equal to its limit, and the two are said to be ultimately
??, a, ratio of cqitality. A line or figure ultimately coin-
cides with the line or figure which is its limit." * The

* Doctrine of Limits, Book II., Definitions and Axioms.

THE PHILOSOPHY OF MATHEMATICS. 183

same view, as we have already seen, is also taken by
Mr. Todhunter in his Differential Calculus. It is, in
fact, the doctrine and the teaching of Cambridge to the
present hour, in spite of all the obscurities, difficulties,
doubts, and objections by which it has never ceased
to be surrounded, to say nothing of the demonstrations
by which it may be refuted.

The views of Mr. Robins respecting the method of
limits appear perfectly just, as far as they go ; yet
nothing, it seems to me, could be more ineffectual than
his attempt to deduce these views from the Principia.
The author of that treatise, says he, " has given such
an interpretation of this method as did no ways re-
quire any such coincidence [between the ultimate form
of the variable and its limit]. In his explication of
this doctrine of prime and ultimate ratios he defines
the ultimate magnitude of any varying quantity to be
the limit of that varying quantity which it can ap-
proach within any degree of nearness, and yet can
never pass. And in like manner the ultimate ratio
of any varying ratio is the limit of that varying
ratio." * Now this fails to make out his case. For
the " ultimate magnitude of any varying quantity" is
one of the magnitudes of that quantity, and if that
magnitude is its limit, then the varying quantity
reaches its limit. Nor is this all. Mr. Robins has
suppressed an important clause in the definition of
Newton. Newton says: " These ultimate ratios with
which quantities vanish are not truly the ratios of
ultimate quantities, but limits towards which the ratios
of quantities decreasing without limit do always con-
verge, and to which they approach nearer than by any

* Eeview of Objections to the Doctrine of Ultimate Proportions.

184 THE PHIL OSOPHY OF MA THEM A TICS.

given difference, but never go beyond, nor in effect
attain to, till the quantities are diminished in infinitum." *
Now here, in the definition of Newton as given by
himself, it is said, that the varying quantity in its
ultimate form attains to its limit. It was reserved
for a later age to establish the truth, that a varying
quantity is never equal to, or coincides with, its limit;
a truth which, as we shall presently see, dispels all
the obscurities of Newton's method, and places that
method on a clear, logical, and immutable basis.

It is, indeed, exceedingly difficult to believe that
Newton intended, by his demonstration, to establish
an ultimate equality or coincidence between the paral-
lelograms and curvilinear spaces of Lemmas II. and
III. ; because such an equality or coincidence seems so
utterly impossible. This was the great difficulty with
Mr. Robins ; rather than believe such a thing of New-
ton, he would explain away the obvious sense of his
most explicit statements. But even at the present day,,
after two centuries of progress in the development of
the calculus and in the perfecting of its principles, the
demonstration of the same paradox is frequently at-
tempted by mathematicians of the highest rank.
This demonstration is worthy of examination, not only
on its own account, but also on account of the light
which it throws on the operations of Newton's mind,
as well as on several passages in the Prineipia. The
demonstration to which I refer is usually found in the
attempt to obtain a general expression or formula for
the differential of a plane area. It is thus given in a
very able and learned work on the Differential and
Integral Calculus:

* Principia, Book II., Section I., Scholium.

THE PHILOSOPHY OF MATHEMATICS.

185

" Prop. To obtain a general formula for the value
of the plane A B C D, included between the curve
D C, the axis O X, and the two parallel ordinates
A D and B C, the curve being referred to rectanglar
co-ordinates.

" Put O E = x, E P ==y, E F = h, F P' = y f , and
the area A E P D = A.

A E F B X

"Then when x receives an increment fi } the area
takes the corresponding increment E P P 7 F, interme-
diate in value between the rectangle F P and the rect-
angle F S. But

. d 77 ll . d? II Jl* .

v + -* . T H -- - . -- }-) etc..

DFP 2/xA y

^ . dy Ji . d*y 1

= 1 H -- - . ~ + = -

,
, etc..

; '

Hence at the limit, when li is indefinitely small, the
area E P P' F, which is always intermediate in value
between F P and F S, must become equal to each of
these rectangles or equal to y X h.

1 6 *

186 ^E FSILOSOPHY OF MATHEMATICS.

in rectan S les F P, FS, and the

mteimed-ate curvdinear space F E P P', are ultimately
equal, or the ultimate ratio of any one of them to the
oher equal to unity. As the same thing is true of
II aniilar parallelograms and the intermediate curvi-
Imear space, so the sum of these parallelograms or
rectangles , 8 equal to the curvilinear space 1 B CD,
Jose value is sough , H ence A =fy dx , the sum '
of all the mscnbed rectangles, such as E F E P Thus

the

as

e ud un

That it M C mm0n baS6 ^ is reduced to 0-

F F P P/ ] * ^ The curvilinear space

Courtenay'a Calculus, p. 330.

THE PHILOSOPHY OF MATHEMATICS. 187

- . d y h . d z if Ji? .

- . - H - . --

. .

flx 1 da? 12y

and this ratio becomes = 1 only when all three areas
vanish, or become identical with the right line F P 3
in consequence of making k = 0. Hence, instead of
proving that the rectangle F S is ever equal to the

I I TT 1 S

rectangle F P. so that = - = 1. the author has only
5 ' DFP ' m J

proved that the right line F P is equal to itself,

F P

F P, so that - = 1 ; a proposition which surely
F P

needed no proof.

But see how adroitly the reasoning is managed.
"Hence at the limit," says the author, "when h is
indefinitely small, the area E P P' F, which is always
intermediate in value between F P and F S ; must be-
come equal to each of these rectangles." Not at all.
It is only when 7i = 0, as we haye just seen in the
preceding line, that the three areas vanish and become
equal to the right line F P. Thus h is made = 0, in
order to prove that the rectangles F S and F P are
equal to each other, and to the curvilinear space
F E P P'. But how will you take the sum of such
rectangles ? How will you take the sum of rectangle's
whose variable altitude is y, and whose base is 0?
Or, in other words, how will you take the sum of
right lines so as to make up an area? The truth is,
as we have seen, that as Ji becomes smaller and smaller,
the rectangles, such as F S and F P, become less and
less in size, and greater and greater in number. Hence
at the limit, when 7t = 0, the rectangles vanish into
right lines, and the number of these linos becomes

188 THE PHILOSOPHY OF MATHEMATICS.

= oo . To take the sum of such rectangles, then, is
only to take the sum of right lines, which throws us
back two centuries, and buries us in the everlasting
quagmire of the method of ^indivisibles.

But the author escapes this difficulty. He makes
f( Ji = 0" in one line, or absolutely nothing, so that all
quantities multiplied by it vanish, and, in the very
next line, he makes h equal to an " indefinitely small"
quantity. This very convenient ambiguity is, indeed,
the logical artifice by which the difficulties of the cal-
culus are usually dodged. In order to evade these
difficulties nothing is more common, in fact, than to
make A = on one side of an equation, and, at the
same time, to make it an " indefinitely small quantity"
on the other side of the same equation. The calculus
before us, as well as some others, is really replete
with sophisms proceeding from the same prolific am-
biguity.

This ambiguity in the ultimate value of li, or in the
method of passing to the limit of the rectangles in
question, is patent and palpable in the above demon-
stration. It is latent and concealed in the demonstra-
tion of Newton. Neither he, nor Cavalieri, nor Robins,
nor Courtenay, nor any other man, could be made to
believe or imagine that the sum of any inscribed paral-
lelogram whatever could be equal to the circum-
scribed curvilinear space, unless some such ambiguity,
either hidden or expressed, had first obscured the
clearness of his mental vision. It is evident, indeed,
from the language of Newton himself, that he failed,
in the demonstration of his lemmas, to effect an escape
from, the conception of indivisibles. It was to effect
such an escape, as he tells us, that he demonstrated

TSE PHIL QSOPHY OF MA THEM A TICS. 189

the lemmas in question, " because the hypothesis of
indivisibles seems somewhat harsh." * But, after all,
it is clear, upon close scrutiny, that his escape from
that hypothesis was far from perfect. Thus, in the
fourth corollary to the third lemma, he tells us, that
" these ultimate figures (as to their perimeters a e E)
are not rectilinear, but curvilinear limits of rectilinear
figures." That is to say, the ultimate form of the
"evanescent parallelograms" (Cor. 1), or of the in-
scribed polygon (Cor. .2), or of the circumscribed poly-
gon (Cor. 3), is not a rectilinear figure, but " the curvi-
linear limit" of such a figure. Now, how can the
ultimate form of a polygon be a a curvilinear limit"
or figure. It becomes so, says Newton, when the sides
of the polygon are " diminished in infinitum." But,
surely, as long as its sides remain right lines it does
not become a curvilinear figure. It is only when its
sides have been " diminished in infaiitum" or ceased to
be right lines, that the polygon can be conceived as
coincident with a curvilinear figure. But is not this
to divide the sides, or to conceive them to be divided,
until they can no longer be divided ? Is not this, in
other words, to fall back on the conception of indivisibles
on the "somewhat harsh hypothesis" of Cavalieri?
And has not the author of the Principia, in spite of
his efforts, failed to extricate his feet at least from the
entanglements of that method? Indeed, it seems
utterly impossible for the human mind to escape from
that method until it abandons the false principle, and
the false demonstrations of the principle, that parallelo-
grams, or polygons, or any other rectilinear figures
whatever, can, by any continual division and subdi-

* Principia, Book I., Section I., Scholium.

190

THE PHILOSOPHY OF MATHEMATICS.

vision, be made to coincide with a curvilinear space.
The thing itself is impossible, and can only be con-
ceived by means of " the absurd hypothesis" of indi-
visibles, as it is called by Carnot.

It is generally, if not universally, asserted by
writers on the theory of the calculus, that the method
of limits is free from the logical fault of a compensa-
tion of errors ; in which respect it is supposed to pos-
sess a decided advantage over the method of Leibnitz.
But this is far from being always the case. If, for
example, we suppose with Sir Isaac Newton, or with
Mr. Courtenay, that the inscribed rectangle, the cir-
cumscribed rectangle, and the intermediate curvilinear
space are ultimately equal to each other, we can, in
many cases, reach an exact conclusion only by means
of a compensation of errors. In order to show this,
let us resume the above general formula : A =Jy dx 9
which signifies that the curvilinear area A is equal to
the sum of all the ultimate rectangles ydx. Now,
for the sake of clearness, let us apply this formula to
the parabolic area O B C, whose vertex corresponds*

C
7,

e
&

*

&

TV

^
*

/

ft,

/
A

B

X

with the origin of co-ordinates 0. Conceive the ab-
scissa O B to be divided into any number of equal
parts, and let each of these parts be denoted by A.
Complete the system of circumscribed rectangles as in

TEE PHILOSOPHY OF MATHER A TICS. 191

the figure. Now it is evident that tlie sura, of those
rectangles is greater than the parabolic area OBC,
and will continue to be greater, however their number
may be increased or their size diminished, provided
only that they do not cease to be rectangles. The
measure of one of these rectangles in its last form is
measured, as we have seen, by y d x, and the whole
area O B C, supposed equal to the sum of these rect-
angles, is fy dx. Now this sum, or fydx, is, I say,
greater than the parabolic area in question. This may
be easily shown.

From the equation of the parabola y 2 = 2jp#, wo
obtain, by differentiation, y 2 + 2y dy + dy* = 2 p x
+ 2pdx, or 2 y dy + dy* = %p d x. Hence

p

By substituting this value of d x in the above formula,
we have the area of

OBC,orA- C(l^

J\ P

Now this is the exact value of the sum of all the in-
definitely small circumscribed rectangles. But it is
greater than the parabolic area OBC; for the first

term above, or C^ c '^- 9 is exactly equal to that area.
+/ P

T? v, Clfdy y s %P x y % .1 n i

Jb or i + -2. = sL. = u, \$ = _ # y the well-known

J p 3p 3p 3 J?
value of the parabolic area OBC.

Now the sum of the parallelograms was made up of

two parts, namely, of C^Jj and of C yd ' f \ The
J p J 2p

192 THE PHILOSOPHY OF MATHEMATICS.

first part alone, C- ^, is exactly equal to the area

O B C ; and, consequently, the part J - , which was

thrown away, must have been exactly equal to the
sum of the little mixtilinear triangles O a b, bod,
d efy etc., by which the sum of the rectangles exceeded
the area of O B C. Hence the exact result O B C =

2

- x y } was obtained by a compensation of errors ; the

3

excess of the sum of the rectangles over the area of

O B C being corrected by the rejection of J ^~^-

/ 2 J9

as nothing. Thus, the method of Newton is not always
free from a secret compensation of errors; a logical
defect which has always been supposed to be exclu-
sively confined to the method of Leibnitz.

The reason of this is, that Newton frequently mixed
up the fundamental conceptions of Leibnitz with his
own clearer principles, and, consequently, failed to
emancipate his method from their darkening influence.
This is evident from the case above considered. In
the method of Leibnitz it is taken for granted that the
rectangle F EP R [Fig. p. 185] may be taken for the
curvilinear space F E P P' ; because they differ from
each other only by the infinitely small quantity P P ; R,
which makes really no difference at all. This is, in
fact, one of the equalities which is specified in the first
postulate of the Marquis de L'H6pital, as we saw
in the last chapter of these reflections. Newton does
not take this equality for granted, but he attempts to
demonstrate it. But no reasoning can ever prove that

THE PHILOSOPHY OF MATHEMATICS. 193

the rectangle FP, however small, is equal to the
curvilinear space F E P P'; even Newton, as we have
seen, failed in his attempt to demonstrate such an im-
possibility. Leibnitz should have said, I commit a
small error in the formation of my equation by taking
F P for F E P P'; but then I will correct this error
by rejecting from my equation certain small quantities;
for this is, in fact, precisely what he did. Newton, in
like manner, should have said, I put IT P in the place
of F E P P 7 , not because they are equal, or can ever
become so, but because they have the same limit ; and,
consequently, in passing to the limit, the same precise
result will be obtained whether the one quantity or
the other be used ; for this is exactly what he did.
But, instead of saying so, or confining their language
to the real processes of their methods, both proceeded
on the false conception that the infinitely small rect-
angle F P is exactly equal to the curvilinear space
F E P P'. The only difference between them was,
that Leibnitz predicated this equality of the two
figures when they were infinitely small, and Newton
when they had reached their ultimate form or value.
Hence in the one system, as in the other, the exact
result was obtained by means of an unsuspected com-
pensation of unsuspected errors.

Again, Sir Isaac Newton wished to avoid, as much \\

as possible, the use of infinitely small quantities in \{

geometry. t( There were some," says Maclaurin, " who t \

disliked the making much use of infinites and infi-
nitesimals in geometry. Of this number was Sir Isaac
Newton (whose caution was as distinguishing a part
of his character as his invention), especially after he
saw that this liberty was growing to so great a
17

194 THE PHILOSOPHY OF MATHEMATICS.

height. 37 * Maclaurin himself entertained the opinion
that " the supposition of an infinitely little magnitude"
is "too bold a postulatum for such a science as geome-
try," f and hence he commends the caution of Newton
in abstaining from the use of such quantities. Indeed,
Newton himself says, " Since we have no ideas of infi-
nitely little quantities, he introduced fluxions, that he
might proceed by finite quantities as much as possi-
ble." J But while he clung to the hypothesis, or
notion, that the variable ultimately coincides with its
limit, he found it impossible to avoid the use of such
quantities, or else something even more obscure and
unintelligible. Thus, as we have seen, he divided the
sides of his inscribed and circumscribed variable poly-
gons until he made them coincide with the limiting
curve. Now, did not this make their sides infinitely
small, or something less? Did it not, in fact, reduce
them to indivisibles or to points ? And if so, did not
their length become infinitely small before it became
nothing ?

Nor is this all. For he says, " Perhaps it may be
objected that there is no ultimate proportion of evan-
escent quantities, because the proportion before the
qualities have vanished is not ultimate, and when
they are vanished, is none. But by the same argu-
ment, it may be alleged, that a body arriving at a
place, and then stopping, has no ultimate velocity,
because the velocity, before the body comes to the
place, is not ultimate; when it has arrived, is none.

* Introduction to Maelnurin's Fluxions, p. 2.
f Preface to Fluxions, p. iv.

% Philosophical Transactions, "So. 342, p. 205 \$ Eobins' Mathemati-
cal Tracts, Vol. II.. p. 96.

THE PHILOSOPHY OF MATHEMATICS. 195

But the answer is easy, for by the ultimate velocity is
meant that with which the body is moved, neither
before it arrives at its last place and the motion ceases,
nor after, but at the very instant it arrives ; that is,
that velocity with which the body arrives at its last
place and with which the motion ceases. And in like
manner, by the ultimate ratio of evanescent quantities
is to be understood the ratio of the quantities not
before they vanish, nor afterwards, but with which
they vanish. In like manner the first ratio of nascent
quantities is that with which they begin to be. And
the first or last sum is that with which they begin or
cease to be (or to be augmented or diminished). There
is a limit which the velocity at the end of the motion
may attain, but not exceed. This is the ultimate
velocity. And there is the like limit in all quantities
and proportions that begin and cease to be." * Thus,
the ultimate, ratio of quantities, as considered by Newton,
is the ratio, not of quantities before they have vanished,
nor after they have vanished, but of somethings some-
where between something and nothing. These some-
things, which exist somewhere in that intermediate
state, is what Bishop Berkeley has ventured to call
" the ghosts of departed quantities." The ultimate
ratio of two rectangles, for example, is their ratio,
neither before nor after they have ceased to be rect-
angles, but while they are somewhere and something
between rectangles and right lines. There may be,
if you please, such things as such ultimate velocities
or departed quantities. But, if introduced into the
domain of mathematical science, will they not bring
with them more of obscurity than of light ?

* Principia, Book I., Section I., Scholium. f ;

i

i

196

THE PHILOSOPHY OF MATHEMATICS.

" D'Alembert," says Carnot, " rejected this explica-
tion, though he completely adopted in other respects
the doctrine of Newton concerning the limits or first
and last ratios of quantities." * And Lagrange said,
" That method has the great inconvenience of con-
sidering quantities in the state in which they cease, so
to speak, to be quantities ; for though we can always
well conceive the ratio of two quantities, as long as
they remain finite, that ratio offers to the mind no
clear and precise idea, as soon as its terms become the
one and the other nothing at the same time." It may
be doubted, then, whether Newton gained anything in
clearness and precision by the rejection of infinitely
small quantities, and the invention of ultimate ones.

In order to take a complete view of Newton's
method, it will be necessary to consider a few more
of his lemmas, and also the object for which such dark
and difficult things are demonstrated. I shall, then,
begin with

LEMMA IV.

"If in two figures AacE, Pp r T, you inscribe
(as before) two ranks of parallelograms, an equal num-

ber in each rank, and when their breadths are dimin-
ished in infinitum, the ultimate ratios of the parallel o-

f Metaphysique, etc., Chap. III., p. 182.

THE PHILOSOPHY OF MATHEMATICS. 197

grams in one figure to those in the other, each to each
respectively, are the same ; I say, that these two figures
A a c E, P p r T, are to one another in that same ratio.

" For as the parallelograms in the one are severally
to the parallelograms in the other, so (by composition)
is the sum of all in the one to the sum of all in the
other, and so the one figure to the other ; because (by
Lemma III.) the former figure to the former sum,
and the latter figure to the latter sum, are both in the
ratio of equality. Q. E. D."

Now this demonstration, it will be perceived, pro-
ceeds on the principle that the inscribed parallelo-
grams exactly coincide with the circumscribed curvi-
linear figure, and if this coincidence were not perfect
then the demonstration would be defective. This
proposition alone is, then, sufficient to show that New-
ton contended for what his words so clearly express;
namely, that the inscribed parallelograms, in their
ultimate form, really and rigidly coincide with the
circumscribed figure. This may be very difficult to
believe, but it is, nevertheless, absolutely demanded
by his demonstration of the fourth lemma, as well as
by his express words. Perhaps such a thing could
not have been believed by any one previously to the in-
troduction of indivisibles, and the darkness which the
overstrained notions of that method introduced into
the minds of the mathematical world. It is certain
that if Euclid or Archimedes could have believed in
such a coincidence between rectilinear and curvilinear
figures, they would have had no occasion to abandon
the principle of supposition, and invent or adopt the
method of exhaustion in order to ascertain the measure
of curvilinear areas.

17*

193 THE PHILOSOPHY OF MATHEMATICS

I have, it may be remembered, demonstrated in a
perfectly clear and unexceptional manner a proposi-
tion similar to the above lemma, without supposing the
variable to reach or coincide with its limit. That is
to say, I have shown that if two variables always
have the same ratio to each other, then, although they
never reach their limits, yet will these limits be in the
same ratio. This proposition, which entirely eschews
and shuns the strained notion that a variable ulti-
mately coincides with its limit, will be found to answer
all the purposes of the fourth lemma of Newton. Even
if that strained notion were true, and could be demon-
strated, it would add nothing but a very unnecessary
obscurity to the demonstrations of the method of
limits. But Newton, as we have seen, has failed to
demonstrate that strained notion, that first and funda-
mental conception of his method. In his attempt to
do so he has, as we have seen, only shown a contra-
diction between two contradictory suppositions. That
conception should, then, it seems to me, be for ever
banished from the domain of mathematical science, as
having perplexed, darkened, and confounded the other-
wise transcendently beautiful method of limits.

LEMMA VI.*

" If any arc AGE, given in position is subtended
by its chord A B, and in any point A, in the middle
of the continued curvature, is touched by a right line

* The fifth lemma is in these words : "In similar figures, all sorts
of homologous sides, whether curvilinear or rectilinear, are propor-
tional, and the areas are in the duplicate ratio of the homologous
sides." It is without a demonstration ; a simple enunciation is all
that the author deemed necessary.

THE PHILOSOPHY OF MATHEMATICS.

199

A. D, produced both ways ; then if the points A and
B approach one another

and meet, I say, the angle _
the chord and the tangent,
will be diminished in infi-
nitum, and will ultimately
vanish.

" For if the angle does
not vanish, the arc A C B
will contain with the tan-
gent A D an angle equal to a rectilinear angle, and
therefore the curvature at the point A will not be con-
tinued, which is against the supposition."

Now this demonstration is merely preliminary to
those which follow. The seventh lemma is in these
words : " The same things being supposed, I say that
the ultimate ratio of the arc, chord, and tangent, any
one to any other, is the ratio of equality." Now this
proposition is demonstrated in order to establish the
practical conclusion, that " in all our reasoning about
ultimate ratios, we may freely use any one of these
lines for any other." [See Cor. III.]

LEMMA VIII.

" If the right lines A E, BE, with the arc A C B,
the chord A B, and the tangent A D, constitute three
triangles E A B, E A C B, E A D, and the point A
and B approach and meet ; I say that the ultimate
form of these evanescent triangles is that of similitude,
and their ultimate ratio that of equality." Now this
lemma is demonstrated, like the last, to establish the
conclusion, that " in all our reasonings about ultimate

200 THE PHIL OSOPHY OF MA THEMA TICS.

ratios we may indifferently use any one of these tri-
angles for any other." [See Cor.] That is to say,
it is concluded that any one of these triangles may
be used for any other; because it has been demon-
strated that they are " ultimately both similar and
equal among themselves."

In this eighth lemma the " ultimate form" of these
several " triangles" is in a single point. Now what,
I would ask, is this " ultimate form?" Perhaps it is
no form at all ; perhaps it is without form and void.
It is certainly contained in a point which has neither
length , breadth, nor thickness. It is not the form of
a triangle, for if it were, it would then be a triangle,
and could not be inscribed in a point. Or, if it were
the form of a triangle, it would then be a triangle that
had not vanished, which is contrary to the very defi-
nition of an " ultimate triangle." Nor is it the form
of a triangle after it has vanished, for then it is nothing,
and has no form. What, then, is this " ultimate
form" of a triangle ? It is not, we are told, a triangle
either before or after it has vanished, but while it is
in the act of vanishing. With what form, then, does
a triangle vanish ? Certainly not with the form of a
triangle, for then, it would still be a triangle, which is
contrary to the definition. Nor with the form of a
point, for then it has ceased to be a triangle, which is
likewise contrary to the definition. Must I conclude,
then, that this " ultimate form" is some unknown
form between that of a triangle and a point? It is
certain that I can no more conceive of " this ultimate
form of the three triangles" which are no longer
triangles before they have vanished, than I can of the
ultimate form of the parallelograms, which, in Lemma

THE PHILOSOPHY OF MATHEMATICS.

201

m

II., are supposed to coincide with the curvilinear
space A a E.

Now all these demonstrations are just as unnecessary
as they are obscure. The sum of the inscribed and
the sum of the circumscribed parallelograms in Lemma
II. are never equal, and all that it is necessary to say
is, not that they are equal, but that they have the
same limit A a E. This is perfectly obvious, and to
go beyond this is a supererogation of darkness and
error. Take, for example, the system of circumscribed
parallelograms, A Z, B m, C n, Do, etc., and if we
obtain an expression for their
sum, we shall find it to consist
of two terms. The one will be
constant, and stand for the in-
variable part of the sum, namely,
the area A a E ; the other will
be variable, and represent the
variable portion of that sum, n
namely, the sum of all the little mixtilinear triangles
a b /, b G m, etc., which is the variable excess of the
parallelograms over the constant area A a E. Hence,
if the variable term which represents the sum of these
little triangles be rejected, the exact area A a E will
be obtained, and this is precisely what is done in pass-
ing to the limit of the expression for the sum of the
parallelograms. Now all this is perfectly plain and
palpable. Hence, if the author had been content to
say that the sum of the parallelograms is never equal
to the area A a E, but that this area is the limit of that
sum, then his method would have been as transparent
and easy of comprehension as it is now dark and diffi-
cult to be apprehended. He saw that in the practical

202 THE PHILOSOPHY OF MATHEMATICS.

application of the calculus it was necessary to use
indifferently the sum of the inscribed and the sum of
circumscribed rectangles for one another, or for the
curvilinear space A a E ; but he justified this procedure
on the wrong ground. He justified it on the ground
that; they were all ultimately equal ; whereas he should
have done so on the ground that the variable sums,
though never equal, have the same limit. This prin-
ciple, which is so clear in the case before us, is general.
For it is evident that " the limit of the sum of infi-
nitely small positive quantities is not changed when
these quantities are replaced by others whose ratios
with them have respectively unity for their limit/'
But this general principle is, if possible, rendered still
more evident by a very short and easy demonstration in
Duhaniel 7 s work.*

The same thing is true in regard to the substitution
of the chord, arc, and tangent for each other in the
application of the calculus whenever such substitution
answers the purpose of the operator. Newton justifies
this substitution on the ground that these several
quantities are all ultimately equal ; but yet, as long as
the arc has any value at all, it is greater than its chord
and less than its tangent. Newton saw this, and hence,
instead of stopping with Leibnitz, who pronounced
these lines equal when they were infinitely small,
he followed them down still further, and pronounced
them equal after they had passed the bounds of the
infinitely small, and ceased to have any magnitude
whatever. But this view, as Lagrange said, has the
great disadvantage of requiring us to consider quan-
tities in the state in which they have ceased to be

Yol. I., Chap. YL, p. 35.

THE PHILOSOPHY OF MATHEMATICS. 203

quantities; and become we know not what. Both
Newton and Leibnitz, however, agreed to justify the
using of " any one of these quantities for any other,"
on the ground that they became equal. The chord,
the arc, and the tangent are coincident and equal
when infinitely small; and hence, in seeking their
ratios, they may be indifferently used the one for the
other. The chord, the arc, and the tangent, said
Newton, are all ultimately coincident and equal ; and
hence, "in all our reasoning about ultimate ratios, we
may freely use any one of these lines for any other."
But if we justify this substitution, or convertibility, on
the true ground, every possible obscurity will vanish
from the process, and Newton himself, if alive, might
well exclaim, " Behold my theory, or method, resumed
with more of clearness and precision than I myself
could put into it ! " *

This true ground is thus stated and demonstrated
by Duhamel :

" SECOND THEOREM. The limit of the ratio of two
quantities indefinitely small is not changed when we re-
place these quantities by others which are not equal, but
of which the ratios with them have unity for their limits.

" Let there be, in fact, two indefinitely small quan-
tities a and /9, a 1 and /?' two other quantities such that

tlia limits of and of may be equal to unity, and

that, consequently, the limits of the inverse ratios

a 1 B f

, t ~- may also be equal to unity ; we shall have iden-

/9

tically

* The exclamation. of Carnot when he saw his own theory of the
method of Leibnitz as propounded by Lagrange.

204 THE PHILOSOPHY OF MATHEMATICS.

/J ~"~ /3' * fl * of*

The limits of equal quantities being equal, the limit
of a product being the product of the limits,* we ob-
tain from the above identity, in designating the limits
by the abbreviation Urn., and observing that

Urn. ~ = 1 and Urn. = 1,
a_ r a!

ClWj llttl, .

which it was necessary to demonstrate."

Now the chord, the arc, and the tangent when con-
sidered as small variables, or infinitesimals, exactly
conform to the conditions of this important theorem.

For as every one knows, the limit of

chord

arc

= 1, the

arc

limit of = 1, and the limit of

tangent tangent

Hence, although these lines are not equal, yet, in seek-
ing the limit of their ratios, any one of them may be
freely used for any other j because this, as just clearly
demonstrated, will make no possible difference in the
result.

The same thing is true of the triangles of Lemma
VIII. For, as may be easily seen, the limit of the
ratio of any two of these triangles = 1. Hence, in
seeking the limit of their ratios, " any one of these
triangles may be freely used for any other," since,
according to the above theorem, this will make no

* See Demonstrations in Chap. II.

THE PHILOSOPHY OF MATHEMATICS.

205

difference in the result. We thus get rid of the
desperate difficulty and darkness of conceiving three
triangles to be inscribed in a single point ; and justify
the substitution of any one of them for any other, even
before they have vanished, and while they are still
finite variable magnitudes, on the ground of a perfectly
clear and unexceptionable principle.

I shall, in conclusion, illustrate these three several
modes of viewing the infinitesimal method by an ex-
ample; and I shall select the question of tangency,
since it was the consideration of that question which
led to the creation of the modern analysis. Let it be
required, then, to determine the tangent line at the
point C of the curve A C c. Now, as we know from

*!

Trigonometry, the tangent of the angle C V B, which
the tangent line V C T makes with the axis of #, is

B C
equal to , and this, from the similar triangles

BV ?

TE

C B V and T E C, is equal to i-^. Hence, if we

C E

T E
find the value of , we shall have the tangent of the

C E'

18

fl 206 THE PHILOSOPHY OF MATHEMATICS.

r required angle B V C, and the tangent line V C T

may be constructed or drawn.

The only question is, then, how to find the value

rr\ -p

of the ratio -. Now T E, which is the increment

C E '

a of B C for the tangent line, when A B is made to

I assume the increment B 6, cannot be found from the

I equation of the tangent line, since that line has to be

j determined before its equation can be known. Hence,

I in all three methods, the line o E is substituted for

T E, in order to find the value of the required ratio

1 T TT

. Now, the ground or principle on which this

i C E

| substitution is justified constitutes precisely the differ-

| ence between the methods of Leibnitz, of Newton, arid

of Duhamel.

-Let us suppose, then, that the line b e moves toward
B C, making the lines C E, C c, and C T continually
smaller and smaller. According to Leibnitz, when
the point c approaches infinitely near to C, so that the
arc C G becomes infinitely small, then the chord C c,
the arc C c, and the tangent C T become coincident
and equal, and consequently c E becomes equal to T E.
Hence, he concluded that c E might be freely and
safely substituted or taken for its equal T E. But,
as we have seen, this was an error which was after-
wards corrected by the opposite and equal error com-
mitted by him in throwing out certain infinitely small
quantities as nothings in comparison with other quan-
tities. Thus, although he reached the true result, find-

T "K*

ing the exact value of -, he did so by means of an
\j JBj

unsuspected compensation of unsuspected errors. His

THE PHIL OS 0PM Y OF MA THEM A TICS. 207

two demands, or suppositions, or postulates, or axioms
were false, and yet his conclusions were correct with-
out his ever having seen why or wherefore. Such was
the method of Leibnitz.

Newton rejected the postulates of Leibnitz. He
refused, as Archimedes had done before him, to con-
sider a curve as a polygon of an infinite number of
sides, or to believe in the absolute coincidence of a
curve and right line, however short the two magni-
tudes. Hence, he denied the coincidence of the two
triangles, c C E, T C E, and the mixtilinear interme-
diate one c C E, as long as c C retained any value
whatever. Accordingly, in order to establish an
identity between the three triangles in question, so as
to justify the taking of c E for T E, he expressly in-
sists, in the introduction to his Quadraturam Curvarum,
that the point c shall not stop short of the point C,
but that these two points shall become exactly coin-
cident, or one and the same point. We are thus re-
quired to believe that a point may be considered as a
triangle, or that a triangle may be inscribed in a point
Nay, that three dissimilar triangles then become
" similar and equal when they have reached their ulti-
mate form in one and the same point." Who would
not be glad to be delivered from the necessity of such
a belief or opinion ?

Duhamel abandons the idea of any such equality.
He supposes e E and T E to remain always unequal.
But he still insists, nay, he demonstrates, that e E may
be used instead of T E, in order to find the value of

T TP G E

, because Urn. = 1. This is evident, for as

CE' TE

the point c approaches the point C, it is obvious that |

j

208 THE PHILOSOPHY OF MA TSEMA TICS.

o E and T E become more and more nearly equal, and
their difference T c approaches more and more to an

equality with zero. Hence Urn. = I, and there-

I hi

TP TT

fore in seeking the value of -r-rr> the line G E may be

C E

used for T E.

Indeed, in the case before us all this is perfectly

evident without the aid of any demonstration what-

nn "p
ever. For - , which is always constant, is evidently

C E

IT
the limit of the variable ratio . For as c ap-

CE

preaches C, the variable ratio - approaches in value

\j SLt,

T E
the constant ratio - , and may be made to approach.

C E

it " nearer than by any given difference." Hence, ac-
cording to the definition of a limit, the limit of

If, therefore, we would find the value of the unknown

T F
ratio - , we only have to obtain from the equation

C E

of the curve an expression for - , and then pass

T F
to its limit, which is the value of - , than which

C E

nothing is more easily done. Behold, then, the method
of limits delivered from its obscurities, and rendered
as transparent as the Elements of Euclid !

CHAPTER VIII.

OF THE SYMBOLS - AND X

IF anything in the whole science of mathematics
should be free from misconception and error, one
would suppose it ought to be the symbol 0, which
usually stands for simply nothing. Yet, in fact, this
is precisely one of those symbols which has most fre-
quently led mathematicians from the pure line of truth,
or kept them from entering upon it. " In the fraction

-," it has been said, " if we suppose a to remain con-

6

stant while b continually increases, the value of the

fraction continually diminishes ; when b becomes very

great in comparison with a, the value of the fraction. p

becomes very small ; finally, when b becomes greater , I

than any assignable quantity, or oo , the value of the

fraction becomes less than any assignable quantity, or

; hence

oo I

, i

I'

This kind of differs analytically from the absolute f j

zero obtained by subtracting a from a, a a = 0. It ( |

is in consequence of confounding the arising from
dividing a by oo with the absolute 0, that so much
confusion has been created in the discussion on the

18 * 209

210 THE PHIL 080PHY OF MA THEM A TICS.

subject. About the absolute there can be no dis-
cussion ; all absolute O's are equal. But the other O's
are nothing else than infinitely small quantities or in-
finitesimals ; and there is no incompatibility in sup-
posing that they differ from each other, and that the
ratio of two such zeros may be a finite quantity." *

Such is the author's interpretation of -. It is not

zero divided by zero at all, it is only one infinitely
small quantity divided by another. If so, why in the
name of common sense did not the reasoner say what
he meant, and, instead of calling an infinitely small
quantity 0, represent it by the symbol i, or some other
different from. 0. Surely, it was just as easy to say

a . . . d ~ i . .

=^ as it is to say = 0. or to write as it is to

oo J oo y

write -. And then there would not have been the

least shadow or appearance of the confusion of which he
complains, and of which he endeavors to explore the
origin.

" Logical accuracy," says the author, " would seem"
to require that some other name should be given "to
one of these zeros [most assuredly] ; but if two mean-
ings of the term are fully understood, no trouble
need arise in retaining the nomenclature which has
been sanctioned by the custom of centuries." But
why introduce such utterly needless ambiguities into
the science of mathematics ? Is it only that they may
be explained in dictionaries, and carefully watched by
mathematicians in order to keep out darkness and
confusion from their reasonings ? The truth is, there

* Dictionary of Mathematical Science, by Davies and Peck

THE PHILOSOPHY OF MA THEMATICS. 21 1
is no use whatever for any such ambiguity, except to
explain the symbol -, and to dodge other difficulties

of the calculus ; causing it to swarm with sophisms

The above explanation is easy, but it does not meet

the difficulties of the symbol - as it arises in the cal-

culus. Indeed, it only deals with that symbol in the
abstract, and not as seen in its necessary connections
in practical operations. The author attempts this in
his well-known work on the Differential Calculus. In
finding the differential co-efficient of u = a x*, he gives
to x the increment h } which makes

u r = a (x + A) 2 = a x* + 2 a x li + a It.
Hence u f u = 2 a x h + a h* 9

,

or - 2 a x + a fi.

h

.Now, he represents "by d x the last value of Ji" that
is, the value of li which cannot be diminished, accord-
ing to the law of change to which h or x is subjected,
without becoming 0, and "by d u the corresponding
difference between u r and u" We then have, says he.

du ~ , ,
= 2ax + a ax.
dx

Now we certainly expected him to say this, but he has
said, we then have

* The same learned disquisition on nothing is also found in
"Davies' Bourdon/' as well as in other works on Algebra.

212 THE PHILOSOPHY OF MATHEMATICS.

du

= 2 ax.

dx

What, then, has become of the term adx*! It ap~"
pears to have vanished without either rhyme or reason.
How is this mystery to be explained?

" It may be difficult," says the author, " to under-
stand why the value which li assumes in passing to
the limiting ratio is represented by d x in the first
member, and made equal to zero in the second." Truly,
this is a most difficult point to understand and needs
explanation. For if h be made absolutely zero or
nothing on one side of the equation, why should it not
also be made zero on the other side ? It may, if you
please, be zero or nothing sometimes, and sometimes
an infinitely small quantity ; but can it be both at one
and the same time, and in the same operation ? It is,
indeed, most convenient to use h in this ambiguous
sense, making it absolutely nothing on one side of an
equation and very small on the other ; for this gives

d u

the true result = 2a x, which might not otherwise
dx

be so easily obtained but has the author anywhere
justified in his Logic of Mathematics a process seem-
ingly so arbitrary ? Or is the Logic of Mathematics
so different from all other logic that so flagrant a
solecism is agreeable to its nature ? In other words,
is the Logic of Mathematics so peculiar in its character
that A, the same identical quantity, may be both some-
thing and nothing at one and the same time? If so,
then, in spite of the author's learned treatise, there is
no telling what may not happen in the Logic of Ma-
thematics.- But, for one, I shrewdly suspect that there

THE PHIL OSOPH Y OF MA THEM A TICS. 213

is no rule in arithmetic, nor in algebra, nor. in geome-
try, nor in the calculus, by which the answer to a
question may be forced without regard to the ordinary
laws of human thought or sound reasoning.

"We have represented by dx" says the author,
"the last value of h." That is, "the last which h can
be made to assume in conformity with the law of its
change or diminution without becoming zero." But
why should Ji, in the second member and not as well
in the first, obey this law of change ? Why should it
there, and there alone, kick out of the traces and be-
come nothing in spite of the law of its existence?
Because (the answer is easy) this is necessary to find
the true result. The author, indeed, assigns another
reason. " By designating this last value by d x" says
he, " we preserve a trace of the letter x, and express
at the same time the last change which takes place in
h as it becomes equal to zero." But why should " a
trace of the letter x" be preserved in the first member
of the equation and not in the second ? The reason
is, just because dx is needed in the first member and
not in the second to enable the operator to proceed
with his work. The author might have fortified his
position by very good authority, since Boucharlat,* as
well as other writers on the Differential Calculus, have
conceived the same laudable desire to preserve "a
trace of the letter x" in one member of all similar
equations, while they unceremoniously eject it from
the other member,

But is this all that can be said by the teachers of

* The intelligent reader, even if he had not been told in the pre-
face, would have known that Dr. Davies had freely used the work
of Boucharlat.

21 4 THE PHIL OSOPHT OF MA THEMA TICS.

the calculus ? Must they be thus for ever foiled in
their attempts to grapple with the difficulties of the
very first differential co-efficient ? Shall they continue
thus grievously to stumble at the very first step in the
path of science, along which they undertake to guide
the thinking and reasoning youth of the rising gene-
ration? Shall they continue to seek and find what
no other rational beings have ever found, namely, that

particular value of " which does not depend on

h

the value of h ?" * That is to say, that particular value
of a fraction which does not depend on its denom-
inator ! f I think it is quite otherwise. Such miscon-
ceptions or blunders may have been unavoidable in
the dim twilight of the science, or before the grand
creations of a Newton or a Leibnitz had so completely
emerged, as at the present day, from the partial chaos
in which their great creators necessarily left them in-
volved. But they are now anything rather than an
honor to the age in which they continue to be repro-
duced. Some, it is to be feared, make haste to become
the teachers before they have become the real students
of those sublime creations. J

* Davies' Differential Calculus, p. 17.

f The same thing is found in Mr. Courtenay's Calculus (p. 61), as
well as in a multitude of others.

J I am sure this was the case with myself. The ignorant boy, If
he has only graduated high in mathematics at West Point, is apt to
presume what, indeed, is more presumptuous than ignorance? that
he is qualified to teach the calculus ; although he may never have
learned its very first lessons aright, or been once taught" and made
to see the rational principles which lie concealed beneath its formula
and enigmas. I had not been a teacher of the calculus long, how-
ever, before I discovered that I had almost everything to learn re-
specting it as a rational system of thought. Difficulties were con-

THE PHIL OSOPH Y OF MA THEM A TICS. 215

One thing appears perfectly evident to my mind,
and that is, that h should be made nothing in both
members of the equation, or else in neither. I must
think this or else refuse to think at all. Hence, we
have

du ~ , ,

dx

or - = 2 a x.

But if we adopt this last form, we escape the illegiti-

mate expression = 2 a z, with all its shuffling
dx

sophisms, only to encounter -, the most formidable of

all the symbols or enigmas in the differential calculus.
This symbol has, in fact, always been a stumbling-
block in the way of the method of limits; the great
and affrightful empusa which has kept thousands from
adopting that method. Even Dulwmel shrinks from
a contact with it, although its adoption seems abso-
lutely necessary to perfect the method of limits. For
if two variables are always equal, then their limits arc
equal. But the limit of 2 ax + ah is 2 a#, and the

tirinally suggested in the eourfio of my reflections on the subject,
nothing. I found, in short, that I had only been tnught in work the
calculus by certain rules without knowing the. real reanoiiH or prin-
ciples of those rules ; pretty much as an engineer, who knows nothing
about the mecha.ninm or principles of an engine, is 1 shown how to
work it, by a few superficial and unexplained rules. This may be a
very useful sort of instruction ; it is certainly not mental training or
education. It may be knowledge; it in not .science.

216 THE PHIL OJSOPHT OF MA THEM A TICS.

limit of is -. Hence, if we are not afraid to

h

trust our fundamental principle or to follow our logic
to its conclusion, we must not shrink from the symbol

This symbol is repudiated by Carnot and La-
grange. It is adopted by Euler and D'Alembert ; but
they do not proceed far before it breaks down under
them. It is, nevertheless, one of the strongholds and
defences of the method of limits, which cannot be sur-
rendered or abandoned without serious and irreparable
loss to the cause.

Carnot thus speaks of this symbol : " The equation

MZ y , , . . -,_. .

__ = 2 found in section (9) is an equation

B, Z a x

always false, though we can render the error as small
as we please by diminishing more and more the quan-
tities M Z, B, Z ; but in order that the error may dis-
appear entirely, it is necessary to reduce these quan-
-.: nn 4-^ oKcnlnte o's but then the equation will reduce

iisen to - = , an equation which we cannot say
a x

is exactly false, but which is insignificant, since - is

an indeterminate quantity. We find ourselves, then,
in the necessary alternative either to commit an error,
however small we may suppose it, or else fall upon a
formula which conveys no meaning; and such is pre-
cisely the knot of the difficulty in the infinitesimal

>-na1irsis."*

^ the problem of quadratures, the only alterna-
d to be either to commit an error with Pas-

* Eeflexions, etc., Chap. I., p. 41.

TEE PHILOSOPHY OF MATHEMATICS. 217

cal by rejecting certain small quantities as zeros, or to
find with Cavalieri the sum of an infinity of nothings,
which, in the modern algorithm, is equivalent to the
symbol X oo ; so in the question of tangency the only
alternative seems to lie between committing a similar
error with Leibnitz, by the arbitrary rejection of infi-
nitely small quantities in the second member of an
equation as nothing, or the recognition and adoption

of the symbol -. J. have already said that, as it seems

to me, there is a profound truth at the bottom of Cava-
lieri's conception, or in the symbol X <*> , which has
never been adequately understood or explained. Pre-
cisely the same thing appears to me perfectly true in
regard to the conception of Newton, which, if properly

understood, is the true interpretation of the symbol -.

Now the objection, which is always urged against
the use of this symbol, or this form of the first differ-
ential co-efficient, is, that - is an indeterminate expres-
sion, and may therefore have one value as well as an-
other. Or, in other words, that it means all things,
and therefore means nothing. This objection is re-
peatedly argued by Carnot, with whom the method of
Leibnitz evidently ranks higher than that of Newton.
(C It seems," says he, " that infinitely small quantities
being variables, nothing prevents us from attributing
to them the value of as well as any other. It is

true that their ratio is -, which may be equally sup-
posed a or 6 ? as well as any other quantity whatever." *

* Reflexions, etc., Chap. III., p. 182.
19 K

218 THE PHILOSOPHY OF MATHEMATICS.

Again, in reply to those who complain of a want of
logical rigor in the method of Leibnitz, Carnot makes
him thus retort in a feigned speech: "All the terms
of their equations vanish at the same time, r.o that
they have only zeros to calculate, or the indeterminate
ratios of to to combine." *

Even those who, by a regard for logical consistency,

have been compelled to adopt the symbol - as the true

expression for the first differential co-efficient, have
utterly failed to emancipate themselves from the influ-
ence of the above difficulty or objection. That (f sym-
bol of indeterrnination," as it is always called, has
still seemed, in spite of all their logic, as vague and
undefined as Berkeley's " ghosts of departed quanti-
ties." Even D'Alembert himself is no exception to
the truth of this remark. For, in his celebrated article
on the metaphysics of the differential calculus in the

Encyclopedic, he says : " Thus -^ is the limit of the

dx
ratio of z to u, and this limit is found by making z='0

in the fraction - . But, it will be said, is it not

2y + z
necessary to make also z = and consequently u =

in the fraction - = , and then we shall have

u 2y + z

==?" That is to say, is it not necessary to make

z = in the first as well as in the second member of
the equation? Most assuredly, in the opinion of
D'Aleinbert, although this should bring us into actual

contact with the symbol ~.
J

* Reflexions, etc., Chap. I v p. 37,

THE PHILOSOPHY OF MA THEMA TICS. 21 9

" But what is it," lie continues, " that this signi-
fies ?" Ay, that is the very question : what is it that
this symbol signifies? Has it any sense behind or
beyond that vague, unmeaning face it wears? and if it
has, what is its real sense ? "I reply," says D' Alem-

bert, " that there is no absurdity in it, for - can be
equal to anything that we please ; hence it can be ."

f7

But no one ever suspected - of having any absurdity

in it; it was only accused of having no signification,
of meaning one thing just as well as another, and, con-
sequently, of meaning nothing to any purpose under

the sun. True, if - may have any value we please,

then it may be equal to , if we so please ; but, then,

. 2y
it is equally true that if we please it may be equal to

any other value just as well as to . But is not this

. 2 y .

it ? If, we ask, what signifies -, Carnot replies, it

signifies anything, a or 6, or , or any other

2y a

quantity we may please to name, and D'Alembert re-
peats the reply! Is that to defend the symbol - or

explain what it signifies ? Or, in other words, is that
to remove the objection that it is a symbol of inde-
termination, which signifies everything, and conse-
quently pothing?

220 TSJE PHIL OSOPHY OF MA THEM A TICS.

M. D'Alembert adds: "Though the limit of the
ratio of z to u is obtained when z = and u = 0, this
limit is not properly the ratio of z = to u = 0, for
that presents no clear idea ; we know not what is a

ratio of which the two terms are both nothing. This

n
limit is the quantity which the ratio - approaches

u

more and more in supposing z and u both real and
decreasing, and which that ratio approaches as near as
we please. Nothing is more clear than this idea; we can
apply it to an infinity of other cases." Now there is
much truth in this second reply; but, if properly
understood and illustrated, this truth will be found
utterly inconsistent with the first reply of D'Alern-

bert. If, then, we would see what the symbol -

really signifies, we must explode the error contained
in D'Alembert' s first reply (or in Carnot's objection),
and bring out into a clear and full light the truth in
his second reply. This will vindicate the true charac-
ter of this all-important and yet much-abused symbol.

The expression - is, as it stands or arises in the

calculus, not a " symbol of indetermination." If viewed
in the abstract, or without reference to the laws or
circumstances to which it owes its origin, then, indeed,
it has no particular meaning or signification. But
nothing, as Bacon somewhere says, can be truly under-
stood if viewed in itself alone, and not in its connec-
tion with other things. This is emphatically true in

regard to the symbol -. If abstracted from all its
connections in the calculus, and viewed in its naked

THE PHILOSOPHY OF MATHEMATICS. 221

form, nothing, it is admitted, could be more indeter-
minate than -. It is, indeed, precisely this unlimited

indetennination of the abstract symbol which consti-
tutes its great scientific value. For, as Carnot himself
says, " It is necessary to observe that the expression
of variable quantities should not be taken in an abso-
lute sense, because a quantity can be more or less in-
determinate, more or less arbitrary ; but it is precisely
upon the different degrees of indetermination of which the
quantity in general is susceptible that every analysis
reposes, and more particularly that branch of it which
we call the infinitesimal analysis" * If such is, then,
the true character of the symbols in every analysis, and
especially in the infinitesimal analysis, why should it
be objected against one symbol and against no other?
Every one knows, for example, that x and y stand for

indeterminate values as well as ~. Why, then, should

this last symbol be objected to on the ground that
it is indeterminate? No one means that its value
may not, in each particular case, be determined,
and if any one should so mean, he might be easily
refuted. The more indeterminate the symbol, says
Carnot, the better, and yet it is seriously objected to

the symbol -, that " it is a quantity absolutely arbi-
trary" or indeterminate ! f

U I have many times," says Carnot, "heard that
profound thinker [Lagrange] say, that the true secret
of analysis consists in the art of seizing the various
degrees of indetermination of which the quantity is

* [Reflexions, etc., Chap. I., p. 18. f Ibid., Chap. III., p. 184*
19*

222 THE PHILOSOPHY OF MATHEMATICS.

susceptible, and with which I was always penetrated,
and which made me regard the method of indeter-
minates of Descartes as the most important of the co-
rollaries to the method of exhaustions." * That is to
say, as the most important of the methods of the infini-
tesimal analysis, for he regards all these methods as
corollaries from the method of exhaustion. Agt ; n,
in his beautiful commentary on the method of Des-
cartes, he says : " It seems to me that Descartes, by
his method of indeterminates, approached very near
to the infinitesimal analysis, or rather, it seems to me,
tJiat the infinitesimal analysis is only a happy applica-
II ' tion of the method of indeterminates" * He then pro-

I ceeds to show that the method of Descartes, and its

|] symbols of indeterminates, lead directly to some of the

f ! most striking and important results of the infinitesimal

;*j analysis. Surely, then, he must have forgotten the

I /? great idea with which he was always so profoundly

.* P penetrated, when he singled out and signalized the

symbol - as objectionable on the ground that it is

indeterminate. It may, it is true, be " either a or 6 ;"
H but so may x and y. These symbols may, as every

one knows, be " a or b" 2 a or 2 6, 3 a or 3 b } and so
on ad infinitum. Yet no one has ever objected to these
symbols that they are indeterminate. On the con-
trary, every mathematician has regarded this indeter-
mination as the secret of their power and utility in
the higher mathematics. This singular crusade of

mathematicians against one poor symbol -, while all
other symbols of indetermination are spared, is certainly

* fteflexions, etc., Chap. III., p. 208. f Ibid., p. 150.

THE PHILOSOPHY OF MATHEMATICS.

223

a very curious fact, and calls for an explanation. It
shall in due time be fully explained.

So far from denying that -, abstractly considered,

is indeterminate, I mean to show that it is, in the words
of the objection to it, "absolutely arbitrary." This
degree of indetermination is, indeed, the very circum-
stance which constitutes its value, and shows the high
rank it is entitled to hold among the indeterminates
of geometry. It is, in other words, its chief excellency
as a mathematical symbol, that it may not only come
to signify "a or 6," but any other value whatever,
covering the whole region of variable ratios from zero
to infinity. Instead of denying this, this is the very
point I intend to establish in order to vindicate the

character of the symbol -.

J

Let S T be a secant cutting the circumference of
the circle in the points A and B, the extremities of
two diameters at right angles to each other. Conceive
this secant to revolve around the point A, so that the

224 THE PHILOSOPHY OF MATHEMATICS.

"O Q

point B shall continually approach A. is equal

A O

to the tangent of the angle B A O, which S T makes
with the line A O, and in each and every successive

position of the secant, such as s t, is equal to the

A o

tangent of the angle which it makes with A O. As
B approaches A, this angle, and consequently its tan-
gent, continually increases. That is to say, although

b o and A o continually decrease, their ratio con-

A o

tinually increases. The limit of the angle 6 A o is the
right angle T 7 A O, whose tangent is equal to infinity,

toward which, therefore, the ratio continually

A o

tends. Hence, when the arc b A becomes indefinitely
small, the angle b A o approaches indefinitely near to

the right angle T 7 A O, and approaches in value

A o

the tangent of that right angle. The secant s t can
never exactly coincide with its limit, the tangent A T ; ,
since that tangent has only one point in common with
the circumference of the circle, while the secant always
has, by its very definition, two points in common with
that circumference. Then, if we pass to the limit by
making A o = 0, and consequently b o = 0, the equation

= tan. b A o will become = tan. T 7 A o = oo .

Ao

Again, if we conceive S T to revolve around the point
B, making A continually approach toward B, we shall

always have = tan. Bro. But, in this case, the

a o

THE PHILOSOPHY OF MATHEMATICS.

225

angle which the secant s t makes with the lineA0 8
has zero for its limit. Hence, if we pass to the limit

the equation = tan. B r o, will become = 0.

a o

Thus, the limit of the ratio of two indefinitely small
quantities may be either infinity or zero. It is easy
to see that it may also be any value between these two
extreme limits, since the tangent which limits the
secant may touch the circumference in any point be-
tween A and B. For example, the tangent of the

angle b P d , which the secant P b makes with P d, or
A O produced, is always equal to , as b approaches

JL Cu

the point of contact P. Hence, if we pass to the

limit. = tan. b P d becomes - = tan. T P d.

T?d

Precisely the same relation is true in regard to every
point of the arc A B. Hence, if the point of contact
P be supposed to move along the arc A B from B to
A, the value of the tangent of the angle T P d, or of

-, will vary from to co . But it should be particu-

K*

226 THE PHILOSOPHY OF MATHEMATICS.

iarly observed, and constantly borne in mind, that if
the question be to find the tangent line to any one

point of the are A B, then - will have only one defi-
nite and fixed value, for this is an all-important fact
in the true interpretation of the symbol in question.
The symbols x and y are indeterminate, just as much

so as -. But if we suppose a particular curve, of which

x and y are the co-ordinates, and make x equal to a, then
y becomes determinate, and both symbols assume defi-
nite and fixed values. Now it is precisely this inde-
termination of the symbols x and y> abstractly con-
sidered, with the capacity to assume, under some
particular supposition, determinate and fixed values,,
that constitutes their great scientific value. Considered
as the co-ordinates of any point of any curve, x and y
nrfi of course indeterminate, absolutely indeterminate;

a given curve they are
.* .aiue. In like manner,

, if considered in a general and
\j

ci point of view, or, in other words, with refer-
to a tangent to any point of any curve, it is in-
ueuu absolutely indeterminate. But the moment you
seek the tangent to a particular point of a given curve,

the r for that point has, and can have, only one value.

There is, then., no more reason why this most useful
symbol should be distrusted, or decried, or rejected
from the infinitesimal analysis as indeterminate, than
t.'K-iv :* tbr the rejection of x and y or any other sym-
bol 01 Indeteriiiiuaiioii from the same analysis. The

TEE PHILOSOPHY OF MATHEMATICS. 227

very quintessence and glory of that analysis, indeed,
consists in the possession and use of precisely snch
symbols of indetermi nation. Why, then, I ask again,
should one be singled out and made the object of
attack?

The explanation of this partial, one-sided, and slip-
shod method of judgment may be easily given. In

the ordinary analysis, or algebra, the symbol - is not

only indeterminate, but it sometimes arises under cir-
cumstances which still leave "it as indeterminate as
ever, failing to acquire any particular value or values
whatever. This is the case in the familiar problem
of the two couriers. If they start from the same
point, travel in the same direction, and with the same
speed, it is evident that they will always be together.
Hence, if in the formula for the time when they will be

together - = t, we make a, or the distance be-

to mn ' '

tween the points of departure, = 0, and m ft, the
difference between the number of miles they travel per
hour, also = 0, we shall have, as we evidently ought
to have,

Now here the symbol - remains indeterminate in the

concrete, or with reference to the facts of the case, as
it was in the abstract, or without reference to any
particular facts or case. And the same thing is true

in all cases in which a fraction, like - , becomes

y m n

228 THE PHIL OJSOPJIY OF MA THEMA TICS.

in consequence of two independent suppositions, the one

causing the numerator and the other the denominator
to become = 0. Thus the student of mathematics
becomes, in his first lessons, familiar with the symbol

- as not only indeterminate in the abstract, but also in

the concrete. That is, he becomes habituated to pro-
nounce it indeterminate, because it has no value in
general, and can have none in the particular cases
considered by him. Hence, from the mere blindness
of custom (for it seems utterly impossible to assign any
other reason), he continues to regard it always and
everywhere in the same light. He spreads, without
reflection, this view of the symbol in question over the
whole calculus, and thereby blots out its real signifi-
cance and utility.

In the infinitesimal analysis the symbol - arises,

not in consequence of two independent suppositions^ but
in consequence of one and the same supposition, which
makes both denominator and numerator = 0. Thus,

in the case considered by D'Alembert - = , z is

u 2y + z

made = 0, and this makes its function u = 0. The

2>

ratio - always tends, as z becomes smaller and smaller.

u J '

toward the limit , and hence in passing to the

2y
limit, by making z = 0, we have

0___ a_
~~ 9, v

value, which is . Hence D'Alembert was in error

THE PHILOSOPHY OF MATHEMATICS. 229

Now in this case - may not have any value as in the

case of the couriers ; for it has, and can have, only one
a_

2?

when he said that since - may have any value, it may

have this particular value as well as any other ; for
this implies that it may have any other value as well

as ; whereas, in the case under consideration, it

V

must have exactly this value, and can not possibly
have any other. Considered in the abstract, then, or
without reference to the facts and circumstances of any

particular case, the symbol - may be said to be inde-
terminate. But yet, in very truth, this symbol never
arises in the calculus without a precise signification or
value stamped on its face. As it appears in the cal-
cnltis, then, it is no longer indeterminate; it is perfectly
clear and fixed in value. It derives this fixed value
from the very law of its origin or existence, and, under
the circumstances to which it owes that existence or
its appearance in practice, it cannot possibly have any
other value whatever.

It seems wonderful that in the very works from

which - is rejected as an unmeaning fe symbol of inde-

termination," there should be methods set forth in
order to find its precise value. Thus in Mr. Courte-
nay's Calculus, as well as in many others that repudi-
ate the symbol in question, there is a method for fin/i

230 THE PHILOSOPHY OF MATHEMATICS.

ing the value of -.* Neither he, nor any one else,

ever found the value of -, except in reference to some

particular case in which it was determinate, having
assumed a concrete form. But, what seems most
wonderful of all, they have a method for finding the

determinate value of - when that value is not obvious,

and yet they assert it has no determinate value when
it appears with one stamped, as it were, on its very
face. Thus, if we seek the trigonometrical tangent of
'the angle which the tangent line to any point of the
common parabola, whose equation is 7/ 2 = a x } makes
with the axis of x, we have

2y

the exact value which is made known by pure geo-
metry. Now here - arises, or appears in the calculus^

with this precise, definite value , and yet the opera-

2y
tor, looking this determinate value in the face, de-

clares that - has no such value. If he could not see

this value, then he would apply his method to find it;
but when it looks him in the face, and does not require
to be found, he declares that it has no existence !
The two variable members of the equation

z a

u 2y + z

* Chap. VII., p. 77.

THE PHILOSOPHY OF MATHEMATICS. 231

are always equal, and hence their limits are equal.

That is to say, the limit of the one - = 3 the limit

2y

of the other. Now here - is, as D'Alembert says, not

the symbol of a fraction, since zero divided by zero
conveys no " clear idea." It is the symbol of a limit.
This is its true character, and it should always be so
understood and interpreted. It is the limit, the con-
stant quantity, (y being the ordinate to the point
2y

of contact), toward which the value of the fraction -

u

continually converges as z, and consequently u, becomes
less and less.

Hence there is no necessity of dodging the symbol

-, as so many mathematicians are accustomed to do.

nil - ni

Having reached the position = 2 a x + a li>

Ji

Dr. Davies could not say, with downright logical
honesty, if we make h = 0, we shall have

- = 2 a x.

On the contrary, he makes Ji = in one member of
his equation, and = d x, or the last value of #, in the
other. By this means he preserves a trace of the letter
u, as well as of the letter x, in one member of his

-. But there was no necessity whatever for any such

232

THE PHILOSOPHY OF MATHEMATICS.

logical legerdemain or jugglery. For if he should
ever have any occasion whatever to use this , he

might just substitute its value, already found, 2 a x,
for it, and have no further difficulty. He might, in

fact, have written his result = 2 a x, provided ho

d x

had understood by - , not the last ratio of ,

d x A

but the limit of that ratio, or- the constant value which
that ratio continually approaches but never reaches.

It would be doing great injustice to Dr. Davies, if
he were represented as standing alone in the perpe-
tration of such logical dexterity. We ought to thank
him, perhaps, for the open and palpable manner in
which he performs such feats, since they are the more
easily detected by every reflecting mind. It is cer-
tain that the same things are done with far greater
circumspection and concealment by others, not de-
signedly, of course, but instinctively; hiding from their
own minds the difficulties they have not been able to
solve. We have a notable example of this in the
solution of the following problem: "To find the gene-
ral differential equation of a line which is tangent to
a plane curve at a given point x', y f .

1 A

THE PHILOSOPHY OF MATHEMATICS. 233

" The equation of the secant line E S passing
through the points x f y f and x n ' y n ', is

or

a But if the secant E S be caused to revolve about
the point P ; , approaching to coincide with the tangent
T V, the point P /; will approach P', and the differences
y ff y f and x rf x' will also diminish, so that at the

limit, where E S and T V coincide, will reduce

x rt x }

dii f

to , and the equation (1) will take the form
dx f

which is the required equation of the tangent line at
the point x 1 y f ." *

Not exactly, for when E S coincides with T V, the
point P ;/ coincides with P 7 , and the two become one
and the same point. Hence, when E S coincides with
T V, or the point P^with the point P x , the equation
(1) takes the form

y y = --(a? a/).

But in order to shun the symbol -, which the author

did not approve, he committed the error of supposing
P" to coincide with P x , without supposing the differ-
ences of their abscissas and ordinates to vanish, or
become = 0. But most assuredly if x f is the abscissa

* Courtonay's Calculus, Part II., Chap. I., p. 148.
20*

234 THE PHIL OSOPHY OF MA THEM A TICS.

of P' and x" is the abscissa of P", then when P /; coin-
cides with P ; , x n will be equal to x f and the difference
x ff x i w [n b e _- Q^ The same thing happens in re-
gard to the difference y n y f , for when the points P ;
and P^coincida, it is clear that the difference of their
ordinates y /f y f = 0. But the author preferred the

inaccurate expression to the symbol -, which, in

dx r

every such case, is perfectly accurate, as well as per-
fectly determinate. And he obtains this inaccurate
expression by means of the false supposition that P 7 and
P ;/ may coincide without causing y ff y' } or x n x f
to become = ; which, in the application of the pro-
cess to any particular curve given by its equation, is
just exactly equivalent to making the increment of
x = on one side of the equation and not on the
other. It is precisely the process of Dr. Davies re-
peated in a more covert form.

I object to the system of Dr. Courtenay, as well as
to that of Dr. Davies, because they both freely use the
terms limit and indefinitely small without having once
denned them. Nor is this all. They habitually pro-
ceed on the false supposition that the variable reaches
or coincides with its limit. Thus, in the example
just noticed, it is supposed that the limit of the vari-

yff yf \$ yf

able ratio y - is its last value ~ ; whereas its

x n x r d x r

real limit lies beyond its last value, and is accurately
found only by making y* y f = 0, and x 2 x f = 0.
For, as we have repeatedly seen, it is no value of the

ratio , which is equal to the tangent of the angle

x x

which the tangent line at the point x f y ! makes with

THE PHILOSOPHY OF MATHEMATICS. 235

the axis of x. That tangent is equal, not to the last

yff yf

value of the ratio * ^-, but to the limit of that

x n x l

ratio ; a quantity which it may approach as near as we
please, but can never reach. Again, they freely speak
of indefinitely small quantities, and yet ; in no part of
their works, have they defined these most important
words. But they habitually use them in a wrong
sense. Instead of regarding indefinitely small quan-
tities as variables which continually decrease, or which
may be supposed to decrease as far as we please with-
out ever being fixed or constant, they consider them
as constant quantities, or as acquiring fixed and un-
alterable values. Thus, in the systems of both, dx,
or the last value of the variable increment of x, is re-
garded as a constant quantity. With such conceptions,
or first principles, or elements, it is impossible for the
ingenuity of man to form a differential calculus free
from inaccuracies and errors. All the works, in fact,
which have been constructed on those principles are,
like the two under consideration, replete with solecisms
and obscurities. It would require much time and toil
to weed them all from the calculus at least the pro-
duction of a volume.

But one more must, in this place, be noticed, both
because it is very important, and because it relates to

the interpretation of the symbol -. In the discussion

of multiple points, at which of course there are several
branches of the curve, and conseqtiently one tangent

for each branch, it is said, that ^- = -, since it " caii-

d x'
not have several values unless it assumes the iudeter-

236 T3LE PHILOSOPHY OF MATHEMATICS.

minate form -"* Now here ; at least, the author

resorts to -, because he cannot proceed without it, and

he gives the wrong reason for its use. The truth is,
if there are two branches of the curve meeting at one

point, then will -, as found from the equation of the

curve foi that point, have exactly two determinate
values precisely as many as are necessary to determine
and fix the positions of the two tangents, and no more.
In like manner, if three or four branches of a curve
meet in the same point, they give rise to a triple or

quadruple point ; then will -, obtained with reference

to that point, have three or four determinate values,
or exactly as many as there are tangent lines to be
determined. If the secant passing through the com-
mon point first cuts one branch of the curve and then

another, the - found for one branch will, of course,

have a different value from the - obtained with refer-

ence to the other branch. Thus, such is the admir-
able adaptation of the symbol - to all questions of tan-

gency, that it will have just as many determinate
values as it ought to have and no more, in order to
effect the complete and perfect solution of the problem.

But it is a manifest error to say that - is indetermi-

J

* Courtenay's Calculus, Part II., Chap. II., p. 191.

THE PHILOSOPHY OF MATHEMATICS. 237

nate in any such case, because it lias two or three or
four determinate values. The truth is, we use it in
such cases, not because it is indeterminate, but just
because it is determinate, having precisely as many
determinate values as there are tangents to be deter-
mined. These are determined and fixed in position,

not by the indeterminate values of -, but by its deter-
minate and determined values.

The above reason for the use of - in the discussion

of multiple points was assigned by Descartes, who, in
the dim twilight of the nascent science, knew not what
else to say; and it has since been assigned by hun-
dreds, simply because it was assigned by Descartes.
But is it not truly wonderful that it should be em-
ployed to determine two or three or more tangents at
the multiple points of a curve, and yet utterly re-
jected as not sufficient to determine one tangent when
there is one curve passing through the point ? Is it
not truly wonderful that it should be thus employed,
because it is indeterminate, and yet rejected for pre-
cisely the same reason ? It is quite too indeterminate
for use, say all such reasoners, when it arises with one
value on its face ; but yet it may, and must be used,
when it arises with two or more values on its face,
just because it is indeterminate ! How long ere such
glaring inconsistencies and grievous blunders shall
cease to disgrace the science of mathematics? Shall
other centuries roll away ere they are exploded and
numbered among the things that are past ? Or may
we not hope that a better era lias dawned an era in

238 THE PHIL OSOPHY OF MA THEM A TICS.

which mathematicians must think, as well as manipu-
late their formulae ?

Only one other point remains to be noticed in regard

to the symbol -. It is said, if we retain this symbol

our operations may be embarrassed or spoiled by the
necessity of multiplying, in certain cases, both mem-
bers of an equation by 0. But the answer is easy.
The first differential co-efficient, if rendered accurate,

always comes out in the form of - ; but it need not

retain this form at all. Whether we use - or ^- in

dx'
writing the differential equation of a tangent line to

the point x f , y f , we shall have to eliminate - in the one

dy'
case, and in the other, in order to make any prac-

dx f
tical application of the formula. Now - is just as

easily eliminated by the substitution of its value in

dy f
any 'particular case as is , and besides its value may

dx f

be found and its form eliminated by substitution with-
out any false reasoning or logical blunder, which is

dv f

more than can be said for the form - .

dx f

For if we write the formula in this form,

y y

and proceed to apply it, we shall have to commit ao

TEE PHILOSOPHY OF MATHEMATICS, 239

dv'

error in the elimination of ^-. Suppose, for exam-

dx'

pie, the question be to find, by means of this general
formula, the tangent line to the point a?', y f of the com-
mon parabola, whose equation is y 1 = 2 p x. If, then,
we would be perfectly accurate, we should have

y f 2y'dx

How shall we, in this case, get rid of the last term

dv f *

- ^ ? Shall we make it zero by making d y f = 0,
2 y r d x !

and yet not consider ~ = -, or shall we throw it out
* dx f

as if it were absolutely nothing, because it is an infinitely
small quantity of the second order ? Both processes
are sophistical, and yet the one or the other must be
used, or some other equivalent device, if we would arrive

at the exact result = ; the result which is found,

dx f y f

or rather forced, in the calculus of Dr. Courtenay,*
as well as in others which have been constructed on
the same principles.

Now, in the second place, suppose the general
formula is written in this form :

y~y f= Q^ x ~ x ^

We here see, by means of x 1 , T/, the point with reference

to which the value of - is to be found. We obtain.
'

as in the last, the expression :

* See Tart II., Chap. I., p. 150.

240 THE PHILOSOPHY OF MATHEMATICS.

dy _ p dy*
dx~~y 2y.dx

in which d y and d x are regarded as the increments
of y and x, which increments are always variables and
never constants. As dx, and consequently dy, be-
comes smaller and smaller, it is evident that the last

dy 2
term - ^ becomes less and less, since dy*, the

C\ 7 X */ 7

2y . ax

square of an indefinitely small quantity, decreases
much more rapidly than its first power. Hence, the
term in question tends continually toward its limit
zero, and if we pass to that limit by making d x, and
consequently dy, 0, we shall have

_.
0~V

or for the point x r 9 y r , we shall have

which substituted for - in the general formula, gives

y y f = p -(x x f \
y f

Thus, precisely the same result is arrived at as in the
former case, and that, too, without the least appear-
ance of a logical blunder, or shadow of obscurity.

The foregoing reflections may be easily extended to
the formula X oo , which is also called a symbol of
indetermi nation. It is, indeed, in many cases nay, in
all cases that arise in practice the symbol of a limit,

THE PHILOSOPHY OF MATHEMATICS.

241

whose exact value may be found. There is, in Dr.
Courtenay's Calculus, as well as in others, a method
for finding the value of X oo when this symbol does
not arise with its value on its face, or on the opposite
side of an equation. In every case in which its value
is thus found O X oo is the limit toward which a
variable quantity continually converges, but never
exactly reaches, as any one may see by referring to the
cases in the calculus of Dr. Courtenay, or of any other
author.

Let us take, for example, the case considered by
Cavalieri, whose conception may be expressed by the
symbol X oo . He considered, as the reader will
remember, the question of the quadrature of any
plane curvilinear area. If we conceive the base A E
of any such area A a E, to
be divided off into equal
parts, and represent each
part by h, and the whole
number of parts by n, and
if we conceive a system of
inscribed parallelograms, or
rectangles, erected on those
equal parts as seen in the
figure, and let y represent
their varying altitudes, we shall have for the sum of
the rectangles the expression

y It X n.

But this sum, as we have seen, is never equal to the
curvilinear area A a E, though by continually dimin-
ishing the size of each rectangle, and consequently
increasing the number of all, the sum may be made

* 21 L

242 THE PHIL OSOPST OF MA THEM A TICS.

to approach as near as we please to the area A a E, or
to differ from it by less than any given area or space.
Hence A a E is, according to the definition of a limit,
the limit of the sum of the rectangles in question. As
li becomes less and less, or converges toward its limit
0, n becomes greater and greater, or tends toward its
limit co and if we pass to the limit by making li abso-
lutely nothing, we shall have for the limit of the sum
of said rectangles X <*> . Now this is not to be
read or understood as zero multiplied by infinity, but
simply as, in this case, the limit of y li X n. Or, in
other words, it is the symbol, not of a product, but of
the limit of a sum of indefinitely small quantities
whose number tends toward oo as their respective mag-
nitudes tend toward 0. Accordingly, if we find the
limit of that sum for any particular quadrature, we
shall find the value of X oo for that case, or, in
other words, the exact value of the area required.
Such was, at bottom, the idea of Cavalieri ; but that
idea was so obscurely perceived by him that he con-
fessed he did not understand it himself. It was cer-
tainly most inadequately expressed by " the sum of
lines," just as if the sum of any number of lines, how-
ever great, could make up an area or surface. Cava-
lieri was right in refusing to say with Eoberval and
Pascal, " the sum of the rectangles," because that sum
is never equal to the required area. But, instead of
his own inadequate expression, he should have said
the limit of that sum, or the value of X oo considered
as the symbol of such limit ; that is to say, provided
utterance of the truth. The mathematical world is,
indeed, scarcely yet prepared for the perfect utterance

THE PHIL SOPHY OF MA THE MA TICS. 243

of the truth in question, so imperfectly has it under-
stood or interpreted the symbol X oo, as well as

the symbol -. To interpret these two symbols truly

is, in fact, to untie all the principal knots of the Dif-
ferential and Integral Calculus, and cause their mani-
fold difficulties and obscurities to disappear.

The symbol X oo may be easily reduced to the

form ~, a transformation which is effected in every

complete treatise on the calculus. Thus, in Dr. Cour-
tenay's work it is transformed : "to find the value of
the function w = PXQ,= F# X /> #, which takes

the form oo X when x = a. Put P = -. Then

P

u = ~z=z- when x = a. the common form."* He thus
P

reduces oo X to the form -, which he truly calls
ee the common form" for all the symbols of indetermina-

tion. He enumerates six such symbols, namely, ,

:oo . .

oo X 0, oo oo , , oo , 1 ; all of which, in suc-

cession, he reduces to the one common form -, and

deals with them in this form. Now, not one of these
symbols has any signification whatever except as the
limit of some variable expression or quantity, and

since they are reducible to the form -, and are dis-
cussed under that form alone, it is clear that it is

* Part I., Chap. VII., p. 85.

244 THE PHILOSOPHY OF MATHEMATICS.

absolutely indispensable to the correct understanding
of the calculus or the doctrine of limits that we should

possess the true interpretation of the symbol -. That

interpretation is, indeed, the key to the calculus, the
solution of all its mysteries. Hence the labor and
pains I have been at in order to perfect that interpre-
tation, which has not been, as some readers may have
felt assured, however, that the mathematician who has
the most profoundly revolved the problems of the cal-
culus in his own mind will the most fully appreciate
my most imperfect labors.

If any one has suspected that in the foregoing re-
flections on the philosophy of the calculus I have
given undue importance to the question of, tangency,
from which nearly all of my illustrations have been
drawn, the answer is found in the words of a cele-
brated mathematician and philosopher. D'Alembert,
in the article already quoted, says with great truth :
" That example suffices for the comprehension of
others. It will be sufficient to become familiar with
the above example of tangents to the parabola, and as
the whole differential calculus may be reduced to the
problem of tangents, it follows that we can always
apply the preceding principles to the different pro-
blems which are resolved by that calculus, such as the
discovery of maxima and minima, of points of inflex-
ion and of " rcbrousscmcnt," etc.* But, after all, the
question of tangents, however general in its applica-
tion, is not the only one considered in the preceding
pages. The question of quadratures is likewise therein

* Encyclopedic, Art. Differential.

THE PHIL OSOPH Y OF MA THEM A TICS. 245

considered and discussed a question which was the
very first to arise in the infinitesimal analysis, and
which agitated the age of Cavalieri. Yet the diffi-
culties attending this question, which Cavalieri turned
over to his successors for a solution, have, so far as I
know, received but little if any attention from writers
on the philosophy, or theory, or rationale (call it what
you please) of the infinitesimal analysis. It is cer-
tainly not even touched by Carnot, or Comte, or Du-
ll amel. Since the invention of the methods of Newton
and of Leibnitz, the attention of such writers seem to
have been wholly absorbed in the consideration of the
theory of the problem of tangents, the one problem of
the differential calculus, leaving the question of quad-
ratures, which belongs to the integral calculus, to
shift for itself, or to find the solution of its own diffi-
culties. It is possible, indeed, to reduce the question
of quadratures to a question of tangents, since, as we
have seen, the symbol X oo may be reduced to the

form - ; but has any one ever discussed the question of

quadratures under this form, or resolved its difficulties
by the use or application of any other form ? Or, in
other words, has any one even attempted to untie the
"Gordian. knot" (as it is called by Cavalieri) of the
problem of quadratures ? Newton, says Maclaurin,
unraveled that "Qordian knot" and "accomplished
what Cavalieri wished for." * But Newton seems to
have excelled all other men in the faculty of inven-
tion, rather than in the faculty of metaphysical specu-
lation, and hence, in his attempts to remove the diffi-
culties of the infinitesimal analysis, he has created

* Introduction to Maclaurin's Fluxions, p. 49.
21*

246 THE PHILOSOPHY OF MATHEMATICS.

more knots than lie has untied. Indeed, his own
method had its Gordian knot, as well as that of
Cavalieri, and it has been the more difficult of solu-
tion, because his followers have been kept in awe and
spell-bound by the authority of his great name.

NOTES.

NOTE A, PAGE 103,

No less a geometer than M. Legendre lias proceeded on the
assumption that one denominate number may be multiplied by
another. "If we have," says lie, "the proportion A : B : : C : D,
we know that the product of the extremes A X D is equal to the
product of the means B X C. This truth is incontestable for
numbers, It is also for any magnitudes whatever, provided they
are expressed or we imagine them, expressed in numbers." Now,
the author does not here explicitly inform us in what sort of num-
bers, abstract or denominate, the magnitudes should be expressed.
But it is certain that they can be expressed only in concrete or
denominate numbers. His meaning is elsewhere still more fully
shown. For he says, " We have frequently used the expression
the product of two or more lines, by which we mean the product of
the numbers that represent the lines." . ..." In the same man-
ner we should understand the product of a surface by a line, of a
surface by a solid, etc. ; it suffices to have established once for all
that these products are, or ought to be, considered as the products
of numbers, each of the hind which agrees with it. Thus the product
of a surface by a solid is no other iking than the product of a, number
of superficial units ly a number of solid units." Hence it appears
that although M. Legendre saw the absurdity of multiplying
magnitudes into each other, he perceived no difficulty in the
attempt to multiply one denominate number by another such as
superficial units by solid units !

NOTE B, PAGE 130.

2THE CLASSIFICATION OF LINES IN GENERAL,

Every equation, between the variables x and ?/, which is em-
braced in. the general form,

A y m + (B x + C) y m ~ l -f (Da; 2 + E x + F) y m ~ 2 +, etc., = 0,

247

248 THE PHIL OSOPHY OF MA THEM A TICS.

is called algebraic, and all others are transcendental. Hence lines
are divided into algebraic and transcendental, according to the nature
of their equations. It is only the first class or algebraic lines
which are usually discussed in Analytical Geometry.

Algebraic lines are arranged in orders according to the degree
of their equations. Thus a line is of the first, second, or third
order when its equation is of the first, second or third degree,
and so on for all higher orders and degrees. Newton, per-
ceiving that equations of the first degree represented only right
lines, called curves of the first order those which are given by equa-
tions of the second degree. There are certainly no simpler curves
than these ; but although Newton has been followed by Maclaurin,
D'Alembert, and a few others, this denomination has not pre-
vailed. By geometers, at the present day, they are universally
called either lines or curves of the second order, though they are
the simplest of all the classes of curves.

As we have said, the right line is the only one which an equa-
tion of the first degree can represent. No equation of the second
degree can be constructed or conceived so as to represent more
than three curves. These remarkable curves, thus constituting
an entire order of themselves, are usually called "the conic sec-
tions' 7 on account of their relation to the cone. No class of curves
could be more worthy of our attention, since the great Architect
of the Universe has been pleased to frame the system of the worlds
around us, as well as countless other systems, in conformity with
the mathematical theory of these most beautiful ideal forms.

But these lines, however important or beautiful, should not be
permitted to exclude all others from works on Analytical Geome-
try. For among lines of tlie third and higher orders there are
many worthy of our most profound attention. If it were other-
wise, it would be strange indeed, since there are only three curves
of the second order, while there are eighty of the third, and thou-
sands of the fourth. This vast and fertile field should not, as
usual, be wholly overlooked and neglected by writers on Analyti-
cal Geometry. The historic interest connected with some of these
curves, the intrinsic beauty of others, and the practical utility of
many in the construction of machinery, should not permit lliem
to be neglected.

PUBLICATIONS OF J. . LIPPINCOTT COMPANY.

VALUABLE EDUCATIONAL WORKS

Selected from Messrs. J. B. LIPPINCOTT & Co.'s Catalogue, which
comprises over Two Thousand Works in all branches of Literature.
Catalogues furnished on application.

Liberal terms will be made for Introduction. Prices marked f ar&
net, and 20 per cent, must be added to cover postage when sent by mail.

HaldemawHs Outlines of Etymology. By S. S*

HALDEMAN, A.M., author of " Analytical Orthography," " Ele-
ments of Latin Pronunciation," etc. I2mo. Fine cloth. 72
cents.

" This is a most scholarly presenta-
tion of the science of etymology, . . .
to which is added an appendix of in-
estimable value to the student of lan-
guage. . . . It is a marvel of con-
ciseness." New England Journal of
Education, July 5, 1877.

"There is. probably, no man living
who has studied and analyzed the Eng-

lish language so thoroughly and so suc-
cessfully as this distinguished savant of
Pennsylvania, and this little treatise is
the latest fruit of his ripe scholarship
and patient research. No newspaper
notice can do justice to the work, for it
cannot be described, and must be studied
Evening Bulletin.

Neelys Elementary Speller and Reader. Con-

taining the Principles and Practice of English Orthography and
Orthoepy systematically developed. Designed to accord with
the "Present Usage of Literary and Well- Bred Society." In
three parts. By Rev. JOHN NEELY. Fourth edition, carefully
revised. 16mo. Boards. 15 cents. j-

Turner on Punctuation. A Hand-Book of

Punctuation, containing the More Important Rules, and an Ex-
position of the Principles upon which they depend. By JOSEPH
A. TURNER, M.D. New, revised edition. i6mo. Limp cloth.
43 cents. -f-

Home Gymnastics. For the Preservation and

Restoration of Health in Children and Young and Old People
of Both Sexes. With a Short Method of Acquiring the Art of
Swimming. By Prof. T. J. HARTELIUS, M.D. Translated
and adapted from the Swedish, by C. L5FVING. With 31 Illus-
trations. 121110. Flexible cloth covers. 60 cents.

" A practical manual of easy gvm-
nastics for persons of all ages. The
physiological effects of every movement
is explained, and the diagrams and di-
rections make the volume very plain and
useful." New York World.

<e It is an admirable little book, and
deserves to be known by those to whom

their health is a valuable consideration."
Brooklyn Eagle.

''An excellent little book upon the
preservation of health by exercise. It
will be found of benefit to almost every
Gazette.

PUBLICATIONS OF % B. LIPPINCOTT COMPANY.

A YALDABLMAUDY REFEREES LIBRARY

CONTAINING

OF FACTS, CHARACTERS, PLOTS, AND REFERENCES.

Hew Dictionary of Quotations

FROM THE GREEK, LATIN, AND MODERN LANGUAGES.

Word, Facts, and pfwases*

A DICTIONARY OF CURIOUS, QUAINT, AND OUT-OF-THI*
WAY MATTERS.

^Worcester's Coanprelieiisiire Dictionary*

CONTAINING PRINCIPLES OP PRONUNCIATION, RULES OF
ORTHOGRAPHY, ETC.

A TREASURY OF ENGLISH WORDS.
Mvc Volumes. Half Morocco. In cloth box. \$13. SO.

PUBLICATIONS OF J. B. LIPPINCOTT COMPANY.

"A LIBRARY IN ITSELF."

CHAMBERS ENCYCLOPEDIA.

A DICTIONARY OF UNIVERSAL KNOWLEDGE FOR
THE PEOPLE.

HOUSEHOLD EDITION.

In Ten Large Octavo Volumes, containing 1 Eight
Thousand Three Hundred and Twenty Pages. Illus-
trated with about Four Thousand Engravings,
printed on good paper. Price per set, in
Cloth, \$15.00; Library Sheep, \$20.00;
Half Morocco, \$25. OO.

The Publishers invite the attention of THE GENERAL READER,
THE TEA<'IIKR, SCHOOLS, and LIBRARIANS to this the Latest Revised
Edition of " Chambers'* Encyclopedia," which is offered at so low
a price that this valuable and popular "Dictionary of Universal
Knowledge" is brought within the means of every reader.

The " Encyclopaedia" is not a mere collection of elaborate
treatises in alphabetical order, but a work to be readily consulted as
a DICTIONARY on every subject on which people generally require
some distinct information.

This edition, embraced in ten volumes, forms the most compreJien*
,,ive and cheapest Encyclopedia ever issued in the English language.

PUBLICATIONS OF J. B. LIPPINCOTT COMPANY.

THE STANDARD DICTIONARY
OF THE- ENGLISH LANGUAGE,

Worcester's

QUARTO DICTIONARY,

Fully Illustrated. With Four Full-page Illuminated Plates. Library
Sheep, Marbled Edges. \$10.00.

RECOMMENDED BY THE MOST EMINENT WRITERS.

ENDORSED BY THE BEST AUTHORITIES,

THE NEW EDITION OF

Contains Thousands of Words not to be found in any other Dictionary.
THE COMPLETE SERIES OF

DICTIONARIES*

QUARTO DICTIONARY. Profusely Illustrated. Library sheep. \$10.00.
ACADEMIC DICTIONARY. Illustrated. Crown 8vo. Half roan. \$i.\$cvj-
COMPREHENSIVE DICTIONARY. Illustrated, xamo. Half roan.. \$1.40.*
NEW COMMON SCHOOL DICTIONARY. Illustrated. i2mo. Half

roan. 90 cents.f

PRIMARY DICTIONARY. Illustrated. i6mo. Half roan. 48 cents.f
POCKET DICTIONARY. Illustrated. 2 4 mo. Cloth. 50 cents.f Roan,

flexible. 69 cents.f Roan, tucks, gilt edges. 78 cents. f