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The nature and utility of mathematics; with the best methods of instruction explained and illustrated

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comes to the place, is not its ultimate velocity ;
when it has arrived, is none. But the answer is

nswered. eas j j f r > 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

Ultimate ceases. And in like manner, by the ultimate
Uo; ratio of evanescent quantities is to be under-
stood the ratio of the quantities not before they
vanish, not afterwards, but with which they
vanish. In like manner, the first ratio of nas-
cent quantities is that with which they begin

Howitia t be. And the first or last sum, is that with
which they begin 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
* t* exceed. This is the ultimate velocity. And there
is the like limit in all quantities and proportions
that begin and cease to be. And since such lim-
its are certain and definite, to determine the same
is a problem strictly geometrical. But whatever
is geometrical we may be allowed to use in de-

CHAP. V.J DIFFERENTIAL CALCULUS. 313

termining and demonstrating any other thing Geometri-
cal,
that is likewise geometrical.

FRUITS OF NEWTON'S THEORY.

337. The main difficulties in the higher ma-
thematics, have arisen from inadequate or erro-
neous notions of ultimate or evanescent quanti-
ties, and of the ratios of such quantities. After
two hundred years of discussion, of experiment
and of trial, opinions yet differ widely in regard
to them, and especially in regard to the forms
of language by which they are expressed.

One cannot approach this subject, which has Difficulty
so long engaged the earnest attention of the
greatest minds known to science, without a feel-
ing of awe and distrust But tapers sometimes O rthe
light corners which the rays of the sun do not
reach ; and as we must adopt a theory in a sys-
tem of scientific instruction, it is perhaps due
to others, that we should assign our reasons
therefor.

338. An ultimate, or evanescent quantity, ultimate
which is the basis of the Newtonian theory, is
not the quantity " before it vanishes, nor after-
wards ; but, with which it vanishes?

314 MATHEMATICAL SCIENCE. [BOOK II.

I have sought, in what precedes and follows,
Very to define this quantity to separate it from all
other quantities to present it to the mind in
important, a crystallized form, and in a language free from
all ambiguity; and then to explain how it be-
comes the key of a sublime science.

As a first step in this process, I have defined
First step continuous quantity (Art. 322), and this is the
only class of quantity to which the Differential
Next step. Calculus is applicable. The next step was to de-
fine consecutive values, and then, the difference
between any two of them (Art. 327). These differ-
uitimate ences are the ultimate or evanescent quantities
of of Newton. They are not quantities of deter-
minate magnitudes, but such as come from va-
riables that have been diminished indefinitely.
They form a class of quantities by themselves,
which have their own language and their own
infinites!- ^ aws ^ change; and they are called, Infinitcsi-
mals - mals, or Differentials.

Since the difference between any two values of

on what a variable quantity, which are near together, but

not consecutive, will depend on the relative

VALUES of the quantities and the law of change,

it is plain, that when we pas? to the limit of this

ofvaiues difference, such limit will also depend for its

value on the variable quantity and the law of

depends, change : and hence, the infinitesimals are un-

CHAP. Y.J DIFFERENTIAL CALCULUS. 315

equal among themselves, and any two of them
may have, the one to the other, any ratio what-
ever.

These infinitesimals will always be quantities Quantities

of the same kind as those from which they were

of

derived ; for the kind of quantity which ex-
presses a difference, is the same, whether the dif- same idnd.
fereuce be great or small

LIMITS.

339. Marked differences of opinion exist Limiu
among men of science in regard to the true Difference
notion of a, limit; and hence, definitions have of
ueen given of it, differing widely from each other. Definitlong
AVe have adopted the views of Newton, so clearly
set forth in the lemmas and scholium which we
have quoted from the Principia. He uses, as HOW defined
stated in the latter part of the scholium, the term
limit, to designate the ultimate or evanescent
value of a variable quantity ; and this value is

by Newtca.

reached under a particular hypothesis. Hence,
our definition (Art. 323).

Let us now refer again to the case of tan-
gency.

Let APB be any curve whatever, and TPF a caw of
tangent touching it at the point P. Draw any
chord of the curve, as PB, and through P and B tangency

816

MATHEMATICAL SCIENCE. [BOOK IL

Secant:

how it

draw the ordinates PD and BH. Also draw PG
parallel to TH.

CF
again con- Then, -p~n= tang. FPC = tang, the angle

PTH, which the tangent line TPF makes with
sidered. the axis TH.

But, -pfj = tangent of the angle BPG.

If now we suppose BH
to move towards PD, the
angle BPG will approach
the angle FPC, which is
its limit. When BH be-
comes consecutive with
PD, B C will reach its ul-
timate value : and since by
Lemma VII., the ultimate ratio of the arc. chord.

becomes a

and tangent, any one to any other, is the ratio of
equality, it follows that they must then all be
equal, each to each. Under this hypothesis the
point B must fall on the tangent line TPF', that
is, the chord and tangent, in their ultimate state,
have two points in common; hence they coin-
cide ; and as the two points of the arc are con-
secutive, it must also coincide with the chord and

and when, tangent.

This, at first sight, seems impossible. But if
be granted that two points of a curve can be

tangent ;

Not
impossible

CHAP. V.] DIFFERENTIAL CALCULUS. 317

consecutive and that a straight line can be drawn
through any two points, we have the solution.
If we deny that two points of the curve can be
consecutive, we deny the law of continuity.

The method of Leibnitz adopted the simple Method
hypothesis that when the point B approached the

point P, infinitely near, the lines CF and CB

of

become infinitely small, and that then, either may

be taken for the other ; under which hypothesis

the ratio of PC to CB, becomes the ratio of PC ** lb " ltx -

toCF.

WHAT THE LEMMAS OF NEWTON PROVE.

340. The first lemma, which is " the corner-
stone and support of the entire system," predi-
cates ultimate equality between any two quanti-
ties which continually approach each other in
value, and under such a law of change, that, in
any finite time they shall approach nearer to
each other than by any given difference. The
common quantity towards which the quantities
separately converge, is the limit of each and both
of them, and this limit is always reached under
a particular supposition.

Lemmas II., III., and IV. indicate the steps by
which we puss from discontinuous to continuous wht
quantity. They introduce us, fully, to the law

318 MATHEMATICAL SCIENCE. [BOOK II.

they of continuity. They demonstrate the great
truth, that the curvilinear space is the common
limit of the inscribed and circumscribed paral-
lelograms, and that this limit is reached under
the hypothesis that the breadth of each parallelo-
gram is infinitely small, and the number of them,
infinitely great. Thus we reach the law of con-
tinuity; and each parallelogram becomes a con-
Links in necting link, in passing from one consecutive
value to another, when we regard the curvilinear
area as a variable. That there might be no mis-
theiawof apprehension in the matter, corollary 1, of Lem-
ma III., affirms, that, "the ultimate sum of these
evanescent parallelograms, will, in all parts, coin-
Continuity. c ide with the curvilinear figure." Corollary 4,
also, affirms that, "therefore, these ultimate fig-
ures (as to their perimeters, acE), are not recti-
linear, but curvilinear limits of rectilinear fig-
Common ures:" that is, the curvilinear area AEa is the
common limit of the inscribed and circumscribed
parallelograms, and the curve Edcba, the corn-
limit. mon limit f their perimeters. This can only
take place when the ordinates, like Dd, Cc, Bk,
become consecutive ; and then, the points o, n,
m and I fall on the curve.

The law of continuity carries with it, neces-

Whatthe sarily, the ideas of the infinitely small and the

law of infinitely great. These are correlative ideas, and

CHAP. \ ] DIFFERENTIAL CALCULUS. 319

in regard to quantity, one is the reciprocal of continuity
the other. The inch of space, as well as the
curved line, or the curvilinear surface of geo- im P hei -
metry, has within it the seminal principles of
this law.

If we regard it as a continuous quantity, hav- continuity,
ing increased from one extremity to the other,
without missing any point of space, we have, the
law of change, the infinitely small (the difference
between two consecutive values, or the link in
the law of continuity), and the infinitely great,
in the number of those values which make up the
entire line.

It has been urged against the demonstrations objection*
of the lemmas, that a mere inspection of the fig-
ures proves the demonstrations to be wrong.
For, say the objectors, there will be, always, ob-
riinixly, "a portion of the exterior parallelograms
lying without the curvilinear space." This is
certainly true for any finite number of parallelo-
grams.

But the demonstrations are made under the objection!
express hypothesis, that, " the breadth of these

parallelograms be supposed to be diminished, and

tally
their number to be augmented, in finitum"

Under this supposition, as we have seen, the

points, o, n, in. and /, fall in the curve, and then an * wered -

the areas named are certainly equal

320

MATHEMATICAL SCIENCE. [BOOK IL,

NEWTON'S METHOD IN HARMONY WITH
THAT OF LEIBNITZ.

341. The method of treating the Infinitesi-
Hannony mal Calculus, by Leibnitz, subsequently ampli-
O f fied and developed by the Marquis L'Hopital, is
Methods, based on two fundamental propositions, or de-
mands, which were assumed as axioms.

I. That if an infinitesimal be added to, or sub-
First tracted from, a finite quantity, the sum or differ-
ence will be the same as the quantity itself. This

demand, demand assumes that the infinitesimal is so small
that it cannot be expressed by numbers.

II. That a curved line may be considered as
8erx>nd. made up of an infinite number of straight lines,

each one of which is infinitely small.

It is proved in Lemma II. that the sum of the
what the ultimate rectangles Ab, Be, Cd, Do, etc., will be
equal to the curvilinear area AaK This can only
be the case when each is " less than any given
space," and their number infinite. "What is
meant by the phrase, "becomes less than any
given space " ? Certainly, a space too small to be
expressed by numbers; for, if we have such a
space, so expressed, we can diminish it by dimin-
ishing the number, which would be contrary to
the hypothesis. This ultimate value, then, of
either of the rectangles, is numerically zero: and

Lemma

proves.

Ultimate

CHAP. V.] DIFFERENTIAL CALCULUS. 321

hence, its addition to, or subtraction from, any value of
finite quantity, would not change the value. The
ultimates of Newton, therefore, conform to the

Rectangles.

first demand of Leibnitz, as indeed they should
do; for, they are not numerical quantities, but
connecting links in the law of continuity.

It is proved in Lemma VII., that the ultimate Ratio
ratio of the arc, chord, and tangent, any one to
any other, is the ratio of equality: hence, their ul-
timate values are equal. When this takes place, of chord.
the two extremities of the chord become consecu-
tive, and the remote extremity of the tangent tangent,
falls on the curve, and coincides with the remote
extremity of the chord : that is, F falls on the and
curve, and PB and PF, coincide with each other,
and with the curve. The length of this arc, arc ' eqna1 '
chord, or tangent, in their ultimate state, is

a value familiar to the most superficial student of
the Calculus.

Behold, then, one side of the inscribed polygon, coinci-
when such side is infinitely small, and the num-
ber of them infinitely great.

That such quantities as we have considered, Quantities
have a conceivable existence as subjects of
thought, and do or may have, proximal ively. an haveareal
actual existence, is clearly stated in the latter

322 MATHEMATICAL SCIENCE. [BOOK II.

part of the scholium quoted from the Principia.
value, It is there affirmed : " This is the ultimate velo-
city. And there is a like limit in all quantities
and proportions which begin and cease to be.
And since such limits are certain and definite,
oitimateiy. to determine the same is a problem strictly geo-
metrical. But whatever is geometrical we may
be allowed to use in determining and demon-
strating any other thing that is likewise geome-
Newton trical." * Hence, the theory of Newton conforms

and

Leibnitz, to the second demand in the theory of Leibnitz.

DIFFERENT DEFINITIONS OF A LIMIT.

342. The common impression that mathe-
Different matics is an exact science, founded on axioms
of limits, too obvious to be disputed, and carried forward
by a logic too luminous to admit of error, is
certainly erroneous in regard to the Infinitesimal
Calculus. From its very birth, about two hun-
dred years ago, to the present time, there have
Different been very great differences of opinion among the
best informed and acutest minds of each gen-
eration, both in regard to its fundamental prin-
ciples and to the forms of logic to be employed
in their development. The conflicting opinions

NOTB. -Tb italics are added ; they are not in the text

CHAP. V.] DIFFERENTIAL CALCULUS. 323

appear, at last, to have arranged themselves into
two classes ; and these differ, mainly, on this
question : "What is the correct apprehension and Difference*.
right definition of the word limit ? All seem to
agree that the methods of treating the Calculus
must be governed by a right interpretation of
this word. The two definitions which involve
this conflict of opinion, are these :

1. The limit of a variable quantity is a quan-
tity towards which it may be made to approach
nearer than any given quantity and which it
reaches under a particular supposition.

And the following definition, from a work on
the Infinitesimal Calculus by M. Duhamel, a
French author of recent date:

2. The limit of a variable is the constant quan-

tity which the variable indefinitely approaches,
but never reaches.

This definition finds its necessary complement
in the following definition by the same author:

"We call," says he, "an infinitely small quan- comple-

ment
tity, or simply, an infinitesimal, every variable

maynilude of which the limit is zero."

The difference between the two definitions is Difference

between

simply this : by the first, the variable, ultimately, definitions
reaches its limit; by the second, it approaches
the limit, but never reaches it. This apparently
slight difference in the definitions, is the divid-

324 MATHEMATICAL SCIENCE. [BOOK II.

ing line between classes of profound thinkers;
and whoever writes a Calculus or attempts to
Difference, teach the subject, must adopt one or the other of
these theories. The first is in harmony with the
theories of Leibnitz and Newton, which do not
differ from each other in any important particu-
Generai lar. It seems also to be in harmony with the
great laws of quantity. In discontinous quan-
tity, especially, we certainly include the limits
in our thoughts, and in the forms of our lan-
whatwe guage. When we speak of the quadrant of a
"them/ circle, we include the arc zero and the arc of
ninety degrees. Of its functions, the limits of
the sine, are zero and radius; zero for the arc
zero, and radius for the arc of ninety degrees.
For the tangents, the limits are zero and infinity;
zero for the arc zero, and infinity for the arc
of ninety degrees ; and similarly, for all the other
For ail functions. For all numbers, the limits are zero
and infinity; and for all algebraic quantities,
minus infinity and plus infinity.

TThen we consider continuous quantity, we
For contin- find the second definition in direct conflict with
towqnan- ^ ^^ Lemma of Newton, which has been
well called, " the corner-stone and foundation
of the Principia." It is very difficult to com-
prehend that two quantities may approach each
other in value, and in any given time become

ClfAl'. V.] DIFFERENTIAL CALCULUS. 325

nearer equal than any given quantity, and yet in conflict
never become equal ; not even when the approach
can be continued to infinity, and when the law wlth
of change imposes no limit to the decrease of

Newton.

their difference. This, certainly, is contrary to
the theory of Xewton.

Take, for example, the tangent line to a curve, Example
at a given point, and through the point of tan-
gency draw any secant, intersecting the curve, in

of the

a second point. If now, the second point be
made to approach the point of tangeucy, both
definitions recognize the angle which the tangent tangent line,
line makes with the axis of abscissas as the limit
of the angles which the secants make with the
same- axis, as the second point of secancy ap-
proaches the tangent point. By the first defini- First
tion, the supposition of consecutive points causes
the secant line to coincide with, and become the definition
tangent. But by the second definition, the se-
cant line can never become the tangent, though
it may approach to it as near as we please. This

to general

is in contradiction to all the analytical methods
of determining the equations of tangent lines to method,
curves. See corollaries 1, 2, 3, and 4 of Lemma
III., in which all the quantities referred to are
supposed to reach their limits.

By the second definition, there would seem 'to second
be an impassable barrier placed between a vari- definition:

326 MATHEMATICAL SCIENCE. [BOOK II.

what it able quantity and its limit. If these two quan-
tities are thus to be forever separated, how can
they be brought under the dominion of a corn-
does, mon law, and enter together into the same equa-
tion ? And if they cannot, how can any prop-
erty of the one be used to establish a property of
the other ? The mere fact of approach, though
Result, infinitely near, would not seem to furnish the
necessary conditions.

The difficulty of treating the subject in this
Difficulty, way is strikingly manifested in the supplement-
ary definition of an infinitesimal. It is defined,
simply, as " every variable magnitude whose limit
is zero"

Now, may not zero be a limit of every variable
Not which has not a special law of change? Is not

definite. . . . - _ . . . , .

this definition too general to give a DEFIXITE
idea of the individual thing defined an infini-
tesimal ? We have no crystallized notions of a
class, till we apprehend, distinctly, the individu-

shouidhe. als of the class their marked characteristics
their harmonies and their differences; and also,
their laws of relation and connection.

Having given and illustrated these definitions,

M.Duha- M. Duhamel explains the methods by which we
can pass from the infinitesimals to their limits;

*

an( j w h eU) an( j under what circumstances, those
limits may be substituted and used for the quan-

('HAP. V.] DIFFERENTIAL CALCULUS. 327

titics themselves. Those methods have not
seemed to me as clear and practical as those of
Newton and Leibnitz.

It is essential to the unity of mathematical unity in
science, that all the definitions, should, as far as
possible, harmonize with each other. In all dis-
continuous quantities, the boundaries are in- Mathematic *
eluded, and are the proper limits. In the hyper-
bola, for example, we say that the asymptote is
the limit of all tangent lines to the curve. But
the asymptote is the tangent, when the point of

contact is at an infinite distance from the ver-

necessary.

tex : and any tangent will become the asymptote,
under that hypothesis.

If denotes any portion of a plane surface, y Differential,
the ordinate and x the abscissa, we have the
known formula :

ds = ydx.

If we integrate between the limits of x = 0,
and x = a, we have, by the language of the Surfac ^
Calculus

a

fds =fydx,
o

which is read, "integral of the surface between ROW read,
limits of x = 0, and x = a" in which both bound-
aries enter into the result.

328 MATHEMATICAL SCIENCE. [BOOK IL

Limits The area, actually obtained, begins where
Area. = 0, and terminates where x = a, and not at
values infinitely near those limits.

WHAT QUANTITIES ARE DENOTED BY 0.

343. Our acquaintance with the character 0,
What quan- begins in Arithmetic, where it is used as a ne-
cessary element of the arithmetical language, and
utiesare where it is entirely without value, meaning, abso-
lutely nothing. Used in this sense, the largest
finite number multiplied by it, gives a product
equal to zero ; and the smallest finite number di-
vided by it, gives a quotient of infinity.

"When we come to consider variable and con-
May not tinuous quantity, the infinitesimal, or element of
change from one consecutive value to another, is
be the oof no t the zero of Arithmetic, though it is smaller
than any number which can be expressed in
terms of one, the base of the arithmetical system.
New Hence, the necessity of a new language. If the
language variable is denoted by x, we express the infini-
tesimal by dx ; if by y, then by dy ; and similarly,
for other variables.

Now, the expressions dx and dy, have no exact
HOW it is synonyms i n the language of numbers. As com-
pared with the unit 1, neither of them can be ex-
amined, pressed by the smallest finite part of it. Hence,

CHAP. V.] DIFFERENTIAL CALCULUS. 329

when it becomes necessary to express such quan-
tities in the language of number, they can be
denoted only by 0. Therefore, this 0, besides its what o

meant.

first function in Arithmetic, where it is an ele-
ment of language, and where the value it denotes
is absolutely nothing, is used, also, to denote the
numerical values of the infinitesimals. Hence, it
is correctly defined as a character which some- sometime*
times denotes absolutely nothing, and sometimes

an

an infinitely small quantity. We now see, clearly,

what appears obscure in Elementary Algebra, inflnitesi

maL

that the quotient of zero divided by zero, may
be zero, a finite quantity, or infinity.

INSCRIBED AND CIRCUMSCRIBED POLYGONS
UNITE ON THE CIRCLE.

344. The theory of limits, developed by New- inscribed

polygon.

ton, is not only the foundation of the higher
mathematics, but indicates the methods of using
the Infinitesimal Calculus in the elementary
branches. This Calculus being unknown to the
ancients, their Geometry was encumbered by the
tedious methods of the reductio ad absurdum.
Newton says in the scholium: "These lemmas K avoids the

are premised to avoid the tediousness of d^du- absurdnm.
cing perplexed demonstrations ad absurdum, ac-
cording to the method of the ancient geometers."

330

MATHEMATICAL SCIENCE. [BOOK II.

Lemma I., which is the " corner-stone and
Lemma i. foundation of the Principia" is also the golden
link which connects geometry with the higher
mathematics.

It is demonstrated in Enclid's Elements, and
also in Davies' Legendre, Book V., Proposition
X., that *' Two regular polygons of the same num-
ber of sides can be constructed, the one circum-
scribed about the circle and the other inscribed
within it, which shall differ from each other by
less than any given surface."

The moment it is proved that the exterior
and interior polygons may be made to differ
from each other by less than any given surface,
Lemma I. steps in and affirms an ultimate equal-
ity between them. And when does that ultimate
equality take place, and when and where do they
become coincident ? Newton, in substance af-
firms, in his lemmas, "on their common limit,
the circle," and under the same hypothesis as
causes the inscribed and circumscribed parallelo-
grams to become equal to their common limits,

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