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NOTE A 247 

NOTEB 247 





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 


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. 


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 


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 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 


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. 


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 


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 

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- 


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 

* 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 


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 
hand, it had its disadvantages ; it was indirect and 
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. 


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 


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. 



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 
of their radii" 

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. 


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- 

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


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. 


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 


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 

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 


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. 


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- 

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, 


ornetry as dirtinot and different 

h ' l > 

be broken up and confounded, as if there 



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 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. 


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 
of a limit? Was this merely to conceal a harsh con- 
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. 


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 


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- 


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 


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 


made to approach as nearly as we please, but which 
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. 


" 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 

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, 



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 



[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. 



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 


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

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 

" 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. 


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. 


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. 


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 


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 

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. 



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. 

Let us return to the original instance of the circle 
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 



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 

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 


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 


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, 


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, 
jad having as little .difference from A as you please; 
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 


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 

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 


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, 



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 

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 



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- 

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 


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 

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 


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 

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. 


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 
other as their radii. 

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 



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- 
ference into its radius. 

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

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 



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- 



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. 



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. 


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 
discoveries are made. 

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, 


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 


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. 


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, 



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 


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 

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 


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. 


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 
answer this question for himself. 

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 


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 
that it is not about an axiom, but only about some- 
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. ^ 




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 
squares of their radii, and these three radii can evi- 
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. 


"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 


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 ; 


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


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 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 . 


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 

No one was more sensible than Cavalieri liinise.i 


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. 



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. 


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- 



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 

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 


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. 

Let us, then, return to the case of the triangle, which 
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 

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 


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 


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 



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 


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

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 





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 



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



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- 

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 
the result, for they made it exactly true instead of 
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- 


val and Pascal. For here two rays of darkness are 
made to produce light. 

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, 


literally understood, has no meaning. For is a 


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


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. 


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


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 


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 


the sum of all the little triangles. For the triangle 

, 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 


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 

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


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= . 


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 


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


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. 


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- 

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 

a BH . 

b = -- h - ; 


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 

the second member of the above equation. Hence, if 

T> TT 

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



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 


infinite, and may be omitted as nothing compared 

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

remembered that, according to the same supposition, 


the first factor becomes = o. Hence the expression 


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 



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 




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 #), 


But, according to a well-known algebraic formula^ 

1 . 2i . o 



Hence, S = . 

or S- 




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 


tends continually more and more toward an equality 


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


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. 


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

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

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- 


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. 



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* 



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 

and we have, from the equation of the curve, 

hence 2yk + k 2 = 2 ph. 


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

I 2_p ' * k x 


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 


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 

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 


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

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



k) sin A 



when Je tends towards zero. That limit is evidently 

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

2.3jp 3 ' 

observing that = A B. 


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 

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. 


"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 



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 



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 
made no progress. 

" 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* 


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. 


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 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. 


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 


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 


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 


such darkening of the very first principles of the 

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- 


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. 


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: 

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 
cal Geometry, would be ready with the answer 1< 
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 


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. 


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 



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

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 




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- 

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* 


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. 


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 


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 

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 


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* 


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 


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 


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 
Analytical Geometry." Why then was this great dis- 
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 


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- 



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- 



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. 




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 












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 


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- 



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- 






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 




other . D ,l * Wlloll y <lependent 


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 



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 



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* 



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); + 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, 





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

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




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 


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- 


lli >. 





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? - ^) 


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


but y 2 = 2jp&'; 

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




tlierefore ; 

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

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 

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 



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. 


is the equation of the curve required. 



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 
which Descartes had laid, and which had introduced 
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. 



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 

" Indeed, since we can suppose x as small as we 
please, we can also render as small as we please the 
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 : 


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 


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

" 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- 
thcm as little as we please from the radius R. Then 
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. 


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 


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 = /?. 


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 

" 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 


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. 



" 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. 
This principle had the advantage of an extreme sim- 
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 


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 


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


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. 


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 
said that he made no attempt to furnish such a reply. 
For it is well known that he did attempt such a reply, 
and also that it was a failure.* " Leibnitz," says M. 
Comte, " urged to answer, had presented an explana- 
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 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 : 


" 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 
we please ; but every intelligent student asks, Will it 
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. 


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. 


" "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. 


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- 


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 


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 


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. 


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 

" 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 < 




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 = TP = TP 

NO ~ 




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 


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 * 



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 

" 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' 


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 


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 


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 

* 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." 



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 
please from equality. 

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, 


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 
unaccountable paradox that mathematicians should 
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 



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 


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 


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 


There was, therefore, an error of 


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 = , 


which followed from the erroneous rule of differences. 

And the measure of this error is = z. Therefore 


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 

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


. 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, 

we throw out the minus quantity ~ as infinitely 

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. 




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- 

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 



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 



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 

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

Fig. 1. 




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. 


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 


- .--,. 

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- 



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 
added and then subtracted, which, of course, made no 
difference in the result. 

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

Fig. 2. 


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 


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. 


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 

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 



dy = p 
dx y' 

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

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' 


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 
to say, we start with the expression instead of 


, 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 

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

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


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 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 



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



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

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 


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 

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- 





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- 


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 

"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, 



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 




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- 


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 

" 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. 




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 : 


"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 


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." 


" 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 
breadth A F is diminished 
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 

"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 * 

f l 



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 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 

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- 


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- 


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- 


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. 


" 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 
defenders of the calculus had the advantage, it must 
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 
the ablest professors, as well as made it my business 
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. 


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 


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. 


, 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. 


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. 


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. 



" 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 

" 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 * 


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 



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. 


- . 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 


= 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- 

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 


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. 



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* 












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 


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 


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 


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 = 


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


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 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 


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. 


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 ; 





" 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 


"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. 


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. 



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. 


" 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. 



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

and meet, I say, the angle _ 
BAD, contained between 
the chord and the tangent, 
will be diminished in infi- 
nitum, and will ultimately 

" 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.] 


" 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 


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 




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 


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. 


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- 



* The exclamation. of Carnot when he saw his own theory of the 
method of Leibnitz as propounded by Lagrange. 


/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 



= 1, the 


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 

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. 



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 ? 


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' 



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 


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 


, because Urn. = 1. This is evident, for as 


the point c approaches the point C, it is obvious that | 



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 


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 

the limit of the variable ratio . For as c ap- 


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 ! 



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- 


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 


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 


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 

" 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 

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 
instead of shining with solutions.* 

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. 


.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. 

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. 



= 2 ax. 


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 

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 


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. 


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 


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- 


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 

du ~ , , 
= 2ax + adx ) 


or - = 2 a x. 

But if we adopt this last form, we escape the illegiti- 

mate expression = 2 a z, with all its shuffling 

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, 
about which I had been taught nothing, and consequently knew 
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. 


limit of is -. Hence, if we are not afraid to 


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 


^ the problem of quadratures, the only alterna- 
d to be either to commit an error with Pas- 

* Eeflexions, etc., Chap. I., p. 41. 


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 


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 

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 ~. 

* Reflexions, etc., Chap. I v p. 37, 


" 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 ." 


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 . 

simply to repeat the objection instead of replying to 

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? 


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 

limit is the quantity which the ratio - approaches 


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 


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* 


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. 



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 -. 


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 


"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 . 


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 



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. 


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- 



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 

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 


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 

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 


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 


ratio - always tends, as z becomes smaller and smaller. 

u J ' 

toward the limit , and hence in passing to the 

limit, by making z = 0, we have 

0___ a_ 
~~ 9, v 

value, which is . Hence D'Alembert was in error 


Now in this case - may not have any value as in the 

case of the couriers ; for it has, and can have, only one 


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 


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 


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 


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- 

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. 


are always equal, and hence their limits are equal. 

That is to say, the limit of the one - = 3 the limit 


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 

of contact), toward which the value of the fraction - 


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> 


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 
equation, and most adroitly escapes the dreaded formula 

-. But there was no necessity whatever for any such 



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 equation of the secant line E S passing 
through the points x f y f and x n ' y n ', is 


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. 


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 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- 


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- 


* Courtenay's Calculus, Part II., Chap. II., p. 191. 


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 


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 

writing the differential equation of a tangent line to 

the point x f , y f , we shall have to eliminate - in the one 

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 



error in the elimination of ^-. Suppose, for exam- 


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. 


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 


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, 



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 

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 


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 
either he or the world had been ready for the exact 
utterance of the truth. The mathematical world is, 
indeed, scarcely yet prepared for the perfect utterance 


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 


u = ~z=z- when x = a. the common form."* He thus 

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. 


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 
suspected, " much ado about nothing." J. have always 
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. 


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. 


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. 


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. 


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, 



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. 




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