Elementary illustrations of the differential and integral calculus

MATH/STAT.

MATH/STATj

ELEMENTARY ILLUSTRATIONS

DIFFERENTIAL AND INTEGRAL
CALCULUS

BY

AUGUSTUS DE MORGAN

NEW EDITION

CHICAGO
THE OPEN COURT PUBLISHING COMPANY

LONDON
KEGAN PAUL, TRENCH, TRUBNER & Co., LTD.

1909

xcy

EDITOR'S PREFACE.

THE publication of the present reprint of De Morgan's Ele
mentary Illustrations of the Differential and Integral Cal
culus forms, quite independently of its interest to professional
students of mathematics, an integral portion of the general educa
tional plan which the Open Court Publishing Company has been
systematically pursuing since its inception, — which is the dissemi
nation among the public at large of sound views of science and of
an adequate and correct appreciation of the methods by which
truth generally is reached. Of these methods, mathematics, by
its simplicity, has always formed the type and ideal, and it is
nothing less than imperative that its ways of procedure, both in
the discovery of new truth and in the demonstration of the neces
sity and universality of old truth, should be laid at the foundation
of every philosophical education. The greatest achievements in
the history of thought — Plato, Descartes, Kant — are associated
with the recognition of this principle.

But it is precisely mathematics, and the pure sciences gener
ally, from which the general educated public and independent
students have been debarred, and into which they have only rarely
attained more than a very meagre insight. The reason of this is
twofold. In the first place, the ascendant and consecutive charac
ter of mathematical knowledge renders its results absolutely un
susceptible of presentation to persons who are unacquainted with
what has gone before, and so necessitates on the part of its devo
tees a thorough and patient exploration of the field from the very
beginning, as distinguished from those sciences which may, so to
speak, be begun at the end, and which are consequently cultivated
with the greatest zeal. The second reason is that, partly through
the exigencies of academic instruction, but mainly through the
martinet traditions of antiquity and the influence of mediaeval

vi EDITOR'S NOTE.

logic-mongers, the great bulk of the elementary text-books of
mathematics have unconsciously assumed a very repellent form, —
something similar to what is termed in the theory of protective
mimicry in biology "the terrifying form." And it is mainly to
this formidableness and touch-me-not character of exterior, con
cealing withal a harmless body, that the undue neglect of typical
mathematical studies is to be attributed.

To this class of books the present work forms a notable ex
ception. It was originally issued as numbers 135 and 140 of the
Library of Useful Knowledge (1832), and is usually bound up with
De Morgan's large Treatise on the Differential and Integral
Calculus (1842). Its style is fluent and familiar; the treatment
continuous and undogmatic. The main difficulties which encom
pass the early study of the Calculus are analysed and discussed in
connexion with practical and historical illustrations which in point
of simplicity and clearness leave little to be desired. No one who
will read the book through, pencil in hand, will rise from its peru
sal without a clear perception of the aim and the simpler funda
mental principles of the Calculus, or without finding that the pro-
founder study of the science in the more advanced and more
methodical treatises has been greatly facilitated.

The book has been reprinted substantially as it stood in its
original form ; but the typography has been greatly improved, and
in order to render the subject-matter more synoptic in form and
more capable of survey, the text has been re-paragraphed and a
great number of descriptive sub-headings have been introduced, a
list of which will be found in the Contents of the book. An index
also has been added.

Persons desirous of continuing their studies in this branch of
mathematics, will find at the end of the text a bibliography of the
principal English, French, and German works on the subject, as
well as of the main Collections of Examples. From the informa
tion there given, they may be able to select what will suit their
special needs.

THOMAS). MCCORMACK.
LA SALLE, 111., August, 1899.

CONTENTS:

PAGE

On the Ratio or Proportion of Two Magnitudes 2

On the Ratio of Magnitudes that Vanish Together .... 4
On the Ratios of Continuously Increasing or Decreasing Quan
tities 7

The Notion of Infinitely Small Quantities n

On Functions 14

Infinite Series 15

Convergent and Divergent Series 17

Taylor's Theorem. Derived Functions 19

Differential Coefficients 22

The Notation of the Differential Calculus 25

Algebraical Geometry 29

On the Connexion of the Signs of Algebraical and the Direc
tions of Geometrical Magnitudes 31

The Drawing of a Tangent to a Curve 36

Rational Explanation of the Language of Leibnitz .... 38

Orders of Infinity 42

A Geometrical Illustration : Limit of the Intersections of Two

Coinciding Straight Lines 45

The Same Problem Solved by the Principles of Leibnitz . . 48

An Illustration from Dynamics ; Velocity, Acceleration, etc. . 52

Simple Harmonic Motion 57

The Method of Fluxions 60

Accelerated Motion 60

Limiting Ratios of Magnitudes that Increase Without Limit. 65

Recapitulation of Results Reached in the Theory of Functions. 74

Approximations by the Differential Calculus 74

Solution of Equations by the Differential Calculus .... 77

Partial and Total Differentials 78

Vlll CONTENTS.

PAGE
Application of the Theorem for Total Differentials to the

Determination of Total Resultant Errors 84

Rules for Differentiation 85

Illustration of the Rules for Differentiation 86

Differential Coefficients of Differential Coefficients .... 88
Calculus of Finite Differences. Successive Differentiation . 88
Total and Partial Differential Coefficients. Implicit Differ
entiation 94

Applications of the Theorem for Implicit Differentiation . . 101

Inverse Functions 102

Implicit Functions 106

Fluxions, and the Idea of Time no

The Differential Coefficient Considered with Respect to Its

Magnitude 112

The Integral Calculus 115

Connexion of the Integral with the Differential Calculus . . 120

Nature of Integration 122

Determination of Curvilinear Areas. The Parabola . . . 124

Method of Indivisibles 125

Concluding Remarks on the Study of the Calculus .... 132
Bibliography of Standard Text-books and Works of Reference

on the Calculus 133

Index 143

DIFFERENTIAL AND INTEGRAL
CALCULUS.

ELEMENTARY ILLUSTRATIONS.

rTvHE Differential and Integral Calculus, or, as it
JL was formerly called in this country [England],
the Doctrine of Fluxions, has always been supposed
to present remarkable obstacles to the beginner. It
is matter of common observation, that any one who
commences this study, even with the best elementary
works, finds himself in the dark as to the real meaning
of the processes which he learns, until, at a certain
stage of his progress, depending upon his capacity,
some accidental combination of his own ideas throws
light upon the subject. The reason of this may be, that
it is usual to introduce him at the same time to new
principles, processes, and symbols, thus preventing
his attention from being exclusively directed to one
new thing at a time. It is our belief that this should
be avoided ; and we propose, therefore, to try the ex
periment, whether by undertaking the solution of
some problems by common algebraical methods, with
out calling for the reception of more than one new
symbol at once, or lessening the immediate evidence
of each investigation by reference to general rules, the
study of more methodical treatises may not be some-

2 ELEMENTARY ILLUSTRATIONS OF

what facilitated. We would not, nevertheless, that
the student should imagine we can remove all ob
stacles ; we must introduce notions, the consideration
of which has not hitherto occupied his mind ; and
shall therefore consider our object as gained, if we
can succeed in so placing the subject before him, that
two independent difficulties shall never occupy his
mind at once.

ON THE RATIO OR PROPORTION OF TWO MAGNITUDES.

The ratio or proportion of two magnitudes is best
conceived by expressing them in numbers of some
unit when they are commensurable ; or, when this is
not the case, the same may still be done as nearly as
we please by means of numbers. Thus, the ratio of
the diagonal of a square to its side is that of 1/2 to 1,
which is very nearly that of 14142 to 10000, and is
certainly between this and that of 14143 to 10000.
Again, any ratio, whatever numbers express it, may
be the ratio of two magnitudes, each of which is as
small as we please ; by which we mean, that if we
take any given magnitude, however small, such as the
line A, we may find two other lines B and C, each
less than A, whose ratio shall be whatever we please.
Let the given ratio be that of the numbers m and n.
Then, P being a line, mP and nP are in the propor
tion of m to n ; and it is evident, that let m, n, and A
be what they may, P can be so taken that mP shall be
less than A. This is only saying that P can be taken
less than the #zth part of A, which is obvious, since A,
however small it may be, has its tenth, its hundredth,
its thousandth part, etc., as certainly as if it were
larger. We are not, therefore, entitled to say that
because two magnitudes are diminished, their ratio is

THE DIFFERENTIAL AND INTEGRAL CALCULUS. 3

diminished ; it is possible that B, which we will sup
pose to be at first a hundredth part of C, may, after
a diminution of both, be its tenth or thousandth, or
may still remain its hundredth, as the following ex
ample will show :

C 3600 1800 36 90

B 36 1 3 9

B=c B==c B==

oo i oo

Here the values of B and C in the second, third, and
fourth column are less than those in the first ; never
theless, the ratio of B to C is less in the second col
umn than it was in the first, remains the same in the
third, and is greater in the fourth.

In estimating the approach to, or departure from
equality, which two magnitudes undergo in conse
quence of a change in their values, we must not look
at their differences, but at the proportions which those
differences bear to the whole magnitudes. For ex
ample, if a geometrical figure, two of whose sides are
3 and 4 inches now, be altered in dimensions, so that
the corresponding sides are 100 and 101 inches, they
are nearer to equality in the second case than in the
first ; because, though the difference is the same in
both, namely one inch, it is one third of the least side
in the first case, and only one hundredth in the sec
ond. This corresponds to the common usage, which
rejects quantities, not merely because they are small,
but because they are small in proportion to those of
which they are considered as parts. Thus, twenty
miles would be a material error in talking of a day's
journey, but would not be considered worth mention
ing in one of three months, and would be called to-

4 ELEMENTARY ILLUSTRATIONS OP

tally insensible in stating the distance between the
earth and sun. More generally, if in the two quanti
ties x and x-}-a, an increase of m be given to x,
the two resulting quantities x -j- m and x -\-m-\- a are
nearer to equality as to their ratio than x and x-\-a,
though they continue the same as to their difference; for

x-\-a . a , x -\-m-\- a a ..*•«.

— ! — =14-- and — ' . = 1 -\ ; — of which

x x x-\- m x -\-m

is less than — , and therefore 1 -\ — is nearer

x-\-m x x-\-m

to unity than 1 -\ . In future, when we talk of an

OC ~' '. .

approach towards equality, we mean that the ratio is
made more nearly equal to unity, not that the differ
ence is more nearly equal to nothing. The second
may follow from the first, but not necessarily; still
less does the first follow from the second.

ON THE RATIO OF MAGNITUDES THAT VANISH TOGETHER.

It is conceivable that two magnitudes should de
crease simultaneously,* so as to vanish or become
nothing, together. For example, let a point A move
on a circle towards a fixed point B. The arc AB will
then diminish, as also the chord AB, and by bringing
the point A sufficiently near to B, we may obtain an
arc and its chord, both of which shall be smaller than
a given line, however small this last may be. But
while the magnitudes diminish, we may not assume
either that their ratio increases, diminishes, or re
mains the same, for we have shown that a diminution
of two magnitudes is consistent with either of these.

* In introducing the notion of time, .ve consult only simplicity. It would
do equally well to write any number of successive values of the two quanti
ties, and place them in two columns.

THE DIFFERENTIAL AND INTEGRAL CALCULUS. 5

We must, therefore, look to each particular case for
the change, if any, which is made in the ratio by the
diminution of its terms.

Now two suppositions are possible in every in
crease or diminution of the ratio, as follows : Let M
and N be two quantities which we suppose in a state
of decrease. The first possible case is that the ratio
of M to N may decrease without limit, that is, M may
be a smaller fraction of N after a decrease than it was
before, and a still smaller after a further decrease,
and so on ; in such a way, that there is no fraction so

small, to which =^ shall not be equal or inferior, if the

decrease of M and N be carried sufficiently far. As
an instance, form two sets of numbers as in the ad
joining table :

..,11
jjt

_
20 400 8000 160000

111 1

1 T T T 16 etc-

Ratio of M to Nl etc.

Here both M and N decrease at every step, but M
loses at each step a larger fraction of itself than N,
and their ratio continually diminishes. To show that
this decrease is without limit, observe that M is at
first equal to N, next it is one tenth, then one hun
dredth, then one thousandth of N, and so on ; by con
tinuing the values of M and N according to the same
law, we should arrive at a value of M which is a
smaller part of N than any which we choose to name ;
for example, -000003. The second value of M beyond
our table is only one millionth of the corresponding
value of N ; the ratio is therefore expressed by -000001

0 ELEMENTARY ILLUSTRATIONS OF

which is less than -000003. In the same law of forma
tion, the ratio of N to M is also increased without limit.
The second possible case is that in which the ratio
of M to N, though it increases or decreases, does not
increase or decrease without limit, that is, continually
approaches to some ratio, which it never will exactly
reach, however far the diminution of M and N may
be carried. The following is an example :

iv/r 111111

M l T T TO 15 21 28etC'

111111
4 T 16 25 36 496t

4 9 16 25 36 49
RatloofMtoNl - - - - - -etc.

The ratio here increases at each step, for -^ is greater

94

than 1, -^-than-pr-, and so on. The difference between
o o

this case and the last is, that the ratio of M to N,
though perpetually increasing, does not increase with
out limit ; it is never so great as 2, though it may be
brought as near to 2 as we please.

To show this, observe that in the successive values
of M, the denominator of the second is 1 -f- 2, that of
the third 1 -|- 2 -|- 3, and so on ; whence the denom
inator of the xth value of M is

+ 2 + 3+ .....

Therefore the xth value of M is —f — r-^-9 and it is

x( '

evident that the x^ value of N is -g, which gives the

M 2*2 2*

* value of the raho = --—, or — or

THE DIFFERENTIAL AND INTEGRAL CALCULUS. 7

.. X 2. If x be made sufficiently great, ^ may
be brought as near as we please to 1, since, being
1 -- — TJ, it differs from 1 by .., which may be

OC j-|~ X OC ' I * X

made as small as we please. But as — — -r-, however

x -\- 1

great x may be, is always less than 1, — — r is always

M x -{- i

less than 2. Therefore (1) -^- continually increases ;

(2) may be brought as near to 2 as we please ; (3) can
never be greater than 2. This is what we mean by

M

saying that -^=- is an increasing ratio, the limit of

N N

which is 2. Similarly of -=-:=-, which is the reciprocal

of -^=-, we may show (1) that it continually decreases ;

(2) that it can be brought as near as we please to ^ ;

(3) that it can never be less than i. This we express

N
by saying that ^ is a decreasing ratio, whose limit

isf

ON THE RATIOS OF CONTINUOUSLY INCREASING OR
DECREASING QUANTITIES.

To the fractions here introduced, there are inter
mediate fractions, which we have not considered.
Thus, in the last instance, M passed from 1 to £ with
out any intermediate change. In geometry and me
chanics, it is necessary to consider quantities as
increasing or decreasing continuously ; that is, a mag
nitude does not pass from one value to another with
out passing through every intermediate value. Thus
if one point move towards another on a circle, both
the arc and its chord decrease continuously. Let AB
(Fig. 1) be an arc of a circle, the centre of which is

8

ELEMENTARY ILLUSTRATIONS OF

O. Let A remain fixed, but let B, and with it the ra
dius OB, move towards A, the point B always remain
ing on the circle. At every position of B, suppose
the following figure. Draw AT touching the circle at
A, produce OB to meet AT in T, draw BM and BN
perpendicular and parallel to OA, and join BA. Bisect
the arc AB in C, and draw OC meeting the chord in
D and bisecting it. The right-angled triangles ODA
and BMA having a common angle, and also right
angles, are similar, as are also BOM and TEN. If
now we suppose B to move towards A, before B

Fig. 1

reaches A, we shall have the following results : The
arc and chord BA, the lines BM, MA, BT, TN, the
angles BOA, COA, MBA, and TBN, will diminish
without limit ; that is, assign a line and an angle,
however small, B can be placed so near to A that the
lines and angles above alluded to shall be severally
less than the assigned line and angle. Again, OT di
minishes and OM increases, but neither without limit,
for the first is never less, nor the second greater, than
the radius. The angles OBM, MAB, and BTN, in
crease, but not without limit, each being always less
than the right angle, but capable of being made as

THE DIFFERENTIAL AND INTEGRAL CALCULUS. Q

near to it as we please, by bringing B sufficiently near
to A.

So much for the magnitudes which compose the
figure : we proceed to consider their ratios, premising
that the arc AB is greater than the chord AB, and
less than BN + NA. The triangle BMA being always
similar to ODA, their sides change always in the same
proportion ; and the sides of the first decrease with
out limit, which is the case with only one side of the
second. And since OA and OD differ by DC, which
diminishes without limit as compared with OA, the
ratio OD -=- OA is an increasing ratio whose limit is 1.
But OD -T- OA = BM -7- BA. We can therefore bring
B so near to A that BM and BA shall differ by as
small a fraction of either of them as we please.

To illustrate this result from the trigonometrical
tables, observe that if the radius OA be the linear
unit, and /BOA = 0, BM and BA are respectively
sine and 2sin|0. Let (9=1°; then sin0= -0174524
and 2sin£0= -0174530; whence 2sin J0-j-sin 6 =
1 • 00003 very nearly, so that BM differs from BA by
less than four of its own hundred-thousandth parts.
If /BO A = 4', the same ratio is 1-0000002, differing
from unity by less than the hundredth part of the
difference in the last example.

Again, since DA diminishes continually and with
out limit, which is not the case either with OD or
OA, the ratios OD -~- DA and OA-r- DA increase with
out limit. These are respectively equal to BM -4- MA
and BA -5- MA ; whence it appears that, let a number
be ever so great, B can be brought so near to A, that
BM and BA shall each contain MA more times than
there are units in that number. Thus if / BOA= 1°,
BM-j-MA = 114-589 and BA -r- MA = 114-593 very

10

ELEMENTARY ILLUSTRATIONS OF

nearly ; that is, BM and BA both contain MA
more than 114 times. If /BO A = 4', BM-r-MA =
1718-8732, and BA~ MA = 1718 -8375 very nearly;
or BM and BA both contain MA more than 1718
times.

No difficulty can arise in conceiving this result, if
the student recollect that the degree of greatness or
smallness of two magnitudes determines nothing as
to their ratio ; since every quantity N, however small,
can be divided into as many parts as we please, and
has therefore another small quantity which is its mil-

lionth or hundred-millionth part, as certainly as if it
had been greater. There is another instance in the
line TN, which, since TBN is similar to BOM, de
creases continually with respect to TB, in the same
manner as does BM with respect to OB.

The arc BA always lies between BA and BN-j-NA,

or BM -j- MA ; hence

chord BA

lies between 1 and

BM MA

BM

BA

BA'

But -~-r- has been shown to approach

BA

MA

to decrease without

continually towards 1, and

arcBA sjr^

limit ; hence •-, — T^FTT continually approaches towards
chord r>A

THE DIFFERENTIAL AND INTEGRAL CALCULUS. 1 1

1. If /BOA-: I°,r=- 0174533 --0174530 =

chord BA
1-00002, very nearly. If ,/BOA = 4', it is less than

1-0000001.

We now proceed to illustrate the various phrases
which have been used in enunciating these and sim
ilar propositions.

THE NOTION OF INFINITELY SMALL QUANTITIES.

It appears that it is possible for two quantities m
and m -}- n to decrease together in such a way, that n
continually decreases with respect to m, that is, be

comes a less and less part of m, so that — also de-

m

creases when n and m decrease. Leibnitz,* in intro
ducing the Differential Calculus, presumed that in
such a case, n might be taken so small as to be utterly
inconsiderable when compared with m, so that m-\- n
might be put for m, or vice versa, without any error at
all. In this case he used the phrase that n is infinitely
small with respect to m.

The following example will illustrate this term.
Since (a -f /fc)2 == 02 -f 2 a h -j- /*2, it appears that if a be
increased by h, a2 is increased by Zah-\-/i'2. But if h
be taken very small, h* is very small with respect to
//, for since \\h\\h\ffi, as many times as 1 contains
h, so many times does h contain h* ; so that by taking

* Leibnitz was a native of Leipsic, and died in 1716, aged 70. His dispute
with Newton, or rather with the English mathematicians in general, about
the invention of Fluxions, and the virulence with which it was carried on,
are well known. The decision of modern times appears to be that both New
ton and Leibnitz were independent inventors of this method. It has, perhaps,
not been sufficiently remarked how nearly several of their predecessors ap
proached the same ground ; and it is a question worthy of discussion, whether
either Newton or Leibnitz might not have found broader hints in writings
accessible to both, than the latter was ever asserted to have received from
the former.

12 ELEMENTARY ILLUSTRATIONS OF

h sufficiently small, h may be made to be as many
times W as we please. Hence, in the words of Leib
nitz, if h be taken infinitely small, h* is infinitely small
with respect to h, and therefore 2ah-\-/fi is the same
as 2 ah; or if a be increased by an infinitely small
quantity h, a1 is increased by another infinitely small
quantity 2 ah, which is to h in the proportion of 2 a
to 1.

In this reasoning there is evidently an absolute
error ; for it is impossible that h can be so small, that
Zah + W and 2ah shall be the same. The word small
itself has no precise meaning ; though the word smaller,
or less, as applied in comparing one of two magnitudes
with another, is perfectly intelligible. Nothing is
either small or great in itself, these terms only imply
ing a relation to some other magnitude of the same
kind, and even then varying their meaning with the
subject in talking of which the magnitude occurs, so
that both terms may be applied to the same magni
tude : thus a large field is a very small part of the
earth. Even in such cases there is no natural point
at which smallness or greatness commences. The
thousandth part of an inch may be called a small dis
tance, a mile moderate, and a thousand leagues great,
but no one can fix, even for himself, the precise mean
between any of these two, at which the one quality
ceases and the other begins. These terms are not
therefore a fit subject for mathematical discussion,
until some more precise sense can be given to them,
which shall prevent the danger of carrying away with
the words, some of the confusion attending their use
in ordinary language. It has been usual to say that
when h decreases from any given value towards noth
ing, h* will become small as compared with h, because,

THE DIFFERENTIAL AND INTEGRAL CALCULUS. 13

let a number be ever so great, h will, before it be
comes nothing, contain ft2 more than that number of
times. Here all dispute about a standard of smallness
is»avoided, because, be the standard whatever it may,
the proportion of h2 to h may be brought under it. It
is indifferent whether the thousandth, ten-thousandth,
or hundred-millionth part of a quantity is to be con
sidered small enough to be rejected by the side of the

whole, for let h be ^ j^ or iw>^>m of the

unit, and h will contain A?, 1000, 10,000, or 100,000,000
of times.

The proposition, therefore, that h can be taken so
small that 2ah-\-h? and 2aA are rigorously equal,
though not true, and therefore entailing error upon
all its subsequent consequences, yet is of this charac
ter, that, by taking h sufficiently small, all errors may
be made as small as we please. The desire of com
bining simplicity with the appearance of rigorous
demonstration, probably introduced the notion of in
finitely small quantities; which was further estab
lished by observing that their careful use never led to
any error. The method of stating the above-mentioned
proposition in strict and rational terms is as follows :
If a be increased by h, a2 is increased by 2 a h -\- h* ,
which, whatever may be the value of h, is to h in the
proportion of 2a-\-h to 1. The smaller h is made,
the more near does this proportion diminish towards
that of 2 a to 1, to which it may be made to approach
within any quantity, if it be allowable to take h as
small as we please. Hence the ratio, increment of <P-±
increment of a, is a decreasing ratio, whose limit is 2 a.

In further illustration of the language of Leibnitz,
we observe, that according to his phraseology, if AB

14 ELEMENTARY ILLUSTRATIONS OF

be an infinitely small arc, the chord and arc AB are
equal, or the circle is a polygon of an infinite num
ber of infinitely small rectilinear sides. This should
be considered as an abbreviation of the proposition
proved (page 10), and of the following: If a polygon
be inscribed in a circle, the greater the number of its
sides, and the smaller their lengths, the more nearly
will the perimeters of the polygon and circle be equal
to one another; and further, if any straight line be
given, however small, the difference between the pe
rimeters of the polygon and circle may be made less
than that line, by sufficient increase of the number of
sides and diminution of their lengths. Again, it would
be said (Fig. 1) that if AB be infinitely small, MA is
infinitely less than BM. What we have proved is,
that MA may be made as small a part of BM as we
please, by sufficiently diminishing the arc BA.

ON FUNCTIONS.

An algebraical expression which contains x in any
way, is called a function of x. Such are x2 -j- a2,

, sin2#. An expression may be a

t

a — x

function of more quantities than one, but it is usual
only to name those quantities of which it is necessary
to consider a change in the value. Thus if in x* -\- a*
x only is considered as changing its value, this is
called a function of x ; if x and a both change, it is
called a function of x and a. Quantities which change
their values during a process, are called variables, and
those which remain the same, constants ; and variables
which we change at pleasure are called independent,
while those whose changes necessarily follow from

THE DIFFERENTIAL AND INTEGRAL CALCULUS. 15

the changes of others are called dependent. Thus in
Fig. 1, the length of the radius OB is a constant, the
arc AB is the independent variable, while BM, MA,
the chord AB, etc., are dependent. And, as in alge
bra we reason on numbers by means of general sym
bols, each of which may afterwards be particularised
as standing for any number we please, unless specially
prevented by the conditions of the problem, so, in
treating of functions, we use general symbols, which
may, under the restrictions of the problem, stand for
any function whatever. The symbols used are the let
ters F,/, <£, <p, ip ; cp(x] and $ (#), or <px and ipx, may
represent any functions of x, just as x may represent
any number. Here it must be borne in mind that cp
and ip do not represent numbers which multiply x, but
are the abbreviated directions to perform certain opera
tions with x and constant quantities. Thus, if <px =
x -\- x2, <p is equivalent to a direction to add x to its
square, and the whole tpx stands for the result of this
operation. Thus, in this case, <p(l) = 2; ^>(2) — 6;
(pa = a-\-a?; <p(x-\- h} = x-{-h-}- (x-\- h^ ; <psin-# =
sin x -f- (sin x) 2. It may be easily conceived that this
notion is useless, unless there are propositions which
are generally true of all functions, and which may be
made the foundation of general reasoning.

INFINITE SERIES.

To exercise the student in this notation, we pro
ceed to explain one of these functions which is of
most extensive application and is known by the name
of Taylor's Theorem. If in cpx, any function of x, the
value of x be increased by h, or x -{- h be substituted
instead of x, the result is denoted by (p(x-\-h}. It

l6 ELEMENTARY ILLUSTRATIONS OF

will generally* happen that this is either greater or
less than <px, and h is called the increment of x, and
cp(x-\-h} — cpx is called the increment of cpx, which is
negative when cp(x + fy<(px. It may be proved
that q)(x-}-h} can generally be expanded in a series
of the form

<px-\-ph + qh*-\-rhl-\- etc. , ad infinitum,
which contains none but whole and positive powers
of h. It will happen, however, in many functions,
that one or more values can be given to x for which
it is impossible to expand f(x -\- h) without introdu
cing negative or fractional powers. These cases are
considered by themselves, and the values of x which
produce them are called singular values.

As the notion of a series which has no end of its
terms, may be new to the student, we will now pro
ceed to show that there may be series so constructed,
that the addition of any number of their terms, how
ever great, will always give a result less than some
determinate quantity. Take the series

l+x + x*-\-x*-\-x*+ etc.,

in which x is supposed to be less than unity. The
first two terms of this series may be obtained by di
viding 1 — x* by 1 — x] the first three by dividing
1 — x* by 1 — x ; and the first n terms by dividing
1 — x" by 1 — x. If or be less than unity, its succes
sive powers decrease without limit ;f that is, there is

*This word is used in making assertions which are for the most part
true, but admit of exceptions, few in number when compared with the other
cases. Thus it generally happens that x% — IO.T -f 40 is greater than 15, with
the exception only of the cas« where x = $. It is generally true that a line
which meets a circle in a given point meets it again, with the exception only
of the tangent.

tThis may be proved by means of the proposition established in theS/«a>
of Mathematics (Chicago : The Open Court Publishing Co., Reprint Edition),

THE DIFFERENTIAL AND INTEGRAL CALCULUS. IJ

no quantity so small, that a power of x cannot be
found which shall be smaller. Hence by taking n

1 xn ^ %n

sufficiently great, or may be

JL — X JL — X i. — X

brought as near to as we please, than which,

— x xn
however, it must always be less, since can never

entirely vanish, whatever value n may have, and there
fore there is always something subtracted from ^ .

It follows, nevertheless, that 1 -f x + x* + etc. , if we
are at liberty to take as many terms as we please, can

be brought as near as we please to = , and in this

sense we say that

= — l-f-^-f'*2 -{-•**-{- etc. , ad infinitum.

CONVERGENT AND DIVERGENT SERIES.

A series is said to be convergent when the sum of
its terms tends towards some limit ; that is, when, by
taking any number of terms, however great, we shall
never exceed some certain quantity. On the other
hand, a series is said to be divergent when the sum of
a number of terms may be made to surpass any quan
tity, however great. Thus of the two series,

and

1+2 + 4 + 8 + etc.

the first is convergent, by what has been shown, and
the second is evidently divergent. A series cannot be
convergent, unless its separate terms decrease, so as,

page 247. For ~X •—• is formed (if m be less than «) by dividing ~ into n
parts, and taking away n — m of them.

1 8 ELEMENTARY ILLUSTRATIONS OF

at last, to become less than any given quantity. And
the terms of a series may at first increase and after
wards decrease, being apparently divergent for a finite
number of terms, and convergent afterwards. It will
only be necessary to consider the latter part of the
series.

Let the following series consist of terms decreas
ing without limit :

which may be put under the form

the same change of form may be made, beginning
from any term of the series, thus :

k + /+ m + etc. =* (1 + L + * .L _|_ etc.).

We have introduced the new terms — -, -y-, etc., or the

a b

ratios which the several terms of the original series
bear to those immediately preceding. It may be shown

(i) that if the terms of the series — , — , — , etc., come

a b c

at last to be less than unity, and afterwards either
continue to approximate to a limit which is less than
unity, or decrease without limit, the series a-\-b-\-
<r-j-etc., is convergent; (2) if the limit of the terms

— , — , etc., is either greater than unity, or if they in
crease without limit, the series is divergent.

(1#). Let — be the first which is less than unity,

W?

and let the succeeding ratios —, etc., decrease, either

/ m n
with or without limit, so that — > — > — , etc. ;

k I m

whence it follows, that of the two series,

THE DIFFERENTIAL AND INTEGRAL CALCULUS.

i + Tj + m + etc->'

the first is greater than the second. But since — is
less than unity, the first can never surpass k X - — r,

& ^

or , and is convergent ; the second is therefore

K /

convergent. But the second is no other than k + /-f
m -j- etc. ; therefore the series a -f- b -\- c -j- etc., is con
vergent from the term k.

(1 £.) Let — be less than unity, and let the succes-

K

I m
sive ratios — , — , etc., increase, never surpassing a

limit A, which is less than unity. Hence of the two
series,

*(!+ A + A A -|- A A A -f etc.),

£H _L ^ J_ ^ m _L ^ m

k k I k I m
the first is the greater. But since A is less than unity,
the first is convergent; whence, as before, a-\-b-\-
r-|-etc., converges from the term k.

(2) The second theorem on the divergence of series
we leave to the student's consideration, as it is not
immediately connected with our object.

TAYLOR'S THEOREM. DERIVED FUNCTIONS.
We now proceed to the series

ph + qh* + r W + -r^4 -f etc.,

in which we are at liberty to suppose h as small as
we please. The successive ratios of the terms to those

-4 -- -1- etc^

"

20 ELEMENTARY ILLUSTRATIONS OF

,. L , a . rh* r .

immediately preceding are — y- or — ft, — — or — h,

pit p qi? q

— TTT or— h, etc. If, then, the terms --, — , — , etc.,
filr-T p q r

are always less than a finite limit A, or become so after

a definite number of terms, — h, — h, etc., will always

P <1
be, or will at length become, less than Aft. And since h

may be what we please, it may be so chosen that Aft
shall be less than unity, for which h must be less than

-r-. In this case, by theorem (1£), the series is con-
A

vergent ; it follows, therefore, that a value of h can
always be found so small that ph-\- qlP -f^8 + etc.,
shall be convergent, at least unless the coefficients
p, q, r, etc., be such that the ratio of any one to the
preceding increases without limit, as we take more
distant terms of the series. This never happens in
the developments which we shall be required to con
sider in the Differential Calculus.

We now return to <p(x + /*), which we have as
serted (page 16) can be expanded (with the exception
of some particular values of #) in a series of the form
q)X-\-ph-\- qh* -f- etc. The following are some in
stances of this development derived from the Differ
ential Calculus, most of which are also to be found in
treatises on algebra :

7,2 JA

(x + k)»=x» +nxM-lft+n(n—l)x«-* — -f n(n—l) (w— 2)**-* — etc.

2 2.O

» 2 JA

a*+*=*a* +kaxh* +&a* — + &a* — etc.

2 2. o

1 1 7*2 2 ft3

s'm(x+ft)=sinx + cosxft — sin*— — — cos x —— etc.

6 a.o

•Here k is the Naperian or hyperbolic logarithm of a; that is, the com
mon logarithm of a divided by .434294482.

tin the last two series the terms are positive and negative in pairs.

THE DIFFERENTIAL AND INTEGRAL CALCULUS. 21

A1 h*

cos(x+h)=cosx— sinxTt — cos* — - + sin x-— etc.

a £•')

It appears, then, that the development of cp(x-\-h)
consists of certain functions of x, the first of which is
cpx itself, and the remainder of which are multiplied

h? h* /z4

by h, -n-j 17 Q-, 3-Q-7» and so on. It is usual to denote
_ — . o Zi. «_>. 4

the coefficients of these divided powers of h by cp'x,
qj'x, qj"x,* etc., where cp', cp", etc., are merely func
tional symbols, as is cp itself ; but it must be recol
lected that cp'x, cp"x, etc., are rarely, if ever, employed

&
to signify anything except the coefficients of h, -^-,

£

etc., in the development of (p(x-\-h). Hence this de
velopment is usually expressed as follows :

p(* + A) = 9>*+ qfxh+<p"x ^ + <p?"x ^ + etc.

Thus, when cpx = xn, (p'x = nxn~l, gj"x = n(n — 1)
xn~2, etc.; when ^»^ = sin^, cp'x = cosxy cp"x =
— sin^, etc. In the first case q)'(x-}-?i} = n(x-[-hy-1,
cp'\x -\- //) = #(# — 1 ) (x -j- /z) n~2 ', and in the second
<p'(x H- ^) = cos (x + /£), cp"(x + h}=— sin(^r -f h}.

The following relation exists between cpx, cp'x,
cp"x, etc. In the same manner as cp'x is the coefficient
of h in the development of cp(x-\-Ji}, so qj'x is the co
efficient of h in the development of cp'(x + ^)> and
<£/".# is the coefficient of h in the development of cp"
(x-\- /£); <piv# is the coefficient of ^ in the development
of cp"'(x -f- ^), and so on.

The proof of this is equivalent to Taylor's Theorem
already alluded to (page 15); and the fact may be
verified in the examples already given. When cpx
= a*, <p'x = ka*, and cpr (x -{- %}= & a*+A = k(a*-\-ka*h
-fete.). The coefficient of h is here k?ax, which is the

* Called derived functions or derivatives. — Ed.

22 ELEMENTARY ILLUSTRATIONS OF

same as cp"x. (See the second example of the pre
ceding table.) Again, <ft'(x-\- h} = & ax+h = & (a* -\-
ka*h-\- etc.), in which the coefficient of h is k*ax, the

same as cp'"x. Again, if cpx = \ogx, cpfx= — , and

1 1 * u

cp (x -{- h) = - -—f = = -f- etc- ' as appears by

x -{- n x x ^

common division. Here the coefficient of h is j,

which is the same as cp"x in the third example. Also
-f- Ji\ = — r-5 = — (x 4- /^)~2, which by the

Binomial Theorem is — (a-2 — 2x~s/i -f etc.). The

2
coefficient of h is 2x~* or — ^, which is cp'"x in the

same example.

DIFFERENTIAL COEFFICIENTS.

It appears, then, that if we are able to obtain the
coefficient of h in the development of any function
whatever of x -(- h, we can obtain all the other coeffi
cients, since we can thus deduce cp'x from cpx, cp"x
from cp'x, and so on. It is usual to call cp'x the first
differential coefficient of cpx, cp"x the second differen
tial coefficient of cpx, or the first differential coefficient
of cp'x; cp'"x the third differential coefficient of cpx,
or the second of cp'x, or the first of cp"x ; and so on.*
The name is derived from a method of obtaining cp'x,
etc., which we now proceed to explain.

Let there be any function of x, which we call cpx,
in which x is increased by an increment h ; the func
tion then becomes

h2 /IB

<px-\- cp'x h -f- cp"x - -f- cp'"x -- -f etc.

*The first, second, third, etc., differential coefficients, as thus obtained,
are also called the first, second, third, etc., derivatives.— Ed.

THE DIFFERENTIAL AND INTEGRAL CALCULUS. 23

The original value cpx is increased by the increment

h? h*

cp'x h + cp"x -^ + <p"'x j-g -I- etc.;

whence (h being the increment of x)

increment of q)x h h*

v- , f = (p'x-\-cp"x -5- -f cp *o-o- + etc.,

increment of x 2 2.3

which is an expression for the ratio which the incre
ment of a function bears to the increment of its vari
able. It consists of two parts. The one, (p'x, into
which h does not enter, depends on x only ; the re
mainder is a series, every term of which is multiplied
by some power of h, and which therefore diminishes
as h diminishes, and may be made as sma.'l as we
please by making h sufficiently small.

To make this last assertion clear, observe that all
the ratio, except its first term cp'x, may be written as
follows :

h (9"x 1 + <?'"* -A. + etc.);

the second factor of which (page 19) is a convergent

series whenever h is taken less than -r-, where A is

A

the limit towards which we approximate by taking
the coefficients cp"x X -o-, <p'"xX IT-K^ etc., and form
ing the ratio of each to the one immediately preced
ing. This limit, as has been observed, is finite in
every series which we have occasion to use ; and
therefore a value for h can be chosen so small, that
for it the series in the last-named formula is conver
gent ; still more will it be so for every smaller value
of h. Let the series be called P. If P be a finite quan
tity, which decreases when h decreases, Ph can be
made as small as we please by sufficiently diminishing

24 ELEMENTARY ILLUSTRATIONS OF

h ; whence (p'x -f- P^ can be brought as near as we
please to cp'x. Hence the ratio of the increments of
cpx and x, produced by changing x into x-\- h, though
never equal to (p'x, approaches towards it as h is di
minished, and may be brought as near as we please
to it, by sufficiently diminishing h. Therefore to find
the coefficient of h in the development of (p(x-\-K),
find <p(x-{- h} — <px, divide it by h, and find the limit
towards which it tends as k is diminished.
In any series such as

a + bh + cft -{-£#• + /#•+!+ w/**+2 + etc.

which is such that some given value of h will make it
convergent, it may be shown that h can be taken so
small that any one term shall contain all the succeed
ing ones as often as we please. Take any one term,
as khn. It is evident that, be h what it may,

khH\lh'+* + mh'+*-\- etc., :: £:M-f-«# + etc.,

the last term of which is /^(/-f-w^-j-etc.). By rea
soning similar to that in the last paragraph, we can
show that this may be made as small as we please,
since one factor is a series which is always finite when

h is less than -^-, and the other factor h can be made
A.

as small as we please. Hence, since k is a given
quantity, independent of h, and which therefore re
mains the same during all the changes of h, the series
h (/_j_ m h _|_ etc. ) can be made as small a part of k as
we please, since the first diminishes without limit,
and the second remains the same. By the proportion
above established, it follows then that lhn+l-\- mhn+<*
-fete., can be made as small a part as we please of
khn. It follows, therefore, that if, instead of the full
development of <p(x-\~A), we use only its two first

THE DIFFERENTIAL AND INTEGRAL CALCULUS. 25

terms cpx -\-qfxh, the error thereby introduced may,
by taking h sufficiently small, be made as small a por
tion as we please of the small term cp'xh.

THE NOTATION OF THE DIFFERENTIAL CALCULUS.

The first step usually made in the Differential Cal
culus is the determination of cp'x for all possible val
ues of cpx, and the construction of general rules for
that purpose. Without entering into these we pro
ceed to explain the notation which is used, and to ap
ply the principles already established to the solution
of some of those problems which are the peculiar
province of the Differential Calculus.

When any quantity is increased by an increment,
which, consistently with the conditions of the prob
lem, may be supposed as small as we please, this in
crement is denoted, not by a separate letter, but by
prefixing the letter d, either followed by a full stop or
not, to that already used to signify the quantity. For
example, the increment of x is denoted under these
circumstances by dx ; that of cpx by d.cpx; that of
xn by d.xn. If instead of an increment a decrement
be used, the sign of dx, etc., must be changed in all
expressions which have been obtained on the suppo
sition of an increment ; and if an increment obtained
by calculation proves to be negative, it is a sign that
a quantity which we imagined was increased by our
previous changes, was in fact diminished. Thus, if
x becomes x -f dx, x2 becomes X* -f d.x2. But this is
also (x+dx}2 or x2 + 2x dx + (dx}2; whence d.x2 =
2x dx-\- (dx']'2. Care must be taken not to confound
d.x1, the increment of x2, with (dx}2, or, as it is often
written, dx2, the square of the increment of x. Again,

26 ELEMENTARY ILLUSTRATIONS OF

if x becomes x-\-dx. — becomes -- \-d. — and the
x xx

,1.1 1 dx . .

change of — is - ; --- or -- ^- - - — ; showing
x x dx x *

that an increment of x produces a decrement in — .

oc

It must not be imagined that because x occurs in
the symbol dx, the value of the latter in any way de
pends upon that of the former : both the first value of
x, and the quantity by which it is made to differ from
its first value, are at our pleasure, and the letter </must
merely be regarded as an abbreviation of the words
"difference of." In the first example, if we divide
both sides of the resulting equation by dx, we have

d.x*

— '- — =2x-\- dx. The smaller dx is supposed to be,
ct oc

the more nearly will this equation assume the form

d.x"2

— — =2x, and the ratio of 2 x to 1 is the limit of the

dx

ratio of the increment of x* to that of x\ to which
this ratio may be made to approximate as nearly as
we please, but which it can never actually reach. In
the Differential Calculus, the limit of the ratio only is
retained, to the exclusion of the rest, which may be
explained in either of the two following ways :

d.x*
(1) The fraction — '- — may be considered as stand-

(IX

ing, not for any value which it can actually have as
long as dx has a real value, but for the limit of all
those values which it assumes while dx diminishes.

d x^
In this sense the equation -^ — =2 x is strictly true.

But here it must be observed that the algebraical
meaning of the sign of division is altered, in such a
way that it is no longer allowable to use the numera
tor and denominator separately, or even at all to con-

THE DIFFERENTIAL AND INTEGRAL CALCULUS. 27

dy
sider them as quantities. If -7— stands, not for the

ratio of two quantities, but for the limit of that ratio,
which cannot be obtained by taking any real value of

dy
dx, however small, the whole -^- may, by convention,

(T OC

have a meaning, but the separate parts dy and dx
have none, and can no more be considered as sep-

dy
arate quantities whose ratio is -~t than the two loops

of the figure 8 can be considered as separate numbers
whose sum is eight. This would be productive of no
great inconvenience if it were never required to sep
arate the two ; but since all books on the Differential
Calculus and its applications are full of examples in
which deductions equivalent to assuming dy=%xdx

dy

are drawn from such an equation as -— —2x, it be
comes necessary that the first should be explained, in
dependently of the meaning first given to the second.
It may be said, indeed, that if y = x*9 it follows that

dy

~=2x-}-dx, in which, if we make dx = Q, the re-

** dy

suit is — — 2x. But if dx = Q, dy also =0, and this
ax Q

equation should be written — =2x, as is actually done

in some treatises on the Differential Calculus,* to the
great confusion of the learner. Passing over the diffi-

cultiesf of the fraction -^-, still the former objection
recurs, that the equation dy = 2xdx cannot be used

*This practice was far more common in the early part of the century
than now, and was due to the precedent of Euler (1755). For the sense in
which Euler's view was correct, see the Encyclopedia Britannica, art. Infin
itesimal Calculus, Vol. XII, p. 14, 2nd column.— Ed.

t See Study of Mathematics (Reprint Edition, Chicago : The Open Court
Publishing Co., 1898), page 126.

28 ELEMENTARY ILLUSTRATIONS OF

(and it is used even by those who adopt this explana
tion) without supposing that 0, which merely implies
an absence of all magnitude, can be used in different
senses, so that one 0 may be contained in another a
certain number of times. This, even if it can be con
sidered as intelligible, is a notion of much too refined
a nature for a beginner.

(2) The presence of the letter d is an indication,
not only of an increment, but of an increment which
we are at liberty to suppose as small as we please.
The processes of the Differential Calculus are intended
to deduce relations, not between the ratios of different
increments, but between the limits to which those ra
tios approximate, when the increments are decreased.
And it may be true of some parts of an equation, that
though the taking of them away would alter the rela
tion between dy and dx, it would not alter the limit
towards which their ratio approximates, when dx
and dy are diminished. For example, dy — 2xdx-\-
(dx)*. If # = 4 and </* = -01, then #=-0801 and

^=8-01. If <**=-0001, ^=-00080001 and ^ =

dx dx

8-0001. The limit of this ratio, to which we shall
come still nearer by making dx still smaller, is 8. The
term (dx)*, though its presence affects the value of dy

dy
and the ratio ~, does not affect the limit of the latter,

dy dx

for in -J- or 2x -j- dx, the latter term dx, which arose
dx

from the term (dx)*, diminishes continually and with
out limit. If, then, we throw away the term (dfc)2,
the consequence is that, make dx what we may, we
never obtain dy as it would be if correctly deduced
from the equation y — x*, but we obtain the limit of
the ratio of dy to dx. If we throw away all powers of

THE DIFFERENTIAL AND INTEGRAL CALCULUS. 2g

dx above the first, and use the equations so obtained,
all ratios formed from these last, or their consequences,
are themselves the limiting ratios of which we are in
search. The equations which we thus use are not abso
lutely true in any case, but may be brought as near as we
please to the truth, by making dy and dx sufficiently
small. If the student at first, instead of using dy =
Zxdx, were to write it thus, dy — 2x dx -{- etc. , the etc.
would remind him that there are other terms ; neces
sary, if the value of dy corresponding to any value of
dx is to be obtained ; unnecessary, if the limit of the
ratio of dy to dx is all that is required.

We must adopt the first of these explanations when
dy and dx appear in a fraction, and the second when
they are on opposite sides of an equation.

ALGEBRAICAL GEOMETRY.

If two straight lines be drawn at right angles to
each other, dividing the whole of their plane into four
parts, one lying in each right angle, the situation
of any point is determined when we know, (1) in
which angle it lies, and (2) its perpendicular distances
from the two right lines. Thus (Fig. 2) the point P
lying in the angle AOB, is known when PM and PN,
or when OM and PM are known ; for, though there
is an infinite number of points whose distance from
OA only is the same as that of P, and an infinite num
ber of others, whose distance from OB is the same as
that of P, there is no other point whose distances
from both lines are the same as those of P. The line
OA is called the axis of x, because it is usual to de
note any variable distance measured on or parallel to
OA by the letter x. For a similar reason, OB is called

30 ELEMENTARY ILLUSTRATIONS OF

the axis of y. The co-ordinates* or perpendicular dis
tances of a point P which is supposed to vary its po
sition, are thus denoted by x and jy; hence OM or PN
is x, and PM or ON is y. Let a linear unit be chosen,
so that any number may be represented by a straight
line. Let the point M, setting out from O, move in
the direction OA, always carrying with it the indef
initely extended line MP perpendicular to OA. While
this goes on, let P move upon the line MP in such a
way, that MP or y is always equal to a given function
of OM or x\ for example, let y = x*, or let the num-

Fig.

\

N P

/

/
Q

~~2

D

0 T M M' A

B

ber of units in PM be the square of the number of
units in OM. As O moves towards A, the point P
will, by its motion on MP, compounded with the mo
tion of the line MP itself, describe a/icurve OP, in
which PM is less than, equal to, or greater than, OM,
according as OM is less than, equal to, or greater
than the linear unit. It only remains to show how
the other branch of this curve is deduced from the
equation y = x2. And to this end we shall first have
to interpolate a few remarks.

*The distances OM and MP are called the co-ordinates of the point P. It
is moreover usual to call the co-ordinate OM, the abscissa, and MP, the ordi-
nate, of the point P.

THE DIFFERENTIAL AND INTEGRAL CALCULUS. 31

ON THE CONNEXION OF THE SIGNS OF ALGEBRAICAL AND
THE DIRECTIONS OF GEOMETRICAL MAGNITUDES.

It is shown in algebra, that if, through misappre
hension of a problem, we measure in one direction, a
line which ought to lie in the exactly opposite direc
tion, or if such a mistake be a consequence of some
previous misconstruction of the figure, any attempt
to deduce the length of that line by algebraical rea
soning, will give a negative quantity as the result.
And conversely it may be proved by any number of
examples, that when an equation in which a occurs
has been deduced strictly on the supposition that a is
a line measured in one direction, a change of sign in
a will turn the equation into that which would have
been deduced by the same reasoning, had we begun
by measuring the line a in the contrary direction.
Hence the change of -j- a into — a, or of — a into -f- #,
corresponds in geometry to a change of direction of
the line represented by a, and vice versa.

In illustration of this general fact, the following
problem may be useful. Having a circle of given ra
dius, whose centre is in the intersection of the axes
of x and y, and also a straight line cutting the axes in
two given points, required the co-ordinates of the
points (if any) in which the straight line cuts the cir
cle. Let OA, the radius of the circle =r, OE = 0,
OF=:£, and let the co-ordinates of P, one of the
points of intersection required, be OM = x, M.P=y.
(Fig. 3.) The point P being in the circle whose ra
dius is r, we have from the right-angled triangle
OMP, x2-\-}/2=r2, which equation belongs to the co
ordinates of every point in the circle, and is called

32 ELEMENTARY ILLUSTRATIONS OF

the equation of the circle. Again, EM : MP : : EO : OF
by similar triangles ; or a — x \y : : a : b, whence ay-\-
bx = ab, which is true, by similar reasoning, for every
point of the line EF. But for a point P' lying in EF
produced, we have EM' : M'P' : : EO : OF, or x+a :y
: : a : b, whence ay — bx = ab, an equation which may
be obtained from the former by changing the sign of
x; and it is evident that the direction of x, in the

Tig.

second case, is opposite to that in the first. Again,
for a point P" in FE produced, we have EM" : M"P" : :
EO : OF, or x— a :y : : a : b, whence bx—ay = ab, which
may be deduced from the first by changing the sign
of y ; and it is evident that y is measured in different
directions in the first and third cases. Hence the
equation ay-\-bx = ab belongs to all parts of the
straight line EF, if we agree to consider M"P" as
negative, when MP is positive, and OM' as negative

THE DIFFERENTIAL AND INTEGRAL CALCULUS. 33

when OM is positive. Thus, if OE = 4, and OF = 5,
and OM — 1, we can determine MP from the equation
ay-\-bx = ab, or 4y-\-5 = 2Q, which gives y or MP =
3J. But if OM' be 1 in length, we can determine
M'P' either by calling OM', 1, and using the equation
ay — bx = ab, or calling OM', — 1, and using the equa
tion ay-\-bx — a&, as before. Either gives M'P' = 6J.
The latter method is preferable, inasmuch as it en
ables us to contain, in one investigation, all the differ
ent cases of a problem.

We shall proceed to show that this may be done
in the present instance. We have to determine the
co-ordinates of the point P, from the following equa
tions :

=. ab,

Substituting in the second the value of y derived from
the first, or ba x \ we have

or «
and proceeding in a similar manner to nndjy, we have

O2
which give

the upper or the lower sign to be taken in both.
Hence when (02-J-£2)r2;>tf2<£2, that is, when r is greater
than the perpendicular let fall from O upon EF, which
perpendicular is

34 ELEMENTARY ILLUSTRATIONS OF

ab

i/a2 + t2'

there are two points of intersection. When (a2 -}- £2)/-2
= a2l>2, the two values of x become equal, and also
those of y, and there is only one point in which the
straight line meets the circle ; in this case EF is a
tangent to the circle. And if (a* -f fiy* < a2&2, the
values of x and j> are impossible, and the straight line
does not meet the circle.

Of these three cases, we confine ourselves to the
first, in which there are two points of intersection.
The product of the values of x, with their proper
sign, is*

and of y,

the signs of which are the same as those of ft — r2,
and a2 — r2. If b and a be both > r, the two values
of x have the same sign ; and it will appear from the
figure, that the lines they represent are measured in
the same direction. And this whether b and a be pos
itive or negative, since ft — r2 and a2 — r2 are both
positive when a and b are numerically greater than r,
whatever their signs may be. That is, if our rule,
connecting the signs of algebraical and the directions
of geometrical magnitudes, be true, let the directions
of OE and OF be altered in any way, so long as OE
and OF are both greater than OA, the two values of
OM will have the same direction, and also those of
MP. This result may easily be verified from the
figure.

* See Study of Mathematics (Chicago : The Open Court Pub. Co.), page 136.

THE DIFFERENTIAL AND INTEGRAL CALCULUS. 35

Again, the values of x and y having the same sign,
that sign will be (see the equations) the same as that
of 20£2 for x, and of 2a2fr for y, or the same as that of
a for x and of b for j>. That is, when OE and OF are
both greater than OA, the direction of each set of co
ordinates will be the same as those of OE and OF,
which may also be readily verified from the figure.

Many other verifications might thus be obtained of
the same principle, viz., that any equation which cor
responds to, and is true for, all points in the angle
AOB, may be used without error for all points lying
in the other three angles, by substituting the proper
numerical values, with a negative sign, for those co
ordinates whose directions are opposite to those of
the co-ordinates in the angle AOB. In this manner,
if four points be taken similarly situated in the four
angles, the numerical values of whose co-ordinates
are # = 4 and y = 6, and if the co-ordinates of that
point which lies in the angle AOB, are called -f- 4 and
-f- 6; those of the points lying in the angle BOC will
be — 4 and +6; in the angle COD — 4 and — 6;
and in the angle DOE -f- 4 and — 6.

To return to Fig. 2, if, after having completed the
branch of the curve which lies on the right of BC,
and whose equation isjy = #2, we seek that which lies
on the left of BC, we must, by the principles estab
lished, substitute — x instead of x, when the numeri
cal value obtained for ( — #)2 will be that of y, and the
sign will show whether y is to be measured in a simi
lar or contrary direction to that of MP. Since ( — #)2
= x'2, the direction and value of y, for a given value
of x, remains the same as on the right of BC; whence
the remaining branch of the curve is similar and equal
in all respects to OP, only lying in the angle BOD.

36

ELEMENTARY ILLUSTRATIONS OF

And thus, if y be any function of x, we can obtain a
geometrical representation of the same, by making y
the ordinate, and x the abscissa of a curve, every or-
dinate of which shall be the linear representation of
the numerical value of the given function correspond
ing to the numerical value of the abscissa, the linear
unit being a given line.

THE DRAWING OF A TANGENT TO A CURVE.

If the point P (Fig. 2), setting out from O, move
along the branch OP, it will continually change the

direction of its motion, never moving, at' one point, in
the direction which it had at any previous point. Let
the moving point have reached P, and let OM=#,
MP=^. Let x receive the increment MM'=//je, in
consequence of which y or MP becomes M'P', and
receives the increment QP'=dy, so thatx-\-dx and
y-\-dy are the co-ordinates of the moving point P,
when it arrives at P'. Join PP', which makes, with

PQ or OM, an angle, whose tangent is ^T or ~.

Since the relation y = x* is true for the co-ordinates of
every point in the curve, we have y-\- dy = (x-\- dx)2,

THE DIFFERENTIAL AND INTEGRAL CALCULUS. 37

the subtraction of the former equation from which
gives 4> — 2xdx+ (dx)*, or •—- = Zx+dx. If the

(lOC

point P' be now supposed to move backwards towards
P, the chord PP' will diminish without limit, and the
inclination of PP' to PQ will also diminish, but not
without limit, since the tangent of the angle P'PQ, or

dv

-j-, is always greater than the limit 2x. If, therefore,

ctoc

a line PV be drawn through P, making with PQ an
angle whose tangent is 2x, the chord PP' will, as P'
approaches towards P, or as dx is diminished, con
tinually approximate towards PV, so that the angle
P'PV may be made smaller than any given angle, by
sufficiently diminishing dx. And the line PV cannot
again meet the curve on the side of PP', nor can any
straight line be drawn between it and the curve, the
proof of which we leave to the student.

Again, if P' be placed on the other side of P, so that
its co-ordinates are x — dx and y — dy, we have j> — dy
= (x — dx)*, which, subtracted from y = x*, gives dy

= 2xdx — (dx)*, or—- =%x — dx. By similar reason-
dx

ing, if the straight line PT be drawn in continuation
of PV, making with PN an angle, whose tangent is
2x, the chord PP' will continually approach to this
line, as before.

The line TPV indicates the direction in which the
point P is proceeding, and is called the tangent of the
curve at the point P. If the curve were the interior
of a small solid tube, in which an atom of matter were
made to move, being projected into it at O, and if all
the tube above P were removed, the line PV is in the
direction which the atom would take on emerging at
P, and is the line which it would describe. The an-

38 ELEMENTARY ILLUSTRATIONS OF

gle which the tangent makes with the axis of x in any
curve, may be found by giving x an increment, find
ing the ratio which the corresponding increment of y
bears to that of x, and determining the limit of that
ratio, or the differential coefficient. This limit is the
trigonometrical tangent* of the angle which the geo
metrical tangent makes with the axis of x. Iiy=(px,
qjx is this trigonometrical tangent. Thus, if the curve
be such that the ordinates are the Naperian loga-
rithmsf of the abscissae, or y = logx, and^-j-^v =

log#-| --- dx — x— g-dfo8, etc., the geometrical tangent
of any point whose abscissa is x, makes with the axis

an angle whose trigonometrical tangent is — .

x

This problem, of drawing a tangent to any curve,
was one, the consideration of which gave rise to the
methods of the Differential Calculus.

RATIONAL EXPLANATION OF THE LANGUAGE OF LEIBNITZ.

As the peculiar language of the theory of infinitely
small quantities is extensively used, especially in
works of natural philosophy, it has appeared right to
us to introduce it, in order to show how the terms
which are used may be made to refer to some natural
and rational mode of explanation. In applying this
language to Fig. 2, it would be said that the curve
OP is a polygon consisting of an infinite number of

* There is some confusion between these different uses of the word tan
gent. The geometrical tangent is, as already defined, the line between which
and a curve no straight line can be drawn ; the trigonometrical tangent has
reference to an angle, and is the ratio which, in any right-angled triangle,
the side opposite the angle bears to that which is adjacent.

t It may be well to notice that in analysis the Naperian logarithms are
the only ones used ; while in practice the common, or Briggs's- logarithms,
are always preferred.

THE DIFFERENTIAL AND INTEGRAL CALCULUS. 39

infinitely small sides, each of which produced is a
tangent to the curve ; also that if MM' be taken in
finitely small, the chord and arc PP' coincide with
one of these rectilinear elements ; and that an infin
itely small arc coincides with its chord. All which
must be interpreted to mean that, the chord and arc
being diminished, approach more and more nearly to
a ratio of equality as to their lengths ; and also that
the greatest separation between an arc and its chord
may be made as small a part as we please of the whole
chord or arc, by sufficiently diminishing the chord.

We shall proceed to a strict proof of this ; but in
the meanwhile, as a familiar illustration, imagine a
small arc to be cut off from a curve, and its extremi
ties joined by a chord, thus forming an arch, of which
the chord is the base. From the middle point of the
chord, erect a perpendicular to it, meeting the arc,
which will thus represent the height of the arch.
Imagine this figure to be magnified, without distortion
or alteration of its proportions, so that the larger fig
ure may be, as it is expressed, a true picture of the
smaller one. However the original arc may be dimin
ished, let the magnified base continue of a given
length. This is possible, since on any line a figure
may be constructed similar to a given figure. If the
original curve could be such that the height of the
arch could never be reduced below a certain part of
the chord, say one thousandth, the height of the mag
nified arch could never be reduced below one thou
sandth of the magnified chord, since the proportions
of the two figures are the same. But if, in the origi
nal curve, an arc can be taken so small that the height
of the arch is as small a part as we please of the
chord, it will follow that in the magnified figure where

4o

ELEMENTARY ILLUSTRATIONS OF

the chord is always of one length, the height of the
arch can be made as small as we please, seeing that
it can be made as small a part as we please of a given
line. It is possible in this way to conceive a whole
curve so magnified, that a given arc, however small,
shall be represented by an arc of any given length,
however great ; and the proposition amounts to this,
that let the dimensions of the magnified curve be any
given number of times the original, however great, an
arch can be taken upon the original curve so small,
that the height of the corresponding arch in the mag
nified figure shall be as small as we please.

Fid.

M

MT

Let PP' (Fig. 4) be a part of a curve, whose equa
tion is y = (p(x), that is, PM may always be found by
substituting the numerical value of OM in a given
function of x. Let OM=x receive the increment
MM' :=*/#, which we may afterwards suppose as small
as we please, but which, in order to render the figure
more distinct, is here considerable. The value of PM
or y is <px, and that of P'M' or y -\- dy is cp(x-\-dx).

Draw PV, the tangent at P, which, as has been
shown, makes, with PQ, an angle, whose trigonomet-

dy

rical tangent is the limit of the ratio -^-, when x is de
creased, or (p'x. Draw the chord PP', and from any

THE DIFFERENTIAL AND INTEGRAL CALCULUS. 41

point in it, for example, its middle point/, draw/?1
parallel to PM, cutting the curve in a. The value of

P'Q, or dy, or <p(x -\-dx~) — (px is

P'Q = ft dx + 9"x - + <p'"x + etc.

But <p'x dx is tan VPQ . PQ = VQ. Hence VQ is the
first term of this series, and P'V the aggregate of the
rest. But it has been shown that dx can be taken so
small, that any one term of the above series shall con
tain the rest, as often as we please. Hence PQ can
be taken so small that VQ shall contain VP' as often
as we please, or the ratio of VQ to VP' shall be as
great as we please. And the ratio VQ to PQ contin
ues finite, being always cp'x ; hence P'V also decreases
without limit as compared with PQ.

Next, the chord PP' or V (dx}* -f (dy)*, or

is to PQ or dx in the ratio of x|l + (-f-\ : 1, which,

^ \axj

as PQ is diminished, continually approximates to that
of 1/1 + (cp'x^ : 1, which is the ratio of PV: PQ.
Hence the ratio of PP': PV continually approaches to
unity, or PQ may be taken so small that the differ
ence of PP' and PV shall be as small a part of either
of them as we please.

Finally, the arc PPf is greater than the chord PP'

arr pp'
and less than PV -f VP'. Hence -J:t^±±__. iies be-

PV VP' chord PP

tween 1 and pp> + pt>T> ^ne former of which two

fractions can be brought as near as we please to unity,
and the latter can be made as small as we please ; for

42 ELEMENTARY ILLUSTRATIONS OF

since P'V can be made as small a part of PQ as we
please, still more can it be made as small a part as we
please of PP', which is greater than PQ. Therefore
the arc and chord PP' may be made to have a ratio as
nearly equal to unity as we please. And because /#
is less than pv, and therefore less than P'V, it follows
that pa may be made as small a part as we please of
PQ, and still more of PP'.

In these propositions is contained the rational ex
planation of the proposition of Leibnitz, that "an in
finitely small arc is equal to, and coincides with, its
chord."

ORDERS OF INFINITY.

Let there be any number of series, arranged in
powers of h, so that the lowest power is first ; let
them contain none but whole powers, and let them all
be such, that each will be convergent, on giving to h
a sufficiently small value : as follows,

Gfc»+ D>*4 + E/^-f-etc. (1)

C'/&8+ D'^4+ EW + etc. (2)

(3)

etc. (4)

etc. etc.

As h is diminished, all these expressions decrease
without limit ; but the first increases with respect to
the second, that is, contains it more times after a de
crease of h, than it did before. For the ratio of (1)
to (2) is that of A + 'Bh -f C#* -f etc. to B'^-f-C'/*2
-}-etc., the ratio of the two not being changed by di
viding both by h. The first term of the latter ratio
approximates continually to A, as h is diminished,
and the second can be made as small as we please,
and therefore can be contained in the first as often as

THE DIFFERENTIAL AND INTEGRAL CALCULUS. 43

we please. Hence the ratio (1) to (2) can be made
as great as we please. By similar reasoning, the ratio
(2) to (3), of (3) to (4), etc., can be made as great as
we please. We have, then, a series of quantities,
each of which, by making h sufficiently small, can be
made as small as we please. Nevertheless this de
crease increases the ratio of the first to the second, of
the second to the third, and so on, and the increase is
without limit.

Again, if we take (1) and h, the ratio of (1) to h is
that of A -f Bh -f Ctf -f etc. to 1, which, by a suffi
cient decrease of h, may be brought as near as we
please to that of A to 1. But if we take (1) and h*,
the ratio of (1) to h* is that of A -f- Bh -f etc. to h,
which, by previous reasoning, may be increased with
out limit ; and the same for any higher power of h.
Hence (1) is said to be comparable to the first power
of h, or of the first order, since this is the only power
of h whose ratio to (1) tends towards a finite limit.
By the same reasoning, the ratio of -(2) to h*, which is
that of B' -f C'h -f etc. to 1, continually approaches
that of B' to 1 ; but the ratio (2) to h, which is that
of B'/£ -j- C7/8 -f- etc. to 1, diminishes without limit, as
h is decreased, while the ratio of (2) to h9, or of B' -J-
C'^-|-etc. to h, increases without limit. Hence (2) is
said to be comparable to the second power of h, or of
the second order, since this is the only power of h whose
ratio to (2) tends towards a finite limit. In the lan
guage of Leibnitz, if h be an infinitely small quan
tity, (1) is an infinitely small quantity of the first or
der, (2) is an infinitely small quantity of the second
order, and so on.

We may also add that the ratio of two series of
the same order continually approximates to the ratio

44 ELEMENTARY ILLUSTRATIONS OF

of their lowest terms. For example, the ratio of Ah*
-f B/i* -f etc. to A'/fc8 + B'/;4 -f etc. is that of A + B^
-fete, to A' -j- ~B'h -\- etc. , which, as h is diminished,
continually approximates to the ratio of A to A', which
is also that of A/i3 to A'/$8, or the ratio of the lowest
terms. In Fig. 4, PQ or dx being put in place of h,

QP', or q/xdx + q/'x-, etc., is of the first or-

22

der, as are PV, and the chord PP'; while P'V, or

(dx^
<p"x -^-— — f- etc., is of the second order.

The converse proposition is readily shown, that if
the ratio of two series arranged in powers of h con
tinually approaches to some finite limit as h is dimin
ished, the two series are of the same order, or the ex
ponent of the lowest power of h is the same in both.
Let Aha and B>6* be the lowest powers of h, whose ra
tio, as has just been shown, continually approximates
to the actual ratio of the two series, as h is diminished.
The hypothesis is that the ratio of the two series, and
therefore that of Ah" to B#>, has a finite limit. This
cannot be if a > b, for then the ratio of Ah" to B^* is
that of Aha~h to B, which diminishes without limit ;
neither can it be when a < b, for then the same ratio
is that of A to ~Bh6~a, which increases without limit ;
hence a must be equal to b.

We leave it to the student to prove strictly a prop
osition assumed in the preceding; viz., that if the
ratio of P to Q has unity for its limit, when h is di
minished, the limiting ratio of P to R will be the same
as the limiting ratio of Q to R. We proceed further
to illustrate the Differential Calculus as applied to
Geometry.

THE DIFFERENTIAL AND INTEGRAL CALCULUS.

45

A GEOMETRICAL ILLUSTRATION.

Let OC and OD (Fig. 5) be two axes at right an
gles to one another, and let a line AB of given length
be placed with one extremity in each axis. Let this
line move from its first position into that of A'B' on
one side, and afterwards into that of A"B" on the
other side, always preserving its first length. The
motion of a ladder, one end of which is against a wall,
and the other on the ground, is an instance.

Let A'B' and A"B" intersect AB in P' and P". If
A"B" were gradually moved from its present position
into that of A'B', the point P" would also move grad-

F1J.S

0 A' A A 0

ually from its present position into that of P', passing,
in its course, through every point in the line P'P".
But here it is necessary to remark that AB is itself
one of the positions intermediate between A' B' and
A" B", and when two lines are, by the motion of one
of them, brought into one and the same straight line,
they intersect one another (if this phrase can be here
applied at all) in every point, and all idea of one dis
tinct point of intersection is lost. Nevertheless P"
describes one part of P"P' before A" B" has come into
the position AB, and the rest afterwards, when it is
between AB and A' B'.

ELEMENTARY ILLUSTRATIONS OF

Let P be the point of separation ; then every point
of P'P", except P, is a real point of intersection of
AB, with one of the positions of A"B", and when
A" B" has moved very near to AB, the point P" will
be very near to P ; and there is no point so near to P,
that it may not be made the intersection of A" B" and
AB, by bringing the former sufficiently near to the
latter. This point P is, therefore, the limit of the in
tersections of A" B" and AB, and cannot be found by
the ordinary application of algebra to geometry, but
may be made the subject of an inquiry similar to those

A" A A. G

which have hitherto occupied us, in the following
manner :

LetOA = rf, OB=l>, AB = A'B' = A"B" = /. Let
AA' = <fo, BB'=<#, whence OA' =
— db. We have then <*2-{-£2 = /2, and
(b — d#)2 = /2; subtracting the former of which from
the development of the latter, we have

2a da + (dd? — 2bdb + (dtf = 0
20 -f da

°r da Zb —

(1)

As A'B' moves towards AB, da and db are diminished
without limit, a and b remaining the same ; hence the

db . 2a a
limit of the ratio — is ^ or — .

da Zb b

THE DIFFERENTIAL AND INTEGRAL CALCULUS. 47

Let the co-ordinates* of P be OW = x and M'P,
•=y. Then (page 32) the co-ordinates of any point in
AB have the equation

ay-\-bx = ab (2)

The point P' is in this line, and also in the one which
cuts off a-}- da and b — db from the axes, whence

(a+da)y+(t — db)x = (a + da) (b — dt>) (3)
subtract (2) from (3) after developing the latter, which
gives

yda — xdb = bda — adb — dadb (4)

If we now suppose A' B' to move towards AB, equa
tion (4) gives no result, since each of its terms dimin
ishes without limit. If, however, we divide (4) by da,

and substitute in the result the value of -^ obtained

da

from (1) we have

2a + da 2a-\-da

y-* v±y- =*— ^g±a -* (&)

From this and (2) we might deduce the values of y
and x, for the point P', as the figure actually stands.
Then by diminishing db and da without limit, and
observing the limit towards which x and y tend, we
might deduce the co-ordinates of P, the limit of the
intersections.

The same result may be more simply obtained, by
diminishing da and db in equation (5), before obtain
ing the values of y and x. This gives

y — -r oc=.b — — or by — ax — fl — a9 (6)
From (6) and (2) we find (Fig. 6)

*The lines OM' and M'P' are omitted, to avoid crowding the figure.

40 ELEMENTARY ILLUSTRATIONS OF

This limit of the intersections is different for every
different position of the line AB, but may be deter
mined, in every case, by the following simple con
struction.

Since (Fig. 6) BP : PN, or OM : : BA : AO, we

1- -Dr, rMV/T , . .,

have BP = OM -r-^- = — — — —; and, similarly,
12 AU l£ a I

PA=— . Let OQ be drawn perpendicular to BA;

then since OA is a mean proportional between AQ

02 #*

and AB, we have AQ=—, and similarly BQ= — .

Hence BP = AQ and AP = BQ, or the point P is
as far from either extremity of AB as Q is from the
other.

O M

THE SAME PROBLEM SOLVED BY THE PRINCIPLES OF
LEIBNITZ.

We proceed to solve the same problem, using the
principles of Leibnitz, that is, supposing magnitudes
can be taken so small, that those proportions may be
regarded as absolutely correct, which are not so in
reality, but which only approach more nearly to the
truth, the smaller the magnitudes are taken. The in
accuracy of this supposition has been already pointed
out ; yet it must be confessed that this once got over,

THE DIFFERENTIAL AND INTEGRAL CALCULUS.

49

the results are deduced with a degree of simplicity
and consequent clearness, not to be found in any other
method. The following cannot be regarded as a dem
onstration, except by a mind so accustomed to the
subject that it can readily convert the various inaccu
racies into their corresponding truths, and see, at one
glance, how far any proposition will affect the final
result. The beginner will be struck with the extra
ordinary assertions which follow, given in their most
naked form, without any attempt at a less startling
mode of expression.

B

B

Pig. 7

Let A'B' (Fig. 7) be a position of AB infinitely
near to it ; that is, let A'PA be an infinitely small
angle. With the centre P, and the radii PA' and PB,
describe the infinitely small arcs A'a, Bfr. An infin
itely small arc of a circle is a straight line perpendic
ular to its radius ; hence A'aA and B£B' are right-
angled triangles, the first similar to BOA, the two
having the angle A in common, and the second simi
lar to B'OA'. Again, since the angles of BOA, which
are finite, only differ from those of B'OA' by the infin
itely small angle A'PA, they may be regarded as

5O ELEMENTARY ILLUSTRATIONS OF

equal; whence A' a A and B'£B are similar to BOA,
and to one another. Also P is the point of which we
are in search, or infinitely near to it ; and since BA =
B'A', of which BP = t>P and aP = A'P, the remain
ders ~B'b and Aa are equal. Moreover, B£ and A'a
being arcs of circles subtending equal angles, are in
the proportion of the radii BP and PA'.

Hence we have the following proportions :

Aa : A'a : : OA : OB : : a : b
B£ : B'J : : OA : OB : : a : b.

The composition of which gives, since A# = B'£:

B£ : A'a : : a9 : P.

Also B£ : A'a : : BP : Pa,

whence BP : Pa : : a? : t>*,

and BP + Pa : Pa ::a* + P : P.

But Pa only differs from PA by the infinitely small
quantity Aa, and BP-j-PA = /, and 02-f-£z = /2;
whence

/:PA::/2:^, or PA= ~,

which is the result already obtained.

In this reasoning we observe four independent
errors, from which others follow : (1) that B£ and A'a
are straight lines at right-angles to Pa-} (2) that BOA
B'OA' are similar triangles ; (3) that P is really the
point of which we are in search ; (4) that PA and Pa
are equal. But at the same time we observe that
every one of these assumptions approaches the truth,
as we diminish the angle A'PA, so that there is no
magnitude, line or angle, so small that the linear or
angular errors, arising from the above-mentioned sup
positions, may not be made smaller.

We now proceed to put the same demonstration

THE DIFFERENTIAL AND INTEGRAL CALCULUS.

in a stricter form, so as to neglect no quantity during
the process. This should always be done by the be
ginner, until he is so far master of the subject as to be
able to annex to the inaccurate terms the ideas neces
sary for their rational explanation. To the former fig
ure add Bj3 and Aa, the real perpendiculars, with
which the arcs have been confounded. Let / A'PA =
d6, PA=/, Aa = dp, BP = ^ B'l> = dq; and
OB=fi, and AB = /. Then* A'a = (p
qdO, and the triangles A'Aor and B'Bfi are similar to

SB

Fig. 7

O JL A

BOA and B'OA'. The perpendiculars A'a and B/?
are equal to PA' sin dO and PBs'mdff, or (/ — dp)
sin dO and q sin dO. Let aa = fA and tfi = r. These
(p. 9) will diminish without limit as compared with
A' or and B/3 • and since the ratios of A' ex to a A and B/3
to /?B' continue finite (these being sides of triangles
similar to AOB and A'OB'), aa and bft will diminish
indefinitely with respect to aA and /?B'. Hence the
ratio Aa to fiB' or dp -\- p to dq-\-v will continually
approximate to that of dp to dq, or a ratio of equality.

*For the unit employed in measuring an angle, see Study of Mathtmatics
(Chicago, 1898), pages 273-277.

52 ELEMENTARY ILLUSTRATIONS OF

The exact proportions, to which those in the last
page are approximations, are as follows :
<# + /" (p — dp)smdd::a : b,

gs'mdd: dq-\-v \\a — da\b + db\
by composition of which, recollecting that dp = dq
(which is rigorously true) and dividing the two first
terms of the resulting proportion by dp, we have

If </0 be diminished without limit, the quantities

</0, </£, and dp, and also the ratios -~- and — as

dp dp

above-mentioned, are diminished without limit, so
that the limit of the proportion just obtained, or the
proportion which gives the limits of the lines into
which P divides AB, is

q-p-.a* :P,

hence q+p = Z-.p :: a« + £» = /» : 03,

the same as before.

AN ILLUSTRATION FROM DYNAMICS.

We proceed to apply the preceding principles to
dynamics, or the theory of motion.

Suppose a point moving along a straight line uni
formly ; that is, if the whole length described be di
vided into any number of equal parts, however great,
each of those parts is described in the same time.
Thus, whatever length is described in the first second
of time, or in any part of the first second, the same
is described in any other second, or in the same part
of any other second. The number of units of length
described in a unit of time is called the velocity \ thus
a velocity of 3-01 feet in a second means that the

THE DIFFERENTIAL AND INTEGRAL CALCULUS. 53

point describes three feet and one hundredth in each
second, and a proportional part of the same in any
part of a second. Hence, if v be the velocity, and
t the units of time elapsed from the beginning of the
motion, vt is the length described ; and if any length
described be known, the velocity can be determined
by dividing that length by the time of describing it.
Thus, a point which moves uniformly through 3 feet
in \\ second, moves with a velocity of 3-j-lJ, or 2
feet per second.

Let the point not move uniformly ; that is, let dif
ferent parts of the line, having the same length, be
described in different times ; at the same time let the
motion be continuous, that is, not suddenly increased
or decreased, as it would be if the point were com
posed of some hard matter, and received a blow while
it was moving. This will be the case if its motion be
represented by some algebraical function of the time,
or if, / being the number of units of time during which
the point has moved, the number of units of length
described can be represented by cpt. This, for ex
ample, we will suppose to be /-j- /2, the unit of time
being one second, and the unit of length one inch ;
so that £ -f J, or | of an inch, is described in the first
half second ; 1 -j- 1, or two inches, in the first second ;
2 -f- 4, or six inches, in the first two seconds, and so on.

Here we have no longer an evident measure of the
velocity of the point ; we can only say that it obvi
ously increases from the beginning of the motion to
the end, and is different at every two different points.
Let the time / elapse, during which the point will de
scribe the distance /-f fl ', let a further time dt elapse,
during which the point will increase its distance to
/ -\- dt -|- (t -f <#)2, which, diminished by /-}-/*, gives

54 ELEMENTARY ILLUSTRATIONS OF

dt+2tdt-\-(dff* for the length described during the
increment of time dt. This varies with the value of
/ ; thus, in the interval dt after the first second, the
length described is 3*# -}- dfl ; after the second second,
it is 5dfr-j-(V/)2, and so on. Nor can we, as in the
case of uniform motion, divide the length described
by the time, and call the result the velocity with which
that length is described ; for no length, however small,
is here uniformly described. If we were to divide a
length by the time in which it is described, and also
its first and second halves by the times in which they
are respectively described, the three results would be
all different from one another.

Here a difficulty arises, similar to that already no
ticed, when a point moves along a curve ; in which,
as we have seen, it is improper to say that it is mov
ing in any one direction through an arc, however
small. Nevertheless a straight line was found at every
point, which did, more nearly than any other straight
line, represent the direction of the motion. So, in
this case, though it is incorrect to say that there is
any uniform velocity with which the point continues
to move for any portion of time, however small, we
can, at the end of every time, assign a uniform ve
locity, which shall represent, more nearly than any
other, the rate at which the point is moving. If we
say that, at the end of the time /, the point is moving
with a velocity v, we must not now say that the length
vdt is described in the succeeding interval of time dt ;
but we mean that dt may be taken so small, that vdt
shall bear to the distance actually described a ratio as
near to equality as we please.

Let the point have moved during the time /, after
which let successive intervals of time elapse, each

THE DIFFERENTIAL AND INTEGRAL CALCULUS. 55

equal to dt. At the end of the times, t, t-\-dt, t+Zdt,
t-\-*&dt, etc., the whole lengths described will be t-\-P,
t -f- dt -f (/ -f <//)2, t + 2dt -f (/ -j- 2<//)2, / + 3<# +
(V-f 3rt7)2, etc.; the differences of which, or dt -\-2tdt
+ (X/)2, /// + 2^/-f3(^/)2, *//+2/<#+ 5 (<//)*, etc.,
are the lengths described in the first, second, third,
etc., intervals dt. These are not equal to one another,
as would be the case if the velocity were uniform ; but
by making dt sufficiently small, their ratio may be
brought as near to equality as we please, since the
terms O//)2, 3(/#)a, etc., by which they all differ from
the common part (1 -|- 2/) dt, may be made as small as
we please, in comparison of this common part. If we
divide the above-mentioned lengths by dt, which does
not alter their ratio, they become \-\-Zt-\-dt, 1-f 2/
-(- 3///, 1 -}- 2/ -{- 5rt#, etc., which may be brought as
near as we please to equality, by sufficient diminution
of dt. Hence 1 -f- 2/ is said to be the velocity of the
point after the time /; and if we take a succession of
equal intervals of time, each equal to dty and suffi
ciently small, the lengths described in those intervals
will bear to (1 -f 2/) dt, the length which would be de
scribed in the same interval with the uniform velocity
1 -f- 2/, a ratio as near to equality as we please. And
observe, that if cpt is /-f- /2, q!t is 1 -f 2/, or the coeffi
cient of h in (/ -j- h] -f (/ -f /%)».

In the same way it may be shown, that if the point
moves so that cpt always represents the length de
scribed in the time /, the differential coefficient of cpt
or cp't, is the velocity with which the point is moving
at the end of the time /. For the time / having elapsed,
the whole lengths described at the end of the times /
and t-\- dt are cpt and cp (/ -j- dt) ; whence the length
described during the time dt is

56 ELEMENTARY ILLUSTRATIONS OF

cp (t 4- df) — q>t, or cp't dt 4- cp"t - 4- etc.

Similarly, the length described in the next interval
dt is

or>

<pt 4- cpt 2dt 4- cp t J 4- etc.

A

— (^>/ -f- ^>'^ dt 4- 9>'V 5i-—£ — f- etc.),

which is

(X/)2
cp' t dt -\-^cp" t — — 4~ etc. ;

the length described in the third interval dt is
4- etc., etc.

2

Now, it has been shown for each of these, that the
first term can be made to contain the aggregate of all
the rest as often as we please, by making dt sufficiently
small; this first term is cp'tdt in all, or the length
which would be described in the time dt by the velo
city cp't continued uniformly : it is possible, therefore,
to take dt so small, that the lengths actually described
in a succession of intervals equal to dt, shall be as
nearly as we please in a ratio of equality with those
described in the same intervals of time by the velocity
cp't. For example, it is observed in bodies which fall
to the earth from a height above it, when the resist
ance of the air is removed, that if the time be taken
in seconds, and the distance in feet, the number of
feet fallen through in / seconds is always at*, where
a = 16^ very nearly ; what is the velocity of a body
which has fallen in vacuo for four seconds? Here cpt
being at*, we find, by substituting t -\- h, or t-\-dt, in
stead of t, that cp't is 2at, or 2 X 16yVX * ', which, at

THE DIFFERENTIAL AND INTEGRAL CALCULUS. 57

the end of four seconds, is 32£ X 4, or 128| feet. That
is, at the end of four seconds a falling body moves at
the rate of 128J feet per second. By which we do
not mean that it continues to move with this velocity
for any appreciable time, since the rate is always
varying ; but that the length described in the interval
dt after the fourth second, may be made as nearly as
we please in a ratio of equality with 128f X<#, by
taking dt sufficiently small. This velocity Zat is said
to be uniformly accelerated ; since in each second the
same velocity 2a is gained. And since, when x is the

space described, cp't is the limit of — , the velocity is

also this limit ; that is, when a point does not move
uniformly, the velocity is not represented by any in
crement of length divided by its increment of time,
but by the limit to which that ratio continually tends,
as the increment of time is diminished.

SIMPLE HARMONIC MOTION.

We now propose the following problem : A point
moves uniformly round a circle ; with what velocities
do the abscissa and ordinate increase or decrease, at
any given point? (Fig. 8.)

Let the point P, setting out from A, describe the
arc AP, etc., with the uniform velocity of a inches
per second. Let

From the first principles of trigonometry

x = rcosO
x — dx = r cos (0 -f- dO) = r cos 0 cos d& — r sin 0 sin dO

ELEMENTARY ILLUSTRATIONS OF

Subtracting the second from the first, and the third
from the fourth, we have

dx — r sin 0 sin dO -\-rcos 0(1 — cos dO) (1)

dy = rcosOsindO-}-rsinO(l — cos</0) (2)

But if dO be taken sufficiently small, sindO, and dO,
may be made as nearly in a ratio of equality as we
please, and 1 — cos dO may be made as small a part
as we please, either of dO or sin dO. These follow from
Fig. 1, in which it was shown that BM and the arc
BA, or (if OA = r and AOB=dO), r sindO and rdO,
may be brought as near to a ratio of equality as we

Fig. 8

O Mx M A

please, which is therefore true of sin dO and dO. Again,
it was shown that AM, or r — rcosdO, can be made
as small a part as we please, either of BM or the arc
BA, that is, either of r s'mdO, or rdQ\ the same is
therefore true of 1 — cosdO, and either sindO or dO.
Hence, if we write equations (1) and (2) thus,

dx = rsinOdO (1) dy = r cosOdO (2),

we have equations, which, though never exactly true,
are such that by making dO sufficiently small, the
errors may be made as small parts of dO as we please.
Again, since the arc AP is uniformly described, so
also is the angle POA ; and since an arc a is described

THE DIFFERENTIAL AND INTEGRAL CALCULUS. 5Q

in one second, the angle — is described in the same

time; this is, therefore, the angular velocity.* If we
divide equations (1) and (2) by dt, we have

dx „ dB dy A dB

these become more nearly true as dt and dB are dimin-

dx

ished, so that if for — , etc., the limits of these ratios
at

be substituted, the equations will become rigorously
true. But these limits are the velocities of x, y, and

B, the last of which is also — ; hence

velocity of x = r sin0 X — =a sin0,

velocity of y=r cos 6 X — =a cos0;

that is, the point M moves towards O with a variable
velocity, which is always such a part of the velocity
of P, as sin0 is of unity, or as PM is of OB ; and the
distance PM increases, or the point N moves from O,
with a velocity which is such a part of the velocity of
P as cos0 is of unity, or as OM is of OA. [The mo
tion of the point M or the point N is called in physics
a simple harmonic motion.']

In the language of Leibnitz, the results of the two
foregoing sections would be expressed thus : If a
point move, but not uniformly, it may still be con
sidered as moving uniformly for any infinitely small

*The same considerations of velocity which have been applied to the
motion of a point along a line may also be applied to the motion of a line
round a point. If the angle so described be always increased by equal angles
in equal portions of time, the angular velocity is said to be uniform, and is
measured by the number of angular units described in a unit of time. By
similar reasoning to that already described, if the velocity with which the
angle increases be not uniform, so that at the end of the time t the angle de
scribed is 0 = $tt the angular velocity is <£7, or the limit of the ratio -j- .

60 ELEMENTARY ILLUSTRATIONS OF

time ; and the velocity with which it moves is the in
finitely small space thus described, divided by the in
finitely small time.

THE METHOD OF FLUXIONS.

The foregoing process contains the method em
ployed by Newton, known by the name of the Method
of Fluxions. If we suppose y to be any function of x,
and that x increases with a given velocity, y will also
increase or decrease with a velocity depending : (1)
upon the velocity of x ; (2) upon the function which
y is of x. These velocities Newton called the fluxions
of y and x, and denoted them by y and x. Thus, if
y = x2, and if in the interval of time dt, x becomes
x -\- dx, and y becomes y -f dy, we have y-{-dy =

and dy = 2x dx + (dx}*, or = 2x ~

-— dx. If we diminish dt, the term — dx will dimin-
at at

ish without limit, since one factor continually ap
proaches to a given quantity, viz., the velocity of x,
and the other diminishes without limit. Hence we
obtain the velocity of y = 2x X the velocity of x, or
y = 2x x, which is used in the method of fluxions in
stead of dy = 2x dx considered in the manner already
described. The processes are the same in both meth
ods, since the ratio of the velocities is the limiting
ratio of the corresponding increments, or, according
to Leibnitz, the ratio of the infinitely small incre
ments. We shall hereafter notice the common objec
tion to the Method of Fluxions.

ACCELERATED MOTION.

When the velocity of a material point is suddenly
increased, an impulse is said to be given to it, and the

THE DIFFERENTIAL AND INTEGRAL CALCULUS. 6l

magnitude of the impulse or impulsive force is in pro
portion to the velocity created by it. Thus, an im
pulse which changes the velocity from 50 to 70 feet
per second, is twice as great as one which changes it
from 50 to 60 feet. When the velocity of the point is
altered, not suddenly but continuously, so that before
the velocity can change from 50 to 70 feet, it goes
through all possible intermediate velocities, the point
is said to be acted on by an accelerating force. Force
is a name given to that which causes a change in the
velocity of a body. It is said to act uniformly, when
the velocity acquired by the point in any one interval
of time is the same as that acquired in any other in
terval of equal duration. It is plain that we cannot,
by supposing any succession of impulses, however
small, and however quickly repeated, arrive at a uni
formly accelerated motion ; because the length de
scribed between any two impulses will be uniformly
described, which is inconsistent with the idea of con
tinually accelerated velocity. Nevertheless, by di
minishing the magnitude of the impulses, and increas
ing their number, we may come as near as we please
to such a continued motion, in the same way as, by
diminishing the magnitudes of the sides of a polygon,
and increasing their number, we may approximate as
near as we please to a continous curve.

Let a point, setting out from a state of rest, in
crease its velocity uniformly, so that in the time /, it
may acquire the velocity v — what length will have
been described during that time / ? Let the time /
and the velocity v be both divided into n equal parts,
each of which is /' and v'\ so that #/'==/, and nv' = v.
Let the velocity v' be communicated to the point at
rest ; after an interval of /' let another velocity v' be

62 ELEMENTARY ILLUSTRATIONS OF

communicated, so that during the second interval f
the point has a velocity 2z/'; during the third interval
let the point have the velocity 3z/, and so on ; so that
in the last or /zth interval the point has the velocity
nv'. The space described in the first interval is, there
fore, v't'-, in the second, 2z;Y; in the third 3z>Y; and
so on, till in the «th interval it is nv'f. The whole
space described is, therefore,

v't' -f 2z>Y + 30Y + . . . -f (n — 1 ) v'f + nv'f

or [1 + 2 + 3 (n — l^ + n^v't'^n.^^v'f

m

tfv't' + wY

_._„__

In this substitute v for »#', and / for nf, which gives
for the space described %v(t-{-f}. The smaller we
suppose /, the more nearly will this approach to \vt.
But the smaller we suppose f, the greater must be n,
the number of parts into which / is divided ; and the
more nearly do we render the motion of the point uni
formly accelerated. Hence the limit to which we ap
proximate by diminishing /without limit, is the length
described in the time / by a uniformly accelerated
velocity, which shall increase from 0 to v in that time.
This is \vt, or half the length which would have been
described by the velocity v continued uniformly from
the beginning of the motion.

It is usual to measure the accelerating force by the
velocity acquired in one second. Let this be g\ then
since the same velocity is acquired in every other sec
ond, the velocity acquired in / seconds will be gtt or
v=gt. Hence the space described is J^/X *» or \£&-
If the point, instead of being at rest at the beginning
of the acceleration, had had the velocity a, the lengths

THE DIFFERENTIAL AND INTEGRAL CALCULUS. 63

described in the successive intervals would have been
tf/'-f- z/Y, at '-\- Zv't', etc. ; so that to the space described
by the accelerated motion would have been added
naf, or at, and the whole length would have been
at-}-%gfi. By similar reasoning, had the force been
a uniformly retarding force, that is, one which dimin
ished the initial velocity a equally in equal times, the
length described in the time / would have been at

Now let the point move in such a way, that the
velocity is accelerated or retarded, but not uniformly ;
that is, in different times of equal duration, let differ
ent velocities be lost or gained. For example, let the
point, setting out from a state of rest, move in such a

Tig. 9
O AB C D

way that the number of inches passed over in / sec
onds is always ts. Here (pt = tB, and the velocity ac
quired by the body at the end of the time /, is the co
efficient of dt in (/-f-///)3, or 3/2 inches per second.
Let the point (Fig. 9) be at A at the end of the time
/; and let AB, BC, CD, etc., be lengths described in
successive equal intervals of time, each of which is dt.
Then the velocities at A, B, C, etc., are 3^, 3(7 -f /#)»,
3(/+2<#)2, etc., and the lengths AB, BC, CD, etc.,
are (/ + <#)3 — /3,
(/+2<#)8, etc.

VELOCITY AT

A 3/2

B 3/2+
C 3/2 +

64 ELEMENTARY ILLUSTRATIONS OF

LENGTH OF

BC 3/V/+ 9/(X/)2-f 7(X/)3
CD 3A// + 15/0//)2 + 19(X/)8

If we could, without error, reject the terms con
taining (X/)2 in the velocities, and those containing
(dt)* in the lengths, we should then reduce the mo
tion of the point to the case already considered, the
initial velocity being 3/2, and the accelerating force 6/.
For we have already shown that a being the initial
velocity, and g the accelerating force, the space de
scribed in the time / is at + \gfi. Hence, 3/2 being
the initial velocity, and 6^ the accelerating force, the
space in the time dt is 3/V/ -f 3/ (dt)2, which is the
same as AB after (dt)* is rejected. The velocity ac
quired is gt, and the whole velocity is, therefore,
&-\-gt\ or making the same substitutions 3P -f- §tdt.
This is the velocity at B, after the term 3(X/)2 is
rejected. Again, the velocity being %fl-^-§tdt, and
the force 6/, the space described in the time dt is
(ZP + §tdt)dt-\- 3/(X/)2, or 3/V/-f9/(X/)2. This is
what the space BC becomes after 7(*//)8 is rejected.
The velocity acquired is Qtdt; and the whole velocity
is3/2-f §tdt+§tdt, or3/2-f I2tdt; which is the velo
city at C after 12 (X/)2 is rejected.

But as the terms involving (dt)* in the velocities,
etc., cannot be rejected without error, the above sup
position of a uniform force cannot be made. Never
theless, as we may take dt so small that these terms
shall be as small parts as we please of those which
precede, the results of the erroneous and correct sup
positions may be brought as near to equality as we
please ; hence we conclude, that though there is no
force, which, continued uniformly, would preserve

THE DIFFERENTIAL AND INTEGRAL CALCULUS. 65

the motion of the point A, so that OA should always
be tz in inches, yet an interval of time may be taken
so small, that the length actually described by A in
that time, and the one which would be described if
the force 6/ were continued uniformly, shall have a
ratio as near to equality as we please. Hence, on a
principle similar to that by which we called 3fl the
velocity at A, though, in truth, no space, however
small, is described with that velocity, we call 6/ the
accelerating force at A. And it must be observed
that §t is the differential coefficient of S^2, or the co
efficient of dt, in the development of 3(/-j- df}*.

Generally, let the point move so that the length
described in any time t is <pt. Hence the length de
scribed at the end of the time /-f dt is cp(t-\-df}, and
that described in the interval dt is cp(t -\- df) — <pt, or

in which dt may be taken so small, that either of the
first two terms shall contain the aggregate of all the
rest, as often as we please. These two first terms are
<p'tdt-\-%<p"t(dt}*, and represent the length described
during dt, with a uniform velocity cp't, and an accel
erating force <p"t. The interval dt may then generally
be taken so small, that this supposition shall represent
the motion during that interval as nearly as we please.

LIMITING RATIOS OF MAGNITUDES THAT INCREASE
WITHOUT LIMIT.

We have hitherto considered the limiting ratio of
quantities only as to their state of decrease : we now
proceed to some cases in which the limiting ratio of
different magnitudes which increase without limit is
investigated.

66 ELEMENTARY ILLUSTRATIONS OF

It is easy to show that the increase of two magni
tudes may cause a decrease of their ratio ; so that, as
the two increase without limit, their ratio may dimin
ish without limit. The limit of any ratio may be found
by rejecting any terms or aggregate of terms (Q) which
are connected with another term (P) by the sign of
addition or subtraction, provided that by increasing
x, Q may be made as small a part of P as we please.

For example, to find the limit of — 0 , , , , when

Ax1" -j- vx

x is increased without limit. By increasing x we can,
as will be shown immediately, cause 2x -f- 3 and §x to
be contained in x2 and 2#2, as often as we please ; re-

x2
jecting these terms, we have ^— ^, or J, for the limit.

The demonstration is as follows : Divide both

numerator and denominator by x*, which gives 1 -{-
o q &

— H — zr, and 2H , for the numerator and denom-

x x2 x

inator of a fraction equal in value to the one proposed.
These can be brought as near as we please to 1 and 2

by making x sufficiently great, or — sufficiently small ;
and, consequently, their ratio can be brought as near

as we please to -^.
2

We will now prove the following : That in any
series of decreasing powers of x, any one term will, if
x be taken sufficiently great, contain the aggregate of
all which follow, as many times as we please. Take,
for example,

ax™ -f bxm~^ -f ex™-* + +px + q

THE DIFFERENTIAL AND INTEGRAL CALCULUS. 67

The ratio of the several terms will not be altered if we
divide the whole by xm, which gives

etc-

It has been shown that by taking — sufficiently small,

oc

that is, by taking x sufficiently great, any term of this
series may be made to contain the aggregate of the
succeeding terms, as often as we please ; which rela
tion is not altered if we multiply every term by tf",
and so restore the original series.

(x \ V\m

It follows from this, that m has unity for its

oc

limit when x is increased without limit. For (x -f- 1)"*
is xm -J- mx™-1 -|- etc. , in which x™ can be made as
great as we please with respect to the rest of the

O-f IV* , mx"-1 -\-etc.

series. Hence v lm } —\-\ -- 5 - , the nu-
x x

merator of which last fraction decreases indefinitely
as compared with its denominator.

In a similar way it may be shown that the limit of

(*+!)•£_*-»' when * is increased, is -i^. For
since (* + l)m+l = xm+1 -\- (m-\- 1) x~ + ±(m-
-\- etc., this fraction is

etc.

in which the first term of the denominator may be
made to contain all the rest as often as we please ;

xm
that is, if the fraction be written thus, T

A can be made as small a part of (#*-j- I)*"* as we

68 ELEMENTARY ILLUSTRATIONS OF

please. Hence this fraction can, by a sufficient in
crease of x, be brought as near as we please to
xm 1

A similar proposition may be shown of the fraction

(x 4. M«
- — — — +1 — j^pp which may be immediately reduced

to the form — — -, where x may be taken

(M-\- l)axm-\- A

so great that xm shall contain A and B any number of
times.

We will now consider the sums of x terms of the
following series, each of which may evidently be made
as great as we please, by taking a sufficient number
of its terms,

. | | ^

1** -j- 2^ -j- 3^ 1 43 i

(x I)3 4-*8

1- 4- 2- 4- 3~ 4- 4" 4- 4. (X — 1)~ 4- «" O)

We propose to inquire what is the limiting ratio of
any one of these series to the last term of the succeed
ing one ; that is, to what do the ratios of (1 4- 2

4- +#) to x2, of (I2 4- 22.... -I-*8) to Ar8, etc.,

approach, when x is increased without limit.

To give an idea of the method of increase of these
series, we shall first show that x may be taken so
great, that the last term of each series shall be as
small a part as we please of the sum of all those which
precede. To simplify the symbols, let us take the
third series I8 4- 28 -f . . . . 4~ •*3> in which we are to
show that x9 may be made less than any given part,

THE DIFFERENTIAL AND INTEGRAL CALCULUS. 69

say one thousandth, of the sum of those which pre
cede, or of !3-j-28 +(* — I)8.

First, x may be taken so great that x* and (x —
1000)8 shall have a ratio as near to equality as we
please. For the ratio of these quantities being the

/, 1000V , 1000 , .
same as that of 1 to (1 J , and being as

small as we please if x may be as great as we please, it

1000 /, 1000\3
follows that 1 , and, consequently, 1 1

x \ x j

may be made as near to unity as we please, or the

ratio of 1 to (1 — : — ] , may be brought as near as

\ x /

we please to that of 1 to 1, or a ratio of equality. But

this ratio is that of x9 to (x — 1000)8. Similarly the
ratios of x* to (x— 999)8, of x8 to (x — 998)8, etc., up
to the ratio of XB to (x — I)3 may be made as near as
we please to ratios of equality ; there being one thou
sand in all. If, then, (x — l)* = axs, (x—2^ = fix*,
etc., up to (x — lOOO)3^^^3, x can be taken so great
that each of the fractions a, /?, etc., shall be as near
to unity, or a -j- ft -f- . . . . -\- GO as near * to 1000 as we

please. Hence — — ^— — which is

ax* -f fix3 -f -f GOX*

--a or

_ 1000V*'

— l)8-f-O — 2

* Observe that this conclusion depends upon the number of quantities a,
ft, etc., being determinate. If there be ten quantities, each of which can be
brought ?s near to unity as we please, their sum can be brought as near to 10
as we please; for, take any fraction A, and make each of those quantities
differ from unity by less than the tenth part of A, then will the sum differ
from 10 by less than A. This argument fails, if the number of quantities be
unlimited.

70 ELEMENTARY ILLUSTRATIONS OF

can be brought as near to Tc™ as we please ; and by

the same reasoning, the fraction

(*—!)• -f- ...... -f O— 1001)8

may be brought as near to nT as we please ; that is,

may be made less than T- Still more then may

_ _

(X — I)' + . . . . + (* — 1001)3 -f . . . . -f 23 -}- 18

be made less than ' or x* ma^ ^e ^ess t*ian

thousandth part of the sum of all the preceding terms.
In the same way it may be shown that a term may
be taken in any one of the series, which shall be less
than any given part of the sum of all the preceding
terms. It is also true that the difference of any two
succeeding terms may be made as small a part of
either as we please. For (x-\-I}m — xm, when devel
oped, will only contain exponents less than m, being

mxm~1 -f- m — 5 — a:"*-2 -f etc. ; and we have shown

a

(page 66) that the sum of such a series may be made
less than any given part of xm. It is also evident
that, whatever number of terms we may sum, if a
sufficient number of succeeding terms be taken, the
sum of the latter shall exceed that of the former in
any ratio we please.

Let there be a series of fractions

__
pa' -f *>'' p

in which a, a', etc., b, V ', etc., increase without limit;
but in which the ratio of b to a, b' to a, etc., dimin
ishes without limit. If it be allowable to begin by

THE DIFFERENTIAL AND INTEGRAL CALCULUS. 71

supposing b as small as we please with respect to a,

or — as small as we please, the first, and all the suc-
a 1

ceeding fractions, will be as near as we please to — -,
which is evident from the equations

a 1 a' 1
__ _____ *»tc

.A,. I L i ' JL. _ I I A' U ' c^*'»

* t + lS

Form a new fraction by summing the numerators and
denominators of the preceding, such as

" + etc., '
the etc. extending to any given number of terms.

This may also be brought as near to — as we please.

P
For this fraction is the same as

etc.

and it can be shown* that

I, -L. y 4. etc.

a -f a' -f etc.
must lie between the least and greatest of the fractions

7 7/

— , —7, etc. If, then, each of these latter fractions
a a

can be made as small as we please, so also can

b + V + etc.

a -j- #' + etc. '

No difference will be made in this result, if we use
the following fraction,

A -f- (a-\-a'-\-a" -f etc. )

B + p (a -}- a' + a" + etc. ) + £ + b' + £" -f etc.

(1)

* See Study of Mathematics (Reprint Edition, Chicago : The Open Court
Publishing Co.), page 270.

72 ELEMENTARY ILLUSTRATIONS OF

A and B being given quantities ; provided that we
can take a number of the original fractions sufficient
to make a-\- a' -}- a" -\- etc., as great as we please,
compared with A and B. This will appear on divid
ing the numerator and denominator of (1) by a-\-ar -f-
a" -f etc.

Let the fractions be

S)3

• . ..,,, etc.

%

The first of which, or — — may, as we have

4x* -f- etc. j

shown, be within any given difference of — . and the

4

others still nearer, by taking a value of x sufficiently
great. Let us suppose each of these fractions to be

within -,AAAAA of —r-. The fraction formed by sum-
100000 4

ming the numerators and denominators of these frac
tions (n in number) will be within the same degree of
nearness to J. But this is

all the terms of the denominator disappearing, except
two from the first and last. If, then, we add #4 to
the denominator, and I3 -f- 23 -f- 38 . . . . -}- jc3 to the nu
merator, we can still take n so great that (x -J- I)3
-f- ____ -f (x -f »)8 shall contain I3 -f- . . -f- xs as often
as we please, and that (x -j- «)4 — .*4 shall contain x*
in the same manner. To prove the latter, observe

that the ratio of (x -\- n)* — x* to x* being l -j --

\ x
can be made as great as we please, if it be permitted

THE DIFFERENTIAL AND INTEGRAL CALCULUS. 73

to take for it a number containing x as often as we
please. Hence, by the preceding reasoning, the frac
tion, with its numerator and denominator thus in
creased, or

may be brought to lie within the same degree of near
ness to J as (2); and since this degree of nearness
could be named at pleasure, it follows that (3) can
be brought as near to J as we please. Hence the
limit of the ratio of (l8-f2»-f- ---- + *») to #*, as x
is increased without limit, is J ; and, in a similar man
ner, it may be proved that the limit of the ratio of
(!*»_{_ 2"»-f ---- +#*") to x*+l is the same as that of

This result will be of use when we come to the
first principles of the integral calculus. It may also

be noticed that the limits of the ratios which x ~ ,

ZJ

x — ^ -- 5 — , etc. , bear to x*, x9, etc., are severally -JT-,

a O Li

^r-n-, etc. ; the limit being that to which the ratios ap-
A-6 x _ i

proximate as x increases without limit. For x — 5 —

a

x— 1 x— Ix — 2 x—lx — Z

-" * = -&-• * ~2- -g- -H^= ~£T -TEr- etc"

... _ i „ _ o

and the limits of — — , - , are severally equal to

unity.

We now resume the elementary principles of the
Differential Calculus.

74 ELEMENTARY ILLUSTRATIONS OF

RECAPITULATION OF RESULTS.

The following is a recapitulation of the principal
results which have hitherto been noticed in the gen
eral theory of functions :

(1) That if in the equation y=(p(x), the variable
x receives an increment dx, y is increased by the se
ries

<p'X dx + 9"X . + <p»X + etc.

(2) That (p"x is derived in the same manner from
cp'x, that cp'x is from <px ; viz., that in like manner as
<p'x is the coefficient of dx in the development of
<p(x-\-dx), so cp"x is the coefficient of dx in the de
velopment of <p' '(x -j- dx')', similarly (p"'x is the coeffi
cient of dx in the development of cp"(x-\-dx), and
so on.

(3) That qfx is the limit of S^ or the quantity to

which the latter will approach, and to which it may
be brought as near as we please, when dx is dimin
ished. It is called the differential coefficient of y.

(4) That in every case which occurs in practice,
dx may be taken so small, that any term of the series
above written may be made to contain the aggregate
of those which follow, as often as we please ; whence,
though qfxdx is not the actual increment produced
by changing x into x -f- dx in the function cpx, yet, by
taking dx sufficiently small, it may be brought as near
as we please to a ratio of equality with the actual in
crement.

APPROXIMATIONS.

The last of the above-mentioned principles is of
the greatest utility, since, by means of it, (p'xdx may

THE DIFFERENTIAL AND INTEGRAL CALCULUS. 75

be made as nearly as we please the actual increment ;
and it will generally happen in practice, that qfxdx
may be used for the increment of (px without sensible
error ; that is, if in (px, x be changed into x -j- dx, dx
being very small, <px is changed into (px+ cp'xdx,
very nearly. Suppose that x being the correct value
of the variable, x-\-h and x-\- k have been succes
sively substituted for it, or the errors h and k have
been committed in the valuation of x, h and k being
very small. Hence <p(x-\-h) and cp(x-\-k) will be
erroneously used for cpx. But these are nearly cpx-\-
<p'xh and cpx-\-<p'xk, and the errors committed in
taking cpx are qfxh and <p'xk, very nearly. These
last are in the proportion of h to k, and hence results
a proposition of the utmost importance in every prac
tical application of mathematics, viz., that if two dif
ferent, but small, errors be committed in the valua
tion of any quantity, the errors arising therefrom at
the end of any process, in which both the supposed
values of x are successively adopted, are very nearly
in the proportion of the errors committed at the be
ginning. For example, let there be a right-angled
triangle, whose base is 3, and whose other side should
be 4, so that the hypothenuse should be 1/32 -}- 42
or 5. But suppose that the other side has been twice
erroneously measured, the first measurement giving
4-001, and the second 4-002, the errors being -001
and -002. The two values of the hypothenuse thus
obtained are

T/32_|_4-0012, or 1/25-008001,

and ]/38 + 4-0022, or 1/25-016004,
which are very nearly 5-0008 and 5-0016. The errors
of the hypothenuse are then -0008 and -0016 nearly ;
and these last are in the proportion of -001 and -002.

76 ELEMENTARY ILLUSTRATIONS OF

It also follows, that if x increase by successive equal
steps, any function of x will, for a few steps, increase
so nearly in the same manner, that the supposition of
such an increase will not be materially wrong. For,
if h, 2h, 3^, etc. , be successive small increments given
to x, the successive increments of (px will be qfxh,
<pxZh, cp'xSh, etc. nearly; which being proportional
to h, Zh, 3^, etc., the increase of the function is nearly
doubled, trebled, etc., if the increase of x be doubled,
trebled, etc.

This result may be rendered conspicuous by ref
erence to any astronomical ephemeris, in which the
positions of a heavenly body are given from day to
day. The intervals of time at which the positions are
given differ by 24 hours, or nearly -g-J-gth part of the
whole year. And even for this interval, though it can
hardly be called small in an astronomical point of view,
the increments or decrements will be found so nearly
the same for four or five days together, as to enable
the student to form an idea how much more near they
would be to equality, if the interval had been less, say
one hour instead of twenty-four. For example, the
sun's longitude on the following days at noon is writ
ten underneath, with the increments from day to day.

Proportion which the differences

1st

158° 30' 35"

58'

2nd

159 28 44

58

3rd

160 26 56

58

4th

161 25 9

5th

162 23 23

The sun's longitude is a function of the time ; that is,
the number of years and days from a given epoch
being given, and called x, the sun's longitude can be

THE DIFFERENTIAL AND INTEGRAL CALCULUS. 77

found by an algebraical expression which may be
called cpx. If we date from the first of January, 1834,
x is -666, which is the decimal part of a year between
the first days of January and September. The incre
ment is one day, or nearly -0027 of a year. Here x is
successively made equal to -666, -666 + -0027, -666-}-
2 X ' 0027, etc. ; and the intervals of the corresponding
values of <pxt if we consider only minutes, are the
same ; but if we take in the seconds, they differ from
one another, though only by very small parts of them
selves, as the last column shows.

SOLUTION OF EQUATIONS.

This property is also used* in finding logarithms
intermediate to those given in the tables ; and may
be applied to find a nearer solution to an equation,
than one already found. For example, suppose it re
quired to find the value of x in the equation <px = Q,
a being a near approximation to the required value.
Let a -f- h be the real value, in which h will be a small
quantity. It follows that cp(a-\- h) = (), or, which is
nearly true, <pa-\- (p'ah = Q. Hence the real value of

h is nearly — — ,— . or the value a — ^—r- is a nearer
cp a cp'a

approximation to the value of x. For example, let
x* -f x — 4 = 0 be the equation. Here cpx = x*-\- x — 4,
and cp(x + h) = (x + ?i)* + x + h — 4 = x* + x— 4 +
(2# + l)£ + /*8; so that <p'x = 2x + l. A near value
of x is 1-57; let this be a. Then <pa = -0349, and

cp'a = 4' 14. Hence — -??- = — • 00843. Hence

cp'a

1 . 57 _ . 00843, or 1 • 56157, is a nearer value of x. If

* See Study of Mathematics (Reprint Edition, Chicago : The Open Court
Publishing Co., 1898), page 169 et seq.

78 ELEMENTARY ILLUSTRATIONS OF

we proceed in the same way with 1-5616, we shall
find a still nearer value of x, viz., 1-561553. We
have here chosen an equation of the second degree,
in order that the student may be able to verify the
result in the common way ; it is, however, obvious
that the same method may be applied to equations
of higher degrees, and even to those which are not
to be treated by common algebraical method, such as
tan x = ax.

PARTIAL AND TOTAL DIFFERENTIALS.

We have already observed, that in a function of
more quantities than one, those only are mentioned
which are considered as variable ; so that all which
we have said upon functions of one variable, applies
equally to functions of several variables, so far as a
change in one only is concerned. Take for example
x2y-{-2xy8. If x be changed into x-\-dx, y remaining
the same, this function is increased by 2xy dx -j- 2y*dx
-{-etc., in which, as in page 29, no terms are con
tained in the etc. except those which, by diminishing
dx, can be made to bear as small a proportion as we
please to the first terms. Again, if y be changed into
y-\-dy, x remaining the same, the function receives
the increment oPdy -\- §xy*dy -f- etc. ; and if x be changed
into x -\-dx, y being at the same time changed into
y-{- dy, the increment of the function is (2xy-\-Zp)dx
_j_ (a* _|_ QXy^dy -f etc. If, then, u = x*y + 2xy*, and
du denote the increment of u, we have the three fol
lowing equations, answering to the various supposi
tions above mentioned,

(1) when x only varies,

du = (2xy + 2/) dx + etc.

THE DIFFERENTIAL AND INTEGRAL CALCULUS. 7Q

(2) when y only varies,

du — O2 -f 6#/) dy -f etc.

(3) when both x and y vary,

du = (2xy + 2/) dx + (x* + 6^2) dy -f etc.

in which, however, it must be remembered, that du
does not stand for the same thing in any two of the
three equations : it is true that it always represents
an increment of u, but as far as we have yet gone, we
have used it indifferently, whether the increment of u
was the result of a change in x only, or y only, or both
together.

To distinguish the different increments of », we
must therefore seek an additional notation, which,
without sacrificing the du that serves to remind us
that it was u which received an increment, may also
point out from what supposition the increment arose.
For this purpose we might use dxu and dyu9 and dXtji,
to distinguish the three ; and this will appear to the
learner more simple than the one in common use,
which we shall proceed to explain. We must, how
ever, remind the student, that though in matters of
reasoning, he has a right to expect a solution of every
difficulty, in all that relates to notation, he must trust
entirely to his instructor ; since he cannot judge be
tween the convenience or inconvenience of two sym
bols without a degree of experience which he evi
dently cannot have had. Instead of the notation above
described, the increments arising from a change in x

and y are severally denoted by -y- dx and — dy, on

the following principle : If there be a number of re
sults obtained by the same species of process, but on
different suppositions with regard to the quantities

80 ELEMENTARY ILLUSTRATIONS OF

used ; if, for example, p be derived from some suppo
sition with regard to a, in the same manner as are q
and r with regard to b and c, and if it be inconvenient
and unsymmetrical to use separate letters /, ^, and r,
for the three results, they may be distinguished by
using the same letter p for all, and writing the three

results thus, — - a, ~ b, — c. Each of these, in com-
a b c

mon algebra, is equal to /, but the letter / does not
stand for the same thing in the three expressions.
The first is the /, so to speak, which belongs to «, the
second that which belongs to b, the third that which
belongs to c. Therefore the numerator of each of the

fractions — -, -£, and — , must never be separated
a o c

from its denominator, because the value of the former
depends, in part, upon the latter ; and one p cannot
be distinguished from another without its denomina
tor. The numerator by itself only indicates what op
eration is to be performed, and on what quantity; the
denominator shows what quantity is to be made use
of in performing it. Neither are we allowed to say

that — divided by -~ is — ; for this supposes that /
a b a

means the same thing in both quantities.

In the expressions - - dx, and - dy, each denotes

that u has received an increment ; but the first points
out that x, and the second that^y, was supposed to in
crease, in order to produce that increment ; while du
by itself, or sometimes d.u, is employed to express
the increment derived from both suppositions at once.
And since, as we have already remarked, it is not the
ratios of the increments themselves, but the limits of
those ratios, which are the objects of investigation in

THE DIFFERENTIAL AND INTEGRAL CALCULUS. 8 1

the Differential Calculus, here, as in page 28, — - dx,

7 (IX

and -j-dyt are generally considered as representing

those terms which are of use in obtaining the limiting
ratios, and do not include those terms, which, from
their containing higher powers of dx or dy than the
first, may be made as small as we please with respect
to dx or dy. Hence in the example just given, where
2xy*, we have

dx = (fry + 2/) dx, 0* = Zx? +

-

du du

du or a.u= ~ ax A — =- ay.

dx d "

The last equation gives a striking illustration of
the method of notation. Treated according to the
common rules of algebra, it is du = du-\- du, which is
absurd, but which appears rational when we recollect
that the second du arises from a change in x only, the
third from a change in y only, and the first from a
change in both. The same equation may be proved
to be generally true for all functions of x and y, if we
bear in mind that no term is retained, or need be re
tained, as far as the limit is concerned, which, when
dx or dy is diminished, diminishes without limit as

compared with them. In using — and -=- as differ

ential coefficients of u with respect to x and y, the ob
jection (page 27) against considering these as the
limits of the ratios, and not the ratios themselves,
does not hold, since the numerator is not to be sep
arated from its denominator.

82 ELEMENTARY ILLUSTRATIONS OF

Let u be a function of x and^y, represented* by
cp(x, y). It is indifferent whether x and y be changed
at once into x -\- dx and jy -f- dy, or whether # be first
changed into x -\- dx, and y be changed into y-\-dy in
the result. Thus, x*y -f- _y3 will become (#-[-d&)2
Oy + d&O + (y -f ^v)3 in either case. If x be changed
into x-\-dx, u becomes u -}- u dx -f- etc. , (where «' is
what we have called the differential coefficient of u
with respect to x, and is itself a function of x and_y;
and the corresponding increment of u is u' dx-\- etc.)
If in this result y be changed into y -\- dy, u will as
sume the form u -f- ut dy -f- etc. , where ut is the differen
tial coefficient of u with respect to y ; and the incre
ment which u receives will be utdy -|- etc. Again,
when^ is changed mto y-\-dy, u', which is a function
of x andjy, will assume the form u' -\-pdy-\-etc. ; and
u -\- u dx -f- etc. becomes u -f- ucly -f- etc. -|- (u' -f- p dy
+ etc. ) dx -f- etc. , or u-\- utdy + u' dx -\- p dx dy -\- etc.,
in which the termfldxdy is useless in finding the limit.
For since dy can be made as small as we please,
pdxdy can be made as small a part of pdx as we please,
and therefore can be made as small a part of dx as
we please. Hence on the three suppositions already
made, we have the following results :

(1) when x only is changed

u receives

the
increment

(2) when y only is changed
into y -f- dy,

(3) when x becomes x-\-dx
and y becomes y-\-dy
at once,

utdy 4- etc.

u'dx -f- utdy -f- etc.

*The symbol $(x,y) must not be confounded with $(xy). The former rep
resents any function of^-and^; the latter a function in which x and y only
enter so far as they are contained in their product. The second is therefore
a particular case of the first ; but the first is not necessarily represented by

THE DIFFERENTIAL AND INTEGRAL CALCULUS. 83

the etc. in each case containing those terms only which
can be made as small as we please, with respect to
the preceding terms. In the language of Leibnitz,
we should say that if x and y receive infinitely small
increments, the sum of the infinitely small increments
of u obtained by making these changes separately, is
equal to the infinitely small increment obtained by
making them both at once. As before, we may cor
rect this inaccurate method of speaking. The several
increments in (1), (2), and (3), maybe expressed by
u'dx -f- P, ut dy -f Q, and u'dx -\- ut dy -f R ; where P,
Q, and R can be made such parts of dx or dy as we
please, by taking dx or dy sufficiently small. The sum
of the two first is u'dx -f- udy -j- P -{- Q, which differs
from the third byP-J-Q — R; which, since each of
its terms can be made as small a part of dx or dy as
we please, can itself be made less than any given part
of dx or dy.

This theorem is not confined to functions of two
variables only, but may be extended to those of any
number whatever. Thus, if z be a function of /, q, r,
and s, we have

dz . , dz . , dz . , dz .
d.z or dz = — - dp 4- — dq -4- -7 - dr -f- — ds 4- etc.
dp dq ar as

in which — dp-\- etc. is the increment which a change

in/ only gives to z, and so on. The etc. is the repre
sentative of an infinite series of terms, the aggregate
of which diminishes continually with respect to dp,
dq, etc., as the latter are diminished, and which, there-

the second. For example, take the function xy + sin jcy, which, though it
contains both x and^, yet can only be altered by such a change in x andjy as
will alter their product, and if the product be called/, will be/ + sin/. This
may properly be represented by $(xy) ; whereas x + ,ry2 cannot be represented
in the same way, since other functions besides the product are contained
in it.

84 ELEMENTARY ILLUSTRATIONS OF

fore, has no effect on the limit of the ratio of d.z to
any other quantity.

PRACTICAL APPLICATION OF THE PRECEDING THEOREM.

We proceed to an important practical use of this
theorem. If the increments dp, dq, etc., be small,
this last-mentioned equation, (the terms included in
the etc. being omitted,) though not actually true, is
sufficiently near the truth for all practical purposes ;
which renders the proposition, from its simplicity, of
the highest use in the applications of mathematics.
For if any result be obtained from a set of data, no
one of which is exactly correct, the error in the result
would be a very complicated function of the errors in
the data, if the latter were considerable. When they
are small, the error in the results is very nearly the
sum of the errors which would arise from the error in
each datum, if all the others were correct. For if /,
q9 r, and s, are the presumed values of the data, which
give a certain value z to the function required to be
found ; and if p -j- dp, q -f- dqt etc., be the correct values
of the data, the correction of the function z will be

very nearly made, if z be increased by — dp -f -=- dq -f-
fly /fv dp dq

-j- dr 4- — ds , being the sum of terms which would
ar as

arise from each separate error, if each were made in
turn by itself.

For example : A transit instrument is a telescope
mounted on an axis, so as to move in the plane of the
meridian only, that is, the line joining the centres of
the two glasses ought, if the telescope be moved, to
pass successively through the zenith and the pole.
Hence can be determined the exact time, as shown by
a clock, at which any star passes a vertical thread,

THE DIFFERENTIAL AND INTEGRAL CALCULUS. 85

fixed inside the telescope so as apparently to cut the
field of view exactly in half, which thread will always
cover a part of the meridian, if the telescope be cor
rectly adjusted. In trying to do this, three errors
may, and generally will be committed, in some small
degree. (1) The axis of the telescope may not be ex
actly level ; (2) the ends of the same axis may not be
exactly east and west; (3) the line which joins the
centres of the two glasses, instead of being perpen
dicular to the axis of the telescope, may be inclined
to it. If each of these errors were considerable, and
the time at which a star passed the thread were ob
served, the calculation of the time at which the same
star passes the real meridian would require compli
cated formulae, and be a work of much labor. But if
the errors exist in small quantities only, the calcula
tion is very much simplified by the preceding princi
ple. For, suppose only the first error to exist, and
calculate the corresponding error in the time of pass
ing the thread. Next suppose only the second error,
and then only the third to exist, and calculate the
effect of each separately, all which may be done by
simple formula?. The effect of all the errors will then
be the sum of the effects of each separate error, at
least with sufficient accuracy for practical purposes.
The formulae employed, like the equations in page 28,
are not actually true in any case, but approach more
near to the truth as the errors are diminished.

RULES FOR DIFFERENTIATION.

In order to give the student an opportunity of ex
ercising himself in the principles laid down, we will
so far anticipate the treatises on the Differential Cal
culus as to give the results of all the common rules

86 ELEMENTARY ILLUSTRATIONS OF

for differentiation ; that is, assuming y to stand for
various functions of x, we find the increment of y aris
ing from an increment in the value of x, or rather,
that term of the increment which contains the first
power of dx. This term, in theory, is the only one
on which the limit of the ratio of the increments de
pends ; in practice, it is sufficiently near to the real
increment of y, if the increment of x be small.

(1) y = xm, where m is either whole or fractional,
positive or negative ; then dy = mx"*'1 dx. Thus the
increment of x$ or the first term 'of (x-\-dx)% — x$

is \x%~idx, or-—. Again, if y = x8, dy = 8x1dx.

When the exponent is negative, or when y = — ,
dy= -- ^-j, or when y = x~m) dy= — mx~m~ldx,

which is according to the rule. The negative sign
indicates that an increase in x decreases the value
of y\ which, in this case, is evident.

(2) y = a*. Here dy = a* log a dx where the log
arithm (as is always the case in analysis, except
where the contrary is specially mentioned) is the Na-
perian or hyperbolic logarithm. When a is the base
of these logarithms, that is when a = 2- 7182818 = e,
or when y = £*, dy = e*dx.

(3) y = logx (the Naperian logarithm). Here

dy= — . If y= common log*, ^ = -4342944 — .

x x

(4) y =
sinxdx; y = ta.nx, dy =

ILLUSTRATION OF THE PRECEDING FORMULA.

At the risk of being tedious to some readers, we
will proceed to illustrate these formulae by examples

THE DIFFERENTIAL AND INTEGRAL CALCULUS. 87

from the tables of logarithms and sines. Let j> = com
mon log x. If x be changed into x-\-dx, the real in
crement of y is

in which the law of continuation is evident. The cor
responding series for Naperian logarithms is to be
found in page 20. From the first term of this the
limit of the ratio of dy to dx can be found ; and if dx
be small, this will represent the increment with suffi
cient accuracy. Let # = 1000, whence y = common
loglOOO = 3; and let dx = ~L, or let it be required to
find the common logarithm of 1000 -f- 1, or 1001. The
first term of the series is therefore -4342 944 Xy^Viy» or
•0004343, taking seven decimal places only. Hence
log 1001 =log 1000+ -0004343 or 3-0004343 nearly.
The tables give 3-0004341, differing from the former
only in the 7th place of decimals.

Again, let y = sinx; from which, by page 20, as
before, if x be increased by dx, sinx is increased by
cosxdx — ^smx(dx)^ — etc., of which we take only
the first term. Let # = 16°, in which case sin# =
•2756374, and cos # = -9612617. Let dx = l', or, as
it is represented in analysis, where the angular unit is
that angle whose arc is equal to the radius*, ^f £^.
Hence sin 16° 1' = sin 16° + • 9612617 X *dHfro =
• 2756374 -f • 0002797 = • 2759171, nearly. The tables
give -2759170. These examples may serve to show
how nearly the real ratio of two increments approaches
to their limit, when the increments themselves are
small.

*See Study of Mathematics (Chicago : The Open Court Pub. Co.), page 273
et seg.

88 ELEMENTARY ILLUSTRATIONS OF

DIFFERENTIAL COEFFICIENTS OF DIFFERENTIAL

COEFFICIENTS.

When the differential coefficient of a function of x
has been found, the result, being a function of x, may
be also differentiated, which gives the differential co
efficient of the differential coefficient, or, as it is called,
the second differential coefficient. Similarly the differ
ential coefficient of the second differential coefficient
is called the third differential coefficient, and so on.
We have already had occasion to notice these succes
sive differential coefficients in page 22, where it ap
pears that cp'x being the first differential coefficient of
cpx, <p"x is the coefficient of h in the development
cp'(x -{• #), and is therefore the differential coefficient
of cp'x, or what we have called the second differential
coefficient of cpx. Similarly cp'"x is the third differ
ential coefficient of cpx. If we were strictly to ad
here to our system of notation, we should denote the
several differential coefficients of cpx or y by

dy

dy .*•?*

J ** ** etc-

dx dx dx

In order to avoid so cumbrous a system of notation,
the following symbols are usually preferred,

dy d*y d*y

dx 2? Z?' '

CALCULUS OF FINITE DIFFERENCES. SUCCESSIVE
DIFFERENTIATION.

We proceed to explain the manner in which this
notation is connected with our previous ideas on the
subject.

THE DIFFERENTIAL AND INTEGRAL CALCULUS. 89

When in any function of x, an increase is given to
x, which is not supposed to be as small as we please,
it is usual to denote it by Ax instead of dx, and the
corresponding increment of y or cpx, by Ay or Acpx,
instead of dy or dcpx. The symbol Ax is called the
difference of x, being the difference between the value
of the variable x, before and after its increase.

Let x increase at successive steps by the same dif
ference ; that is, let a variable, whose first value is x,
successively become x-}-Ax, x + ZAx, x-}-%Ax, etc.,
and let the successive values of cpx corresponding to
these values of x be y, yi, y%, j8, etc. ; that is, cpx is
called^, <p(x-\-Ax} is y\, <p(x-}-2Ax) is y*, etc., and,
generally, cp(x -\-mAx] isym. Then, by our previous
definition y\ — y is Ay, y^ — y\ is Ay\, J3 — y% is Ay^
etc., the letter A before a quantity always denoting
the increment it would receive if x-\-Ax were substi
tuted for x. Thus yi or cp(x-\-*&Ax) becomes <p(x-{-
Ax -\- 3 Ax), or cp(x -j- ±Ax), when x is changed into
x + Ax, and receives the increment cp(x -J- ±Ax) —
cp(x -\-ZAx), or j>4 — j8. If y be a function which de
creases when x is increased, y\ — y, or Ay is negative.

It must be observed, as in page 26, that Ax does
not depend upon x, because x occurs in it j the sym
bol merely signifies an increment given to x, which
increment is not necessarily dependent upon the value
of x. For instance, in the present case we suppose
it a given quantity; that is, when x-\-Ax is changed
into x -\-Ax-\- Ax, or x -\-2Ax, x is changed, and Ax
is not.

In this way we get the two first of the columns un
derneath, in which each term of the second column is
formed by subtracting the term which immediately
precedes it in the first column from the one which im-

ELEMENTARY ILLUSTRATIONS OF

mediately follows. Thus Ay is_yi — y, Ay\ is jy2 — y\,
etc.

>.+ ^)....j

(p(x

Ay
Ay\

Ay*

etc.

In the first column is to be found a series of suc
cessive values of the same function cpx, that is, it con
tains terms produced by substituting successively in
cpx the quantities x, x-\-Ax, x -\-2Ax, etc., instead of
x. The second column contains the successive values
of another function <p(x-\-Ax) — cpx, or A cpx, made by
the same substitutions ; if, for example, we substitute
x -\-ZAx for x, we obtain cp(x -\-3Ax) — (p(x -\-2Ax),
or ys — J2, or Ay*. If, then, we form the successive
differences of the terms in the second column, we ob
tain a new series, which we might call the differences
of the differences of the first column, but which are
called the second differences of the first column. And
as we have denoted the operation which deduces the
second column from the first by A, so that which de
duces the third from the second may be denoted by
A A, which is abbreviated into A*. Hence as y\ — y
was written Ay, Ay\ — Ay is written A Ay, or A^y. And
the student must recollect, that in like manner as A
is not the symbol of a number, but of an operation,
so A9 does not denote a number multiplied by itself,
but an operation repeated upon its own result ; just
as the logarithm of the logarithm of x might be writ
ten Iog2#; (logjc)2 being reserved to signify the square
of the logarithm of x. We do not enlarge on this no
tation, as the subject is discussed in most treatises on

THE DIFFERENTIAL AND INTEGRAL CALCULUS. QI

algebra.* Similarly the terms of the fourth column,
or the differences of the second differences, have the
prefix AAA abbreviated into A*, so that A^y\ — A*y —
A*y, etc.

When we have occasion to examine the results
which arise from supposing Ax to diminish without
limit, we use dx instead of Ax, dy instead of Ay, d^y in
stead of A*y, and so on. If we suppose this case, we
can show that the ratio which the term in any column
bears to its corresponding term in any preceding col
umn, diminishes without Hmit. Take for example,
d*y and dy. The latter is <p(x -f- dx) — cpx, which, as
we have often noticed already, is of the form / dx -f-
q(dxf -\- etc., in which p, q, etc., are also functions
of x. To obtain d*y, we must, in this series, change
x into x-\-dx, and subtract pdx-\- q(dx)* -\- etc. from
the result. But since p, q, etc., are functions of x,
this change gives them the form

f + fdx + etc., ?4Y*fcr-f etc.;
so that d*y is

(p +p'dx + etc. ) dx -f (q -f q'dx -f etc. ) (dx)* -f etc.
— (pdx -f- q (dx)* + etc. )

in which the first power of dx is destroyed. Hence
(pages 42-44), the ratio of d*y to dx diminishes with
out limit, while that of d*y to (dx)* has a finite limit,
except in those particular cases in which the second
power of dx is destroyed, in the previous subtraction,
as well as the first. In the same way it may be shown
that the ratio of dzy to dx and (dx)* decreases without
limit, while that of d*y to (dx)* remains finite ; and so

*The reference of the original text is to " the treatise on Algebraical Ex
pressions," Number 105 of the Library of Useful Knowledge, — the same series
in which the present work appeared. The first six pages of this treatise are
particularly recommended by De Morgan in relation to the present point.— Ed.

92 ELEMENTARY ILLUSTRATIONS OF

TT t - dy d*y d*y

on. Hence we have a succession of ratios -f-, ~» -^=.

dx dx* dx*

etc., which tend towards finite limits when dx is di
minished.

We now proceed to show that in the development
of cp(x -f A), which has been shown to be of the form

h* h*

<px-\- cp'xh -f cp"x -£ + <jt"x g-g -f etc.,

in the same manner as <p'x is the limit of -j- (page 23),

d^v d^y

so (p"x is the limit of — ^, <p'"x is that of -^~, and so

forth.

From the manner in which the preceding table
was formed, the following relations are seen imme

diately :

Ay i = Ay

etc.

etc.

Hence y\, y^ etc., can be expressed in terms of y, Ay,
A*y, etc. For yi =y+Ay\ y* =yi + Ayi = (y -f Ay} -f
(Ay -^-A^f) =y + 24y -f A^y. In the same way Ay* =
Ay -f- 2 A^y -\- A*y ; hence _y8 = j
J«y) + (J_y -f 2 A*y + A*y) =y
Proceeding in this way we have

A*y

, etc.

from the whole of which it appears that yn or cp(x -f
nAx} is a series consisting of y, Ay, etc., up to Any,
severally multiplied by the coefficients which occur in
the expansion (1 -f «)"» °r

THE DIFFERENTIAL AND INTEGRAL CALCULUS. 93

Let us now suppose that x becomes x -f- h by n

h 2/t nh

equal steps ; that is, x, x -j , x -\ , etc x -f- —

or x -f- h, are the successive values of x, so that
nAx — h. Since the product of a number of factors is
not altered by multiplying one of them, provided we
divide another of them by the same quantity, multiply
every factor which contains n by Ax, and divide the
accompanying difference of y by Ax as often as there
are factors which contain n, substituting h for nAx,

which gives

Ay A nAx — Ax A*y
cp(x -f n Ax} =y -f n Ax -—- -f nAx

Zj OC

nAx — Ax nAx — 2 Ax

*•

If h remain the same, the more steps we make be
tween x and x-\- h, the smaller will each of those
steps be, and the number of steps may be increased,
until each of them is as small as we please. We can
therefore suppose Ax to decrease without limit, with
out affecting the truth of the series just deduced.
Write dx for Axt etc., and recollect that h — dx,
h — 2dx, etc., continually approximate to h. The se
ries then becomes

dy . , d*yh* , d*y »

94 ELEMENTARY ILLUSTRATIONS OF

in which, according to the view taken of the symbols

~- etc. in pages 26-27, S- stands for the limit of the
ax ax

dy d^y

ratio of the increments, -f- is cp'x, -.— is cp"x, etc.

uX uX

According to the method proposed in pages 28-29,
the series written above is the first term of the devel
opment of cp(x-}-fr), the remaining terms (which we
might include under an additional -f- etc.) being such
as to diminish without limit in comparison with the
first, when dx is diminished without limit. And we

d*y

may show that the limit of -^4 is the differential co-
J dx*

dy

efficient of the limit of -/-; or if by these fractions

dx di

themselves are understood their limits, that -^ is the

dy dx

differential coefficient of -^- : for since dy, or (p(x -j- dx)

itOC

— cpx, becomes dy -f- d*y, when x is changed into
x -\- dx ; and since dx does not change in this process,

dy dy d^y d^y

-4- will become -^- -f- -r-t or its increment is —=-. The

dx dx dx dx

d^y
ratio of this to dx is ^2, the limit of which, in the

definition of page 22, is the differential coefficient of

-£. Similarly the limit of -^ is the differential co-
dx dx*

d*y

efficient of the limit of -~\ and so on.
dx*

TOTAL AND PARTIAL DIFFERENTIAL COEFFICIENTS.
IMPLICIT DIFFERENTIATION.

We now proceed to apply the principles laid down,
to some cases in which the variable enters into its
function in a less direct and more complicated man
ner.

THE DIFFERENTIAL AND INTEGRAL CALCULUS. 95

For example, let z be a given function of x and y,
and let y be another given function of x ; so that z
contains x both directly and indirectly ; the latter as
it contains y, which is a function of x. This will be
the case if z = x\ogy, where y = sin x. If we were to
substitute for y its value in terms of x, the value of z
would then be a function of x only ; in the instance
just given it would be xlogs'mx. But if it be not con
venient to combine the two equations at the beginning
of the process, let us first consider z as a function of
x andy, in which the two variables are independent.
In this case, if x and y respectively receive the incre
ments dx and dy, the whole increment of z, or d.z, (or
at least that part which gives the limit of the ratios)
is represented by

dz dz ,

— dx 4- -=- dy.
dx dy •

liy be now considered as a function of x, the conse
quence is that dy, instead of being independent of dx,
is a series of the form pdx -f- q (dxf -f etc., in which p
is the differential coefficient of y with respect to x.
Hence

dz , dz d.z dz dz

d. z = -3— dx -f -j- pdx or — - = \- -=- p,

dx dy * dx dx dy r

in which the difference between — ^— and -y- is this,

dx ax

that in the second, x is only considered as varying
where it is directly contained in z, or z is considered
in the form in which it first appeared, as a function of
x and y, where y is independent of x ; in the first, or

-^— , the total variation of z is denoted, that is, y is

(IOC

now considered as a function of x, by which means if
x become x -j- dx, z will receive a different increment

96 ELEMENTARY ILLUSTRATIONS OF

from that which it would have received, had y been
independent of x. In the instance above cited, where
z = x\ogy andy = s'mx, if the first equation be taken,
and x becomes x + dx, y remaining the same, z be
comes x logy -f- logydx or — is log_y. If y only varies,

dX

since (page 20) z will then become

dy
x\Q%y + x-±— etc.,

-j- is — . And -2- is cos* when y = s'mx (page 20)
dy y dx

dz dz dz dz dy . , x

HenCe -& + Tyt' °r TX + Ty TX 'S Iog' + 7 C°S*'

or log sin AH — : — cosx. This is — =^— , which might
sm x dx

have been obtained by a more complicated process, if
sinx had been substituted lory, before the operation
commenced. It is called the complete or total differen
tial coefficient with respect to x, the word total indi
cating that every way in which z contains x has been

used ; in opposition to -7-, which is called the partial

uX

differential coefficient, x having been considered as
varying only where it is directly contained in z.

Generally, the complete differential coefficient of z
with respect to x, will contain as many terms as there
are different ways in which z contains x. From look
ing at a complete differential coefficient, we may see
in what manner the function contained its variable.
Take, for example, the following,

d.z dz dz dy dz da dy dz da

dx dx dy dx da dy dx da dx*

Before proceeding to demonstrate this formula, we
will collect from itself the hypothesis from which it

THE DIFFERENTIAL AND INTEGRAL CALCULUS. Q7

must have arisen. When x is contained in z, we shall
say that z is a direct* function of x. When x is con
tained in y, and y is contained in z, we shall say that
z is an indirect function of x through y. It is evident
that an indirect function may be reduced to one which
is direct, by substituting for the quantities which con
tain x, their values in terms of x.

The first side of the equation —^— is shown by the

point to be a complete differential coefficient, and in
dicates that z is a function of x in several ways; either
directly, and indirectly through one quantity at least,
or indirectly through several. If z be a direct function
only, or indirectly through one quantity only, the

symbol — , without the point, would represent its

total differential coefficient with respect to x.
On the second side of the equation we see :

(1) -=- : which shows that z is a direct function of
dx

x, and is that part of the differential coefficient which
we should get by changing x into x-\-dx throughout
Zj not supposing any other quantity which enters into
z to contain x.

(2) — -f-: which shows that z is an indirect func-
J dy dx

tion of x through y. If x and y had been supposed to
vary independently of each other, the increment of 5,
(or those terms which give the limiting ratio of this

increment to any other,) would have been — dx-\-

dz . dx

-j- dy, in which, if dy had arisen from y being a func-

*It may be right to warn the student that this phraseology is new, to the
best of our knowledge. The nomenclature of the Differential Calculus has
by no means kept pace with its wants ; indeed the same may be said of alge
bra generally. [Written in 1832.— Ed.}

98 ELEMENTARY ILLUSTRATIONS OF

tion of x, dy would have been a series of the form
pdx -f- q (dx^f -J- etc., of which only the differential co
efficient/ would have appeared in the limit. Hence

dz . dz dz dy

— dy would have given -r-/, or — ~.
dy ' dy* dy dx

(3) — -- — : this arises from z containing a, which

J da dy dx

contains y, which contains x. If z had been differen
tiated with respect to a only, the increment would

have been represented by 7 da ; if da had arisen from

da

an increment of y, this would have been expressed by

dz da .

— - dy ; if y had arisen from an increment given to
da dy ' //////

x. this would have been expressed by — —r- dx.

J da dy dx

which, after dx has been struck out, is the part of the
differential coefficient answering to that increment.

(4) — — - : arising from a containing x directly,

and z therefore containing x indirectly through a.

Hence z is directly a function of x, y, and a, of
which y is a function of x, and a ot y and x.

If we suppose x, y and a to vary independently,
we have

d.z= ^ dx + ^ dy+ ^ da + etc. (pages 28-29).

But as a varies as a function of y and x,

da da

da— — dx-\- — dy.

dx dy '

If we substitute this instead of da, and divide by dx,
taking the limit of the ratios, we have the result first
given.

For example, let (1) z = x*ya*, (2)y=x*, and (3)
a = x6y. Taking the first equation only, and substi-

THE DIFFERENTIAL AND INTEGRAL CALCULUS. Q9

tuting x -f dx for x etc. , we find — = 2xya9, -=- = x2a?,

s7w {I V <*

and — = 3xf*ya2. From the second — - = 2x, and from

da da •* ja dx

the third — =3jc2y, and -3- =xs. Substituting these

dx d z y

in the value of — '— , we find
ax

d.z dz dz dy dz da dy dz da

dx dx dy dx da dy dx da dx

2xya* + *2a3 X 2x + Zx2ya? X & X 2* + 3^2j'a2 X

If for j> and # in the first equation we substitute their
values x1 and x*y, or x5, we have z = x19, the differen
tial coefficient of which 19^tr18. This is the same as
arises from the formula just obtained, after x2 and x5
have been substituted for y and a ; for this formula
then becomes

2 *w __ 6 *" 9 ^c18 or 19 x16.

In saying that 2 is a function of x and^, and that
y is a function of x, we have first supposed x to vary,
jy remaining the same. The student must not imagine
that y is then a function of x ; for if so, it would vary
when x varied. There are two parts of the total dif
ferential coefficient, arising from the direct and indi
rect manner in which z contains x. That these two
parts may be obtained separately, and that their sum
constitutes the complete differential coefficient, is the

theorem we have proved. The first part — is what

would have been obtained if y had not been a function
of x ; and on this supposition we therefore proceed to

find it. The other part -j- -j~- is the product (1) of

— , which would have resulted from a variation of y
dy%

only, not considered as a function of x; and (2) of

100 ELEMENTARY ILLUSTRATIONS OF

dy

-f-, the coefficient which arises from considering^ as a

ax

function of x. These partial suppositions, however
useful in obtaining the total differential coefficient,
cannot be separately admitted or used, except for this
purpose; since if y be a function of x, x and^ must
vary together.

If z be a function of x in various ways, the theorem
obtained may be stated as follows :

Find the differential coefficient belonging to each
of the ways in which z will contain x, as if it were the
only way ; the sum of these results (with their proper
signs) will be the total differential coefficient.

Thus, if z only contains x indirectly through y,

dz . dz dy ,, . . . , ^ . , , . ,

-j- is -r -f-. If * contains a, which contains b, which
dx dy dx

dz dz da db

contains x, -=- = -=- -77 -r.

dx da db dx

This theorem is useful in the differentiation of com
plicated functions; for example, let z = log(x* -}-a?).

If we makejy=#2-f a2, we have s = log^y, and-y-= — ;

while from the first equation -~ —2x. Hence — or

2x dx dx

s—
2-\-

If s = log log sin#, or the logarithm of the loga
rithm of sin#, let sin#=j> and logy = a; whence
z:=log0, and contains x, because a contains^, which
contains x. Hence

dz _ dz da dy ^
dx dady~dxy
but since z = loga,

~da

THE DIFFERENTIAL AND INTEGRAL CALCULUS. IOI

since a — logy,

** -~L

dy ~ y>

and since y = sin x,

dy
dx~
Hence

dz dz da dy 11 cos*

-. - = -=- -=- -f- = COSX= -: : : .

dx da dy dx ay logsm^sin*

We now put some rules in the form of applications
of this theorem, though they may be deduced more
simply.

APPLICATIONS OF THE PRECEDING THEOREM.

(1) Let z = ab, where a and b are functions of x.
The general formula, since z contains x indirectly
through a and b, is (in this case as well as in those
which follow,)

dz dz da dz db
dx da dx db dx'

We must leave — and — as we find them, until we
dx dx

know what functions a and b are of x\ but as we
know what function z is of a and 3, we substitute for

— and — . Since z = ab, if a becomes a-}- da, g be-
da db jz

comes ab -\- bda, whence —-=b. In this case, and part

of the following, the limiting ratio of the increments
is the same as that of the increments themselves.

Similarly -jr=a, whence from z = ab follows
ao

dz , da . db

IO2 ELEMENTARY ILLUSTRATIONS OF

(2) Letz = --. If a become a-\-da, z becomes

a-\-da a , da . da . 1 _. .
— - - or — -f — , and — is -7-. If b become b -}- db, z
b bo da b

a a adb . dz . a

becomes . , ., or - --- 7=- -f- etc. , whence — is — -75.
b b P db b1

Hence from z= — follows
b

i da dt>

dz 1 da a db ° -£< — a irx

(3) Let z-=a*. Here (a + da)* = a* + fa*-* da
-j-etc. (page 21), whence -^ = bab~l. Again, ab+db =
abadb = ab(\ -f log^t^ -f- etc.) whence — =«*
Therefore from z = a* follows

dz , . da . . . db

~- = bab~^ -j- + ab log a —.
dx dx dx

INVERSE FUNCTIONS.

If y be a function of x, such as y=cpx, we may,
by solution of the equation, determine x in terms of
y, or produce another equation of the form x = $y.
For example, when y = x2, x=y%. It is not neces
sary that we should be able to solve the equation
y=cpx in finite terms, that is, so as to give a value
of x without infinite series ; it is sufficient that x can
be so expressed that the value of x corresponding to
any value of y may be found as near as we please
from x — i/jy, in the same manner as the value of y
corresponding to any value of x is found from y = cpx.

The equations y = g>x, and x = ip>y, are connected,
being, in fact, the same relation in different forms ;
and if the value of y from the first be substituted in

THE DIFFERENTIAL AND INTEGRAL CALCULUS. 103

the second, the second becomes x = tp((px>), or as it is
more commonly written, ty(px. That is, the effect of
the operation or set of operations denoted by ip is de
stroyed by the effect of those denoted by cp ; as in the
instances (#2)^, (#8)£, &°sx, angle whose sine is (sin#),
etc., each of which is equal to x.

By differentiating the first equation y=<px, we ob

tain -=- = qjx, and from the second — — tb'y. But
dx dy

whatever values of x and y together satisfy the first
equation, satisfy the second also ; hence, if when x be
comes x -\- dx in the first, y becomes y -f- dy ; the same
y-^dy substituted for y in the second, will give the

same x-\- dx. Hence -- as deduced from the second,

and -J- as deduced from the first, are reciprocals for
doc

every value of dx. The limit of one is therefore the
reciprocal of the limit of the other ; the student may

easily prove that if a is always equal to —, and if a
continually approaches to the limit a, while b at the

same time approaches the limit y#, a is equal to -^ .

dx P.

But — or if>'y, deduced from x — rpy, is expressed in

d v
terms of y, while -j- or cp'x, deduced from y = (px is

ctoc

expressed in terms of x. Therefore ip'y and cp'x are
reciprocals for all such values of x and y as satisfy
either of the two first equations.

For example let y = £*, from which x = logy. From

the first (page 20) -J- = £* ; from the second — = — ;
and it is evident that s* and — are reciprocals, when

ever y = £*.

If we differentiate the above equations twice, we get

104 ELEMENTARY ILLUSTRATIONS OF

-~~ = q>"x, and -j-^ =tb"x. There is no very obvious

dx1 dy*

analogy between -~ and —^ ; indeed no such appears
dx dy

from the method in which these coefficients were first
formed. Turn to the table in page 90, and substitute
d for A throughout, to indicate that the increments
may be taken as small as we please. We there sub
stitute in (px what we will call a set of equidistant val
ues of x, or values in arithmetical progression, viz.,
x, x-\-dx, x -\-2dx, etc. The resulting values of y,
or y, y\, etc., are not equidistant, except in one func
tion only, when y=ax-\-b, where a and b are con
stant. Therefore dy, dy\, etc., are not equal ; whence
arises the next column of second differences, or d*y,
d*yi, etc. The limiting ratio of d*y to (dfcr)8, expressed

d2y
by TTJI is the second differential coefficient of y with

respect to x. If from y — cpx we deduce x = $y, and
take a set of equidistant values of y, viz., y, y-\-dy,
y-^-2dy, etc., to which the corresponding values of x
are x, x\, x%, etc., a similar table may be formed,
which will give dx, dxi, etc., d2x, d*x\, etc., and the

d2x
limit of the ratio of d*x to (/#>)8 or —^- is the second

differential coefficient of x with respect to y. These
are entirely different suppositions, dx being given in
the first table, and dy varying ; while in the second dy
is given and dx varies. We may show how to deduce
one from the other as follows :

When, as before, y=cpx and x = fiy, we have

dy _ , 1 1
dx-V* "ft — p'

if ty'y be called /. Calling this u, and considering it

THE DIFFERENTIAL AND INTEGRAL CALCULUS. IO5

as a function of x from containing /, which contains
y, which contains x, we have

du dp dy

dp dy dx

for its differential coefficient with respect to x. But
since

_ ^

therefore

du= _1.

dp~ f

since p — rp'y, therefore

and if>"y is the differential coefficient of i/>'y, and is

or or

(*)!
\W

Hence the differential coefficient of u or -^-, with re-

d*y . dx

spect to x, which is - is also

dx dy* dx \dx) dy* '

If y = e*, whence # = logjy, we have--=€* and

d*y _ dx 1 , //8^ 1 _, .
•n = €*• But — = — and -ry = ^. Therefore

8 ^^. _/ 1\ &* $*

or — s- or -«-»

y

which is fi*, the value just found for -

IO6 ELEMENTARY ILLUSTRATIONS OF

d^y
In the same way -~ might be expressed in terms

x d*x , d*x
-— , -}-=-, and -T-J:-
dy dy* dy*

, dx ,

of -— , -}-=-, and -T-J:-; and so on.
* *

IMPLICIT FUNCTIONS.

The variable which appears in the denominator of
the differential coefficients is called the independent
variable. In any function, one quantity at least is
changed at pleasure ; and the changes of the rest,
with the limiting ratio of the changes, follow from the
form of the function. The number of independent
variables depends upon the number of quantities
which enter into the equations, and upon the number
of equations which connect them. If there be only
one equation, all the variables except one are inde
pendent, or may be changed at pleasure, without ceas
ing to satisfy the equation ; for in such a case the
common rules of algebra tell us, that as long as one
quantity is left to be determined from the rest, it can
be determined by one equation ; that is, the values of
all but one are at our pleasure, it being still in our
power to satisfy one equation, by giving a proper
value to the remaining one. Similarly, if there be
two equations, all variables except two are independ
ent, and so on. If there be two equations with two
unknown quantities only, there are no variables ; for
by algebra, a finite number of values, and a finite
number only, can satisfy these equations ; whereas it
is the nature of a variable to receive any value, or at
least any value which will not give impossible values
for other variables. If then there be m equations con
taining n variables, (n must be greater than m), we
have n — m independent variables, to each of which

THE DIFFERENTIAL AND INTEGRAL CALCULUS. IOJ

we may give what values we please, and by the equa
tions, deduce the values of the rest. We have thus
various sets of differential coefficients, arising out of
the various choices which we may make of independ
ent variables.

If, for example, a, b, x, y, and z, being variables,
we have

<p(a, b, x,y, *,) = °»

if>(a, b, x,y, *,) = 0,

X(a, t>, x, y, *,) = 0,

we have two independent variables, which may be
either x and y, x and z, a and b, or any other com
bination. If we choose x and^, we should determine
a, b, and z in terms of x and y from the three equa
tions ; in which case we can obtain

da da db
Jx~' ~fy> Jx~' e

Wheny is a function of x, as in y— <px, it is called
an explicit function of x. This equation tells us not
only that y is a function of x, but also what function
it is. The value of x being given, nothing more is
necessary to determine the corresponding value of y,
than the substitution of the value of x in the several
terms of (px.

But it may happen that though y is a function of
x, the relation between them is contained in a form
from which y must be deduced by the solution of an
equation. For example, in #2 — xy-\-y'2==at when #
is known, y must be determined by the solution of an
equation of the second degree. Here, though we know
that y must be a function of x, we do not know, with
out further investigation, what function it is. In this
case y is said to be implicitly a function of x} or an im-

108 ELEMENTARY ILLUSTRATIONS OF

plicit function. By bringing all the terms on one side
of the equation, we may always reduce it to the form
cp(x, y) = 0. Thus, in the case just cited, we have

We now want to deduce the differential coefficient

dy

-j- from an equation of the form q>(x, y) = 0. If we

(t£

take the equation u = cp(x, jv), in which when x and y
become x -|- dx and y -f dy, u becomes u -{- du, we have,
by our former principles,

du = u'dx -\- utdy -}- etc. , (page 82),

in which »' and ut can be directly obtained from the
equation, as in page 82. Here x and y are independ
ent, as also dx and dy ; whatever values are given to
them, it is sufficient that u and du satisfy the two last
equations. But if x and y must be always so taken
that u may =0, (which is implied in the equation
<p(x, y) = 0, ) we have # = 0, and du = §\ and this,
whatever may be the values of dx and dy. Hence dx
and dy are connected by the equation

0 = u'dx -f utdy -f etc. ,

and their limiting ratio must be obtained by the equa
tion

y and x are no longer independent ; for, one of them
being given, the other must be so taken that the equa
tion (p(x, _y) = 0 maybe satisfied. The quantities u'

and u we have denoted by -3— and -=-, so that
' dx dy

_
dx

THE DIFFERENTIAL AND INTEGRAL CALCULUS. IOQ

We must again call attention to the different mean
ings of the same symbol du in the numerator and de
nominator of the last fraction. Had du, dx, and dy
been common algebraical quantities, the first meaning
the same thing throughout, the last equation would
not have been true until the negative sign had been
removed. We will give an instance in which du shall
mean the same thing in both.

Let u = <p(x), and let u = t/y, in which two equa
tions is implied a third <px = ipy, and y is a function
of x. Here, x being given, u is known from the first
equation ; and u being known, y is known from the
second. Again, x and dx being given, du, which is
<p(x-\-dx} — <px is known, and being substituted in
the result of the second equation, we have du —
^(y ~\~ dy} — tyy, which dy must be so taken as to
satisfy. From the first equation we deduce du =
qj 'x dx -j- etc. and from the second du — il>'ydy-\- etc. ,
whence

qjx dx -f- etc. = ip'y dy -j- etc. ;

the etc. only containing terms which disappear in find
ing the limiting ratios. Hence,

Q — ^—'ZL t*\

dx~~ Wy " *±

dy

a result in accordance with common algebra.

But the equation (1) was obtained from u = (p(x,y\
on the supposition that x and y were always so taken
that u should =0, while (2) was obtained from « =
<p(x) and u = Sy, in which no new supposition can be
made ; since one more equation between u, x, and y
would give three equations connecting these three
quantities, in which case they would cease to be vari
able (page 106).

110 ELEMENTARY ILLUSTRATIONS OF

As an example of (1) let xy — * = 1, or xy — x—
1 = 0. From u = xy — x — 1 we deduce (page 81)

du . du

— —y — if =x; whence, by equation (1),

dx~ x

By solution of xy — x = l, we fmdy = l-\ -- , and

dy 1

Hence ~ (meaning the limit) is -- -v which will also

be the result of (3) if 1 H -- be substituted for.y.

FLUXIONS, AND THE IDEA OF TIME.

To follow this subject farther would lead us be
yond our limits ; we will therefore proceed to some
observations on the differential coefficient, which, at
this stage of his progress, may be of use to the stu
dent, who should never take it for granted that be
cause he has made some progress in a science, he un
derstands the first principles, which are often, if not
always, the last to be learned well. If the mind were
so constituted as to receive with facility any perfectly
new idea, as soon as the same was legitimately ap
plied in mathematical demonstration, it would doubt
less be an advantage not to have any notion upon a
mathematical subject, previous to the time when it is
to become a subject of consideration after a strictly
mathematical method.

This not being the case, it is a cause of embarrass
ment to the student, that he is introduced at once to a
definition so refined as that of the limiting ratio which

* See page 26.

THE DIFFERENTIAL AND INTEGRAL CALCULUS. Ill

the increment of a function bears to the increment of
its variable. Of this he has not had that previous ex
perience, which is the case in regard to the words
force, velocity, or length. Nevertheless, he can easily
conceive a mathematical quantity in a state of con
tinuous increase or decrease, such as the distance be
tween two points, one of 'which is in motion. The
number which represents this line (reference being
made to a given linear unit) is in a corresponding
state of increase or decrease, and so is every function
of this number, or every algebraical expression in the
formation of which it is required. And the nature of
the change which takes place in the function, that is,
whether the function will increase or decrease when
the variable increases ; whether that increase or de
crease corresponding to a given change in the vari
able will be smaller or greater, etc., depends on the
manner in which the variable enters as a component
part of its function.

Here we want a new word, which has not been in
vented for the world at large, since none but mathe
maticians consider the subject ; which word, if the
change considered were change of place, depending
upon change of time, would be velocity. Newton
adopted this word, and the corresponding idea, ex
pressing many numbers in succession, instead of at
once, by supposing a point to generate a straight line
by its motion, which line would at different instants
contain any different numbers of linear units.

To this it was objected that the idea of time is in
troduced, which is foreign to the subject. We may
answer that the notion of time is only necessary, in
asmuch as we are not able to consider more than one
thing at a time. Imagine the diameter of a circle di-

I 1 2 ELEMENTARY ILLUSTRATIONS OF

vided into a million of equal parts, from each of which
a perpendicular is drawn meeting the circle. A mind
which could at a view take in every one of these lines,
and compare the differences between every two con
tiguous perpendiculars with one another, could, by
subdividing the diameter still further, prove those
propositions which arise from supposing a point to
move uniformly along the diameter, carrying with it
a perpendicular which lengthens or shortens itself so
as always to have one extremity on the circle. But
we, who cannot consider all these perpendiculars at
once, are obliged to take one after another. If one
perpendicular only were considered, and the differen
tial coefficient of that perpendicular deduced, we might
certainly appear to avoid the idea of time ; but if all
the states of a function are to be considered, corre
sponding to the different states of its variable, we
have no alternative, with our bounded faculties, but
to consider them in succession ; and succession, dis
guise it as we may, is the identical idea of time intro
duced in Newton's Method of Fluxions.

THE DIFFERENTIAL COEFFICIENT CONSIDERED WITH RE
SPECT TO ITS MAGNITUDE.

The differential coefficient corresponding to a par
ticular value of the variable, is, if we may use the
phrase, the index of the change which the function
would receive if the value of the variable were in
creased. Every value of the variable, gives not only
a different value to the function, but a different quan
tity of increase or decrease in passing to what we may
call contiguous values, obtained by a given increase of
the variable.

If, for example, we take the common logarithm of

THE DIFFERENTIAL AND INTEGRAL CALCULUS. 113

x, and let x be 100, we have common log 100 = 2. If
x be increased by 2, this gives common log 102 =
2-0086002, the ratio of the increment of the function
to that of the variable being that of • 0086002 to 2, or
•0043001. In passing from 1000 to 1003, we have the
logarithms 3 and 3-0013009, the above-mentioned ra
tio being -0004336, little more than a. tenth of the
former. We do not take the increments themselves,
but the proportion they bear to the changes in the
variable which gave rise to them ; so in estimating
the rate of motion of two points, we either consider
lengths described in the same time, or if that cannot
be done, we judge, not by the lengths described in
different times, but by the proportion of those lengths
to the times, or the proportions of the units which
express them.

The above rough process, though from it some
might draw the conclusion that the logarithm of x is
increasing faster when # = 100 than when # = 1000,
is defective; for, in passing from 100 to 102, the
change of the logarithm is not a sufficient index of the
change which is taking place when x is 100 ; since,
for any thing we can be supposed to know to the con
trary, the logarithm might be decreasing when # =
100, and might afterwards begin to increase between
#==100 and # = 102, so as, on the whole, to cause
the increase above mentioned. The same objection
would remain good, however small the increment
might be, which we suppose # to have. If, for ex
ample, we suppose # to change from # = 100 to # =
100-00001, which increases the logarithm from 2 to
2-00000004343, we cannot yet say but that the log
arithm may be decreasing when # = 100, and may be
gin to increase between # = 100 and # = 100 -00001.

114 ELEMENTARY ILLUSTRATIONS OF

In the same way, if a point is moving, so that at
the end of 1 second it is at 3 feet from a fixed point,
and at the end of 2 seconds it is at 5 feet from the
fixed point, we cannot say which way it is moving at
the end of one second. On the whole, it increases its
distance from the fixed point in the second second ;
but it is possible that at the end of the first second it
may be moving back towards the fixed point, and may
turn the contrary way during the second second. And
the same argument holds, if we attempt to ascertain
the way in which the point is moving by supposing
any finite portion to elapse after the first second. But
if on adding any interval, however small, to the first
second, the moving point does, during that interval,
increase its distance from the fixed point, we can then
certainly say that at the end of the first second the
point is moving from the fixed point.

On the same principle, we cannot say whether the
logarithm of x is increasing or decreasing when x in
creases and becomes 100, unless we can be sure that
any increment, however small, added to x, will in
crease the logarithm. Neither does the ratio of the
increment of the function to the increment of its vari
able furnish any distinct idea of the change which is
taking place when the variable has attained or is pass
ing through a given value. For example, when x
passes from 100 to 102, the difference between log 102
and log 100 is the united effect of all the changes
which have taken place between # = 100 and x =
100^; # = 100^ and # = 100^, and so on. Again,
the change which takes place between # = 100 and
# = 100^ may be further compounded of those which
take place between x = 100 and x = lOOyJ^ ; x =
and # = 100^, and so on. The objection

THE DIFFERENTIAL AND INTEGRAL CALCULUS. 115

becomes of less force as the increment diminishes,
but always exists unless we take the limit of the ratio
of the increments, instead of that ratio.

How well this answers to our previously formed
ideas on such subjects as direction, velocity, and
force, has already appeared.

THE INTEGRAL CALCULUS.

We now proceed to the Integral Calculus, which
is the inverse of the Differential Calculus, as will after
wards appear.

We have already shown, that when two functions
increase or decrease without limit, their ratio may either
increase or decrease without limit, or may tend to
some finite limit. Which of these will be the case de
pends upon the manner in which the functions are re
lated to their variable and to one another.

This same proposition may be put in another form,
as follows : If there be two functions, the first of which
decreases without limit, on the same supposition which
makes the second increase without limit, the product
of the two may either remain finite, and never exceed
a certain finite limit ; or it may increase without limit,
or diminish without limit.

For example, take cos 0 and tan 0. As the angle 6
approaches a right angle, cos0 diminishes without
limit ; it is nothing when 6 is a right angle ; and any
fraction being named, 0 can be taken so near to a
right angle that cos0 shall be smaller. Again, as 6
approaches to a right angle, tan0 increases without
limit ; it is called infinite when 0 is a right angle, by
which we mean that, let any number be named, how
ever great, 0 can be taken so near a right angle that
tan0 shall be greater. Nevertheless the product cos0X

Il6 ELEMENTARY ILLUSTRATIONS OF

tan 0, of which the first factor diminishes without limit,
while the second increases without limit, is always
finite, and tends towards the limit 1 ; for cos#X tan0
is always sin0, which last approaches to 1 as 0 ap
proaches to a right angle, and is 1 when 0 is a right
angle.

Generally, if A diminishes without limit at the
same time as B increases without limit, the product
AB may, and often will, tend towards a finite limit.
This product AB is the representative of A divided by

^g- or the ratio of A to -=-. If B increases without
-tJ -i r>

limit, •=- decreases without limit ; and as A also de-

1
creases without limit, the ratio of A to -^ may have a

finite limit. But it may also diminish without limit ;
as in the instance of cos20 X tan0, when 0 approaches
to a right angle. Here cos20 diminishes without limit,
and tan0 increases without limit; but cos20X*an0
being cos0Xsin0, or a diminishing magnitude multi
plied by one which remains finite, diminishes without
limit. Or it may increase without limit, as in the case
of cos0 X tan20, which is also sin 0 X tan0 ; which last
has one factor finite, and the other increasing without
limit. We shall soon see an instance of this.

If we take any numbers, such as 1 and 2, it is evi
dent that between the two we may interpose any num
ber of fractions, however great, either in arithmetical
progression, or according to any other law. Suppose,
for example, we wish to interpose 9 fractions in arith
metical progression between 1 and 2. These are 1^,
1-jj^, etc., up to 1-^j- ; and, generally, if m fractions in
arithmetical progression be interposed between a and
a -\- h, the complete series is

THE DIFFERENTIAL AND INTEGRAL HALCULUS.

mh
up to a -\ r-

The sum of these can evidently be made as great as
we please, since no one is less than the given quan
tity a, and the number is as great as we please. Again,
if we take <px, any function of x, and let the values
just written be successively substituted for x, we shall
have the series

................... up to <p(a + R) (2);

the sum of which may, in many cases, also be made
as great as we please by sufficiently increasing the
number of fractions interposed, that is, by sufficiently
increasing m. But though the two sums increase with
out limit when m increases without limit, it does not
therefore follow that their ratio increases without
limit ; indeed we can show that this cannot be the
case when all the separate terms of (2) remain finite.
For let A be greater than any term in (2), whence,
as there are (w-j-2) terms, (w + 2)A is greater than
their sum. Again, every term of (1), except the first,
being greater than a, and the terms being m-\-2 in
number, (m -|- 2)0 is less than the sum of the terms in
(1). Consequently,

(m -L- 2) A . . sum of terms in (2)

T - — ~ — is greater than the ratio - -f— - : — ^,
(m -f 2)# sum of terms in (1)

since its numerator is greater than the last numerator,
and its denominator less than the last denominator.
But

Il8 ELEMENTARY ILLUSTRATIONS OF

A

(m + 2)a ~a'

which is independent of m, and is a finite quantity.
Hence the ratio of the sums of the terms is always
finite, whatever may be the number of terms, at least
unless the terms in (2) increase without limit.

As the number of interposed values increases, the
interval or difference between them diminishes ; if,
therefore, we multiply this difference by the sum of
the values, or form

— T \(pa -j- <p{ a -\ -- — - 4-
+\\J ^\ an n

m-\-

we have a product, one term of which diminishes, and
the other increases, when m is increased. The pro
duct may therefore remain finite, or never pass a cer
tain limit, when m is increased without limit, and \\e
shall show that this is the case.

As an example, let the given function of x be ^2,
and let the intermediate values of x be interposed be
tween x = a and x = a4-h. Let v= =-, whence

m + l

the above-mentioned product is

................ -f tf4-(^+l>

(m + 2) va* -f- 2av* { 1 + 2 + 3 + . . + (m + 1) }

of which, l + 2 + ....-K« + l)==Kai+l)(»»-f2)
and (page 73), !2-f 22 + . . . . -f (» + 1)« approaches
without limit to a ratio of equality with $(;«-}- 1)8,
when m is increased without limit. Hence this last
sum may be put under the form |(>-f 1)8(1 + a),

THE DIFFERENTIAL AND INTEGRAL CALCULUS. IIQ

where a diminishes without limit when m is increased
without limit. Making these substitutions, and put

ting for v its value — r-=t the above expression be-
m -j- 1

comes

in which — ~— has the limit 1 when m increases with-

m-\-I

out limit, and 1-f- a has also the limit 1, since, in that
case, a diminishes without limit. Therefore the limit
of the last expression is

ha* + tfa+- or

This result may be stated as follows : If the vari
able x, setting out from a value a, becomes succes
sively a-^-dx, a -\-2dx, etc., until the total increment
is h, the smaller dx is taken, the more nearly will the
sum of all the values of x^dx, or a?dx -\-(a-\- dx^f dx -f-
(a-\-2dx)2dx-\-etc.t be equal to

and to this the aforesaid sum may be brought within
any given degree of nearness, by taking dx sufficiently
small.

This result is called the integral of x*dx, between
the limits a and a -\- h, and is written fx^dx, when it
is not necessary to specify the limits, andy],a hx*dx,
or* fx^dx?**, or fx2dx%^+h in the contrary case. We

*This notation f3?dx£+h appears to me to avoid the objections which
may be raised against J'^i^dx as contrary to analogy, which would require
that /" jrdxr should stand for the second integral of x^dx. It will be found
convenient in such integrals &sfzdx*dy&x. There is as yet no general agree
ment on this point of notation.— Zte Morgan, 1832.

I2O ELEMENTARY ILLUSTRATIONS OF

now proceed to show the connexion of this process
with the principles of the Differential Calculus.

CONNEXION OF THE INTEGRAL WITH THE DIFFERENTIAL

CALCULUS.

Let x have the successive values a, a 4 dx, a -f- Zdx,
etc. , . . . . up to a 4- mdx, or a -f ht h being a given
quantity, and dx the /0th part of h, so that as m is in
creased without limit, dx is diminished without limit.
Develop the successive values (px, or cpa, cp(a -f- dx} —
(page 21),

4, a

* + etc.

>"a + ^'"« -f etc.

+ etc.

+ etc.

If we multiply each development by dx and add the
results, we have a series made up of the following
terms, arising from the different columns,

<pa X mdx

cp'a X(l +2 +3 +...+*) (</*)»

etc.

and, as in the last example, we may represent (page

73),

1 4.2 4-3 -f ..... +m by£w2(14-tf)
12 _|_ 22 4- 32 + ..... 4- m* . . \m* (1 4- ft)

- »4l- etc-

THE DIFFERENTIAL AND INTEGRAL CALCULUS. 121

where a, /?, y, etc., diminish without limit, when m
is increased without limit. If we substitute these val

ues, and also put — instead of dxt we have, for the

M

sum of the terms,

<f/a ~ (1 + «) + <p"a ^ (1 + fS)

2

which, when m is increased without limit, in conse
quence of which a, ft, etc., diminish without limit,
continually approaches to

h* h* h*

(pah + (p'a^-\- cp"a ^ + (p'"a ^-^ + etc.

which is the limit arising from supposing x to increase
from a through a-\-dx, a -\-2dx, etc., up to a-\-h,
multiplying every value of <px so obtained by dxt sum
ming the results, and decreasing dx without limit.

This is the integral of cpxdx from x = a to x =
a-\-h. It is evident that this series bears a great re
semblance to the development in page 21, deprived
of its first term. Let us suppose that fya is the func
tion of which (pa is the differential coefficient, that is,
that fy'a=<pa. These two functions being the same,
their differential coefficients will be the same, that is,
il)"a = (p'a. Similarly if>'"a = (p"a) and so on. Sub
stituting these, the above series becomes

*/>'<*& + fa £ + fa ^~ + $»a ^-^ + etc.

which is (page 21) the same as ip(a-\-Ji) — i/>a. That
is, the integral of cpxdx between the limits a and a-\-h,
is il>(a-\-?f) — fa, where $x is the function, which,

122 ELEMENTARY ILLUSTRATIONS OF

when differentiated, gives (px. For a -j- h we may
write £, so that ?/•£ — ipa is the integral of <pxdx from
x = a to x = b. Or we may make the second limit in
definite by writing x instead of b, which gives ipx — ipa,
which is said to be the integral of (pxdx, beginning
when x — a, the summation being supposed to be con
tinued from x = a until x has the value which it may
be convenient to give it.

NATURE OF INTEGRATION.

Hence results a new branch of the inquiry, the re
verse of the Differential Calculus, the object of which
is, not to find the differential coefficient, having given
the function, but to find the function, having given
the differential coefficient. This is called the Integral
Calculus.

From the definition given, it is obvious that the
value of an integral is not to be determined, unless
we know the values of x corresponding to the begin
ning and end of the summation, whose limit furnishes
the integral. We might, instead of defining the in
tegral in the manner above stated, have made the
word mean merely the converse of the differential co
efficient ; thus, if (px be the differential coefficient of
ipx, ipx might have been called the integral of (pxdx.
We should then have had to show that the integral,
thus defined, is equivalent to the limit of the summa
tion already explained. We have preferred bringing
the former method before the student first, as it is
most analogous to the manner in which he will deduce
integrals in questions of geometry or mechanics.

With the last-mentioned definition, it is also obvi
ous that every function has an unlimited number of
integrals. For whatever differential coefficient fyx

THE DIFFERENTIAL AND INTEGRAL CALCULUS. 12$

gives, C -}- tyx will give the same, if C be a constant,
that is, not varying when x varies. In this case, if x
become x -f- h, C -}- i/>x becomes C -j- if>x -f- fy'x h -f etc. ,
from which the subtraction of the original form C -f i/>x
gives ffi'x ft -\-etc.; whence, by the process in page 23,
i/j'x is the differential coefficient of C + i/?x as well as
of i/>x. As many values, therefore, positive or nega
tive, as can be given to C, so many different integrals
can be found for fy'x \ and these answer to the various
limits between which the summation in our original
definition may be made. To make this problem def
inite, not only ip'x the function to be integrated, must
be given, but also that value of x from which the sum
mation is to begin. If this be a, the integral of ip'x is,
as before determined, ipx — ipa, and C = — ipa. We
may afterwards end at any value of x which we please.
If x = a, tpx — i/}a = Q, as is evident also from the
formation of the integral. We may thus, having given
an integral in terms of x, find the value at which it
began, by equating the integral to zero, and finding
the value of x. Thus, since x2, when differentiated,
gives 2x, x2 is the integral of 2x, beginning at x = Q j
and x2 — 4 is the integral beginning at x = 2.

In the language of Leibnitz, an integral would be
the sum of an infinite number of infinitely small quan
tities, which are the differentials or infinitely small in
crements of a function. Thus, a circle being, accord
ing to him, a rectilinear polygon of an infinite number
of infinitely small sides, the sum of these would be
the circumference of the figure. As before (pages
13-14, 38 et seq., 48 et seq.) we proceed to interpret
this inaccuracy of language. If, in a circle, we suc
cessively describe regular polygons of 3, 4, 5, 6, etc.,
sides, we may, by this means, at last attain to a poly-

124 ELEMENTARY ILLUSTRATIONS OF

gon whose side shall differ from the arc of which it is
the chord, by as small a fraction, either of the chord
or arc, as we please (pages 7-11). That is, A being
the arc, C the chord, and D their difference, there is
no fraction so small that D cannot be made a smaller
part of C. Hence, if m be the number of sides of the
polygon, mC -f- mD or mA is the real circumference ;
and since mD is the same part of mC which D is of C,
niD may be made as small a part of mC as we please ;
so that mC, or the sum of all the sides of the polygon,
can be made as nearly equal to the circumference as
we please.

As in other cases, the expressions of Leibnitz are
the most convenient and the shortest, for all who can
immediately put a rational construction upon them ;
this, and the fact that, good or bad, they have been,
and are, used in the works of Lagrange, Laplace,
Euler, and many others, which the student who really
desires to know the present state of physical science,
cannot dispense with, must be our excuse for contin
ually bringing before him modes of speech, which,
taken quite literally, are absurd.

DETERMINATION OF CURVILINEAR AREAS. THE PARABOLA.

We will now suppose such a part of a curve, each
ordinate of which is a given function of the corre
sponding abscissa, as lies between two given ordi-
nates ; for example, MPP'M'. Divide the line MM'
into a number of equal parts, which we may suppose
as great as we please, and construct Figure 10. Let
O be the origin of co-ordinates, and let OM, the value
of x, at which we begin, be a ; and OM', the value
at which we end, be b. Though we have only divided

THE DIFFERENTIAL AND INTEGRAL CALCULUS. 125

MM' into four equal parts in the figure, the reasoning
to which we proceed would apply equally, had we di
vided it into four million of parts. The sum of the
parallelograms Mr, mr, m'r", and m"R, is less than
the area MPP'M', the value of which it is our object
to investigate, by the sum of the curvilinear triangles
Prp, pr'p', //'/', and /'RP'. The sum of these tri
angles is less than the sum of the parallelograms Qr,
qr', q'r", and ^"R ; but these parallelograms are to-

p'

R

gether equal to the parallelogram q"w, as appears by
inspection of the figure, since the base of each of the
above-mentioned parallelograms is equal to m"M', or
/'P', and the altitude P'w is equal to the sum of the
altitudes of the same parallelograms. Hence the sum
of the parallelograms Mr, mr m'r", and m"R, differs
from the curvilinear area MPP'M' by less than the
parallelogram q"w. But this last parallelogram may
be made as small as we please by sufficiently increas
ing the number of parts into which MM' is divided ;

126 ELEMENTARY ILLUSTRATIONS OF

for since one side of it, P'w, is always less than P'M',
and the other side P'/', or m"M', is as small a part as
we please of MM' the number of square units in g"w,
is the product of the number of linear units in P'w
and PV", the first of which numbers being finite, and
the second as small as we please, the product is
as small as we please. Hence the curvilinear area
MPP'M' is the limit towards which we continually
approach, but which we never reach, by dividing MM'
into a greater and greater number of equal parts, and
adding the parallelograms Mr, mr', etc., so obtained.
If each of the equal parts into which MM' is divided
be called dx, we have OM = a, Om = a-\-dx, Omr =
a -\-2Jx, etc. And MP, mp, m'/, etc., are the values
of the function which expresses the ordinates, corre
sponding to a, a-\-dx, a -\-2dx, etc., and may there
fore be represented by (pa, (p(a-{-dx), cp(a -\-Zdx),
etc. These are the altitudes of a set of parallelo
grams, the base of each of which is dx\ hence the
sum of their area is

(pa dx -f- (p(a -\- dx) dx -j- cp(a -J- 2dx) dx -f- etc. ,
and the limit of this, to which we approach by dimin
ishing dx, is the area required.

This limit is what we have defined to be the in
tegral of <pxdx from x = a to x = fr; or if ipx be the
function, which, when differentiated, gives cpx, it is
fyb — i/>a. Hence, y being the ordinate, the area in
cluded between the axis of x, any two values of y, and
the portion of the curve they cut off, is fydx, begin
ning at the one ordinate and ending at the other.

Suppose that the curve is a part of a parabola
of which O is the vertex, and whose equation* is

*If the student has not any acquaintance with the conic sections, he must
nevertheless be aware that there is some curve whose abscissa and ordinate

THE DIFFERENTIAL AND INTEGRAL CALCULUS. 127

therefore y* =px where p is the double ordinate which
passes through the focus. Here y=p*x%, and we
must find the integral of p^x\dx, or the function
whose differential coefficient is p%x*, p% being a con
stant. If we take the function cxnt c being independ
ent of x, and substitute x-\-h for x, we have for the
development cxn -\- cnxn~l h + etc. Hence the differ
ential coefficient of cxn is cnxn~l ; and as c and « may
be any numbers or fractions we please, we may take
them such that en shall =/i and n — 1— J, in which
case n = % and c=^p*. Therefore the differential co
efficient of %p%x% is p%x^, and conversely, the integral
oip^x^dx is f/^rl

The area MPP'M' of the parabola is therefore
^p\b\ — -|/W. If we begin the integral at the vertex
O, in which case a = 0, we have for the area OM'P',
where £ = OM'. This is f/W X bt which, since
M'P' is f P'M' X OM', or two-thirds of the rect
angle* contained by OM' and M'P'.

METHOD OF INDIVISIBLES.

We may mention, in illustration of the preceding
problem, a method of establishing the principles of
the Integral Calculus, which generally goes by the
name of the Method of Indivisibles. A line is consid
ered as the sum of an infinite number of points, a
surface of an infinite number of lines, and a solid of
an infinite number of surfaces. One line twice as long
as another would be said to contain twice as many

are connected by the equation y% =Ar. This, to him, must be the definition
of parabola; by which word he must understand, a curve whose equation is
yi =Ar.

*This proposition is famous as having been discovered by Archimedes
at a time when such a step was one of no small magnitude.

128 ELEMENTARY ILLUSTRATIONS OF

points, though the number of points in each is unlim
ited. To this there are two objections. First, the
word infinite, in this absolute sense, really has no
meaning, since it will be admitted that the mind has
no conception of a number greater than any number.
The word infinite* can only be justifiably used as an
abbreviation of a distinct and intelligible proposition -,

for example, when we say that a -\ is equal to a

when x is infinite, we only mean that as x is increased,

a-\ becomes nearer to a, and may be made as near

oc

to it as we please, if x may be as great as we please.
The second objection is, that the notion of a line
being the sum of a number of points is not true, nor
does it approach nearer the truth as we increase the
number of points. If twenty points be taken on a
straight line, the sum of the twenty-one lines which
lie between point and point is equal to the whole line :
which cannot be if the points by themselves constitute
any part of the line, however small. Nor will the sum
of the points be a part of the line, if twenty thousand
be taken instead of twenty. There is then, in this
method, neither the rigor of geometry, nor that ap
proach to truth, which, in the method of Leibnitz,
may be carried to any extent we please, short of abso
lute correctness. We would therefore recommend to
the student not to regard any proposition derived
from this method as true on that account ; for false
hoods, as well as truths, may be deduced from it. In
deed, the primary notion, that the number of points
in a line is proportional to its length, is manifestly in
correct. Suppose (Fig. 6, page 48) that the point Q

*See Study of Mathematics (Chicago : The Open Court Publishing Co ),
page 123 et seq.

THE DIFFERENTIAL AND INTEGRAL CALCULUS. I2Q

moves from A to P. It is evident that in whatever
number of points OQ cuts AP, it cuts MP in the same
number. But PM and PA are not equal. A defender
of the system of indivisibles, if there were such a per
son, would say something equivalent to supposing
that the points on the two lines are of different sizes,
which would, in fact, be an abandonment of the
method, and an adoption of the idea of Leibnitz, us
ing the word point to stand for the infinitely small
line.

This notion of indivisibles, or at least a way of
speaking which looks like it, prevails in many works
on mechanics. Though a point is not treated as a
length, or as any part of space whatever, it is consid
ered as having weight ; and two points are spoken of
as having different weights. The same is said of a
line and a surface, neither of which can correctly be
supposed to possess weight. If a solid be of the same
density throughout, that is, if the weight of a cubic
inch of it be the same from whatever part it is cut, it
is plain that the weight may be found by finding the
number of cubic inches in the whole, and multiplying
this number by the weight of one cubic inch. But if
the weight of every two cubic inches is different, we
can only find the weight of the whole by the integral
calculus.

Let AB (Fig. 11) be a line possessing weight, or
a very thin parallelepiped of matter, which is such,
that if we were to divide it into any number of equal
parts, as in the figure, the weight of the several parts
would be different. We suppose the weight to vary
continuously, that is, if two contiguous parts of equal
length be taken, as pq and qr, the ratio of the weights

130 ELEMENTARY ILLUSTRATIONS OF

of these two parts may, by taking them sufficiently
small, be as near to equality as we please.

The density of a body is a mathematical term, which
may be explained as follows : A cubic inch of gold
weighs more than a cubic inch of water ; hence gold
is denser than water. If the first weighs 19 times as
much as the second, gold is said to be 19 times more
dense than water, or the density of gold is 19 times
that of water. Hence we might define the density by
the weight of a cubic inch of the substance, but it is
usual to take, not this weight, but the proportion
which it bears to the same weight of water. Thus,
when we say the density, or specific gravity (these terms
are used indifferently), of cast iron is 7-207, we mean
that if any vessel of pure water were emptied and
filled with cast iron, the iron would weigh 7-207 times
as much as the water.

If the density of a body were uniform throughout,
we might easily determine it by dividing the weight
of any bulk of the body, by the weight of an equal
bulk of water. In the same manner (pages 52 et seq.)
we could, from our definition of velocity, determine
any uniform velocity by dividing the length described
by the time. But if the density vary continuously,
no such measure can be adopted. For if by the side
of AB (which we will suppose to be of iron) we placed
a similar body of water similarly divided, and if we
divided the weight of the part pq of iron by the weight
of the same part of water, we should get different
densities, according as the part/^ is longer or shorter.
The water is supposed to be homogeneous, that is,
any part of it pr, being twice the length of pq, is twice
the weight of pq, and so on. The iron, on the con
trary, being supposed to vary in density, the doubling

THE DIFFERENTIAL AND INTEGRAL CALCULUS. 131

the length gives either more or less than twice the
weight. But if we suppose q to move towards/, both
on the iron and the water, the limit of the ratio pq of
iron to pq of water, may be chosen as a measure of
the density of /, on the same principle as in pages
54-55, the limit of the ratio of the length described to
the time of describing it, was called the velocity. If
we call k this limit, and if the weight varies contin
uously, though no part pq, however small, of iron,
would be exactly k times the same part of water in
weight, we may nevertheless take pq so small that
these weights shall be as nearly as we please in the
ratio of k to 1.

Let us now suppose that this density, expressed
by the limiting ratio aforesaid, is always oc* at any

Fig. 11.

point whose distance from A is x feet ; that is, the
density at q, 2 feet distance from A, is 4, and so on.
Let the whole distance AB=#. If we divide a into
n equal parts, each of which is dx, so that ndx = a,
and if we call b the area of the section of the paral
lelepiped, (b being a fraction of a square foot,) the
solid content of each of the parts will be bdx in
cubic feet ; and if w be the weight of a cubic foot of
water, the weight of the same bulk of water will be
wbdx. If the solid AB were homogeneous in the im
mediate neighborhood of the point /, the density being
then x9, would give x* X bwdx for the weight of the
same part of the substance. This is not true, but can
be brought as near to the truth as we please, by tak
ing dx sufficiently small, or dividing AB into a suffi-

132 ELEMENTARY ILLUSTRATIONS OF

cient number of parts. Hence the real weight of pq
may be represented by bwx^dx -j- a, where a may be
made as small a part as we please of the term which
precedes it.

In the sum of any number of these terms, the sum
arising from the term a diminishes without limit as
compared with the sum arising from the term bwx^dx ;
for if a be less than the thousandth part of/, a' less
than the thousandth part of p't etc., then a-}- a'-f- etc.
will be less than the thousandth part of / -j- p' -}- etc. :
which is also true of any number of quantities, and of
any fraction, however small, which each term of one
set is of its corresponding term in the other. Hence
the taking of the integral of bwx^dx dispenses with
the necessity of considering the term a ; for in taking
the integral, we find a limit which supposes dx to
have decreased without limit, and the integral which
would arise from a has therefore diminished without
limit.

The integral of bw x^dx is \bwx*, which taken from
x = Q to x = a is \bwcP. This is therefore the weight
in pounds of the bar whose length is a feet, and whose
section is b square feet, when the density at any point
distant by x feet from the beginning is x2 ; w being
the weight in pounds of a cubic foot of water.

CONCLUDING REMARKS ON THE STUDY OF THE CALCULUS.

We would recommend it to the student, in pur
suing any problem of the Integral Calculus, never for
one moment to lose sight of the manner in which he
would do it, if a rough solution for practical purposes
only were required. Thus, if he has the area of a
curve to find, instead of merely saying that y, the
ordinate, being a certain function of the abscissa x,

THE DIFFERENTIAL AND INTEGRAL CALCULUS. 133

fydx within the given limits would be the area re
quired ; and then proceeding to the mechanical solu
tion of the question : let him remark that if an ap
proximate solution only were required, it might be
obtained by dividing the curvilinear area into a num
ber of four-sided figures, as in Figure 10, one side of
which only is curvilinear, and embracing so small an
arc that it may, without visible error, be considered
as rectilinear. The mathematical method begins with
the same principle, investigating upon this supposi
tion, not the sum of these rectilinear areas, but the
limit towards which this sum approaches, as the sub
division is rendered more minute. This limit is shown
to be that of which we are in search, since it is proved
that the error diminishes without limit, as the subdi
vision is indefinitely continued.

We now leave our reader to any elementary work
which may fall in his way, having done our best to
place before him those considerations, something
equivalent to which he must turn over in his mind be
fore he can understand the subject. The method so
generally followed in our elementary works, of lead
ing the student at once into the mechanical processes
of the science, postponing entirely all other considera
tions, is to many students a source of obscurity at
least, if not an absolute impediment to their progress ;
since they cannot imagine what is the object of that
which they are required to do. That they shall un
derstand everything contained in these treatises, on
the first or second reading, we cannot promise ; but
that the want of illustration and the preponderance of
technical reasoning are the great causes of the difficul
ties which students experience, is the opinion of many
who have had experience in teaching this subject.

BRIEF BIBLIOGRAPHY.*

STANDARD TEXT-BOOKS AND TREATISES ON
THE CALCULUS.

ENGLISH.

Perry, John : Calculus for Engineers, Second edition, London
and New York: Edward Arnold. 1897. Price, 75. 6d. ($2.50).

Extract from Author's Preface : "This book describes what has
for many years been the most important part of the regular course in
the Calculus for Mechanical and Electrical Engineering students at
the Finsbury Technical College. The students in October knew only
the most elementary mathematics, many of them did not know the
Binomial Theorem, or the definition of the sine of an angle. In July
they had not only done the work of this book, but their knowledge
was of a practical kind, ready for use in any such engineering prob
lems as I give here."

Especially good in the character and number of practical exam
ples given.

Lamb, Horace : Infinitesimal Calculus. New York : The Mac-
millan Co. 1898. Price, $3.00.

Extract from Author's Preface: "This book attempts to teach
those portions of the Calculus which are of primary importance in
the application to such subjects as Physics and Engineering. . . .
Stress is laid on fundamental principles. . . . Considerable attention
has been paid to the logic of the subject."

*The information given regarding the works mentioned in this list is de
signed to enable the reader to select the books which are best suited to his
needs and his purse. Where the titles do not sufficiently indicate the char
acter of the books, a note or extract from the Preface has been added. The
American prices have been supplied by Messrs. Lemcke & Buechner, 812
Broadway, New York, through whom the purchases, especially of the foreign
books, may be conveniently made. — Ed.

136 BIBLIOGRAPHY.

Edwards, Joseph : An Elementary Treatise on the Differential
Calculus. Second edition, revised. 8vo, cloth. New York
and London: The Macmillan Co. 1892. Price, $3.50. —
Differential Calculus for Beginners. 8vo, cloth. 1893. The
Integral Calculus for Beginners. 8vo, cloth. (Same Pub
lishers.) Price, $1.10 each.

Byerly, William E. : Elements of the Differential Calculus. Bos
ton: Ginn & Co. Price, $2.15. — Elements of the Integral
Calculus. (Same Publishers.) Price, $2.15.

Rice, J. M., and Johnson, W. W. : An Elementary Treatise on
the Differential Calculus Founded on the Method of Rates
or Fluxions. New York: John Wiley & Sons. 8vo. 1884.
Price, $3.50. Abridged edition, 1889. Price, $"1.50.

Johnson, W. W. : Elementary Treatise on the Integral Calculus
Founded on the Method of Rates or Fluxions. 8vo, cloth.
New York: John Wiley & Sons. 1885. Price, $1.50.

Greenhill, A. G. : Differential and Integral Calculus. With ap
plications. 8vo, cloth. Second edition. New York and Lon
don: The Macmillan Co. 1891. Price, gs. ($2.60).

Price : Infinitesimal Calculus. Four Vols. 1857-65. Out of
print and very scarce. Obtainable for about $27.00.

Smith, William Benjamin: Infinitesimal Analysis. Vol. I., Ele
mentary : Real Variables. New York and London : The Mac
millan Co. 1898. Price, $3.25.

" The aim has been, within a prescribed expense of time and
energy to penetrate as far as possible, and in as many directions, into
the subject in hand,— that the student should attain as wide knowl
edge of the matter, as full comprehension of the methods, and as clear
consciousness of the spirit and power of analysis as the nature of the
case would admit." — From Author's Preface.

Todhunter, Isaac : A Treatise on the Differential Calculus. Lon
don and New York: The Macmillan Co. Price, ics. 6d.
($2.60). A Treatise on the Integral Calcuhis. (Same pub
lishers.) Price, IDS. 6d. ($2.60).

Todhunter' s text-books were, until recently, the most widely used
in England. His works on the Calculus still retain their standard
character, as general manuals.

BIBLIOGRAPHY. 137

Williamson : Differential and Integral Calculus. London and
New York : Longmans, Green, & Co. 1872-1874. Two Vols.
Price, $3. 50 each.

De Morgan, Augustus: Differential and Integral Calculus. Lon
don : Society for the Diffusion of Useful Knowledge. 1842.
Out of print. About $6.40.

The most extensive and complete work in English. " The object
has been to contain within the prescribed limits, the whole of the
students' course from the confines of elementary algebra and trigo
nometry, to the entrance of the highest works on mathematical phys
ics " (Author's Preface). Few examples. In typography, and gen
eral arrangement of material, inferior to the best recent works. Val
uable for collateral study, and for its philosophical spirit.

FRENCH.

Sturm: Cours d" analyse de ?& cole Poly technique. 10. Edition,
revue et corrige par E. Prouhet, et augmentee de la theorie
e'le'mentaire des fonctions elliptiques, par H. Laurent. 2 vol
umes in — 8. Paris: Gauthier-Villars et fils. 1895. Bound,
16 fr. 50 c. $4.95.

One of the most widely used of text-books. First published in
1857. The new tenth edition has been thoroughly revised and brought
down to date. The exercises, while not numerous, are sufficient, those
which accompany the additions and complementary chapters of M.
De Saint Germain having been taken from the Collection of M. Tis-
serand, mentioned below.

Duhamel: Elements de calcul infinitesimal. 4. edition, revue et
annotee par J. Bertrand. 2 volumes in — 8 ; avec planches.
Paris : Gauthier-Villars et fils. 1886. 15 fr. $4.50.

The first edition was published between 1840 and 1841. " Cordially
recommended to teachers and students" by De Morgan. Duhamel
paid great attention to the philosophy and logic of the mathematical
sciences, and the student may also be referred in this connexion to
his Mlthodes dans les sciences de raisonnement. 5 volumes. Paris :
Gauthier-Villars et fils. Price, 25.50 francs. $7.65.

Lacroix, S.-F. : Traite elementaire de calcul differentiel et de
calcul integral, g. edition, revue et augmented de notes par
Hermite et Serret. 2 vols. Paris : Gauthier-Villars et fils.
1881. 15 fr. $4.50.

A very old work. The first edition was published in 1797. It was
the standard treatise during the early part of the century, and has
been kept revised by competent hands.

138 BIBLIOGRAPHY.

Appell, P.: Elements d' analyse mathematique . A 1'usage des
inge"nieurs et des physiciens. Cours professe a I'fecole Cen-
trale des Arts et Manufactures, i vol. in — 8, 720 pages, avec
figures, cartonne a 1'anglaise. Paris : Georges Carre & C.
Naud. 1899. Price, 24 francs. $7.20.

Boussinesq, J. : Cours d' analyse infinitesimal. A 1'usage des
personnes qui e"tudient cette science en vue de ses applications
mecaniques et physiques, 2 vols. , grand in-8, avec figures.
Tome I. Calcul differentiel. Paris, 1887. 17 fr. ($5.10).
Tome II. Calcul integral. Paris: Gauthier-Villars et fils.
1890. 23 fr. 50 c. ($7.05).

Hermite, Ch. : Cours d' analyse de r£cole Poly technique. 2 vols.
Vol.1. Paris: Gauthier-Villars et fils. 1897.

A new edition of Vol. I. is in preparation (1899). Vol. II. has not
yet appeared.

Jordan, Camille : Cours d"1 analyse de fAcole Poly technique. 3
volumes. 2. Edition. Paris : Gauthier-Villars et fils. 1893
1898. 51 fr. $14.70.

Very comprehensive on the theoretical side. Enters deeply into
the metaphysical aspects of the subject.

Laurent, H.: Traite d 'analyse. 7 vols in — 8. Paris: Gauthier-
Villars et fils. 1885-1891. 73 fr. $21.90.

The most extensive existing treatise on the Calculus. A general
handbook and work of reference for the results contained in the
more special works and memoirs.

Picard, £mile : Traite" d' analyse. 4 volumes grand in-8. Paris:
Gauthier-Villars et fils. 1891. 15 fr. each. Vols. I.— III.,
$14.40. Vol. IV. has not yet appeared.

An advanced treatise on the Integral Calculus and the theory of
differential equations. Presupposes a knowledge of the Differential
Calculus.

Serret, J.-A. : Cours de calcul differentiel et integral. 4. edi
tion, augmente'e d'une note sur les fonctions elliptiques, par
Ch. Hermite. 2 forts volumes in — 8. Paris : Gauthier-Villars
et fils. 1894. 25 fr. $7.50.

A Rood German translation of this work by Axel Harnack has
passed through its second edition (Leipsic: Teubner, 1885 and 1897).

BIBLIOGRAPHY. 139

Hoiiel, J.: Cours de calcul infinitesimal. 4 beaux volumes grand
in — 8, avec figures. Paris : Gauthier-Villars et fils. 1878-
1879-1880-1881. 50 fr. $15.00.

Bertrand, J. : Traite de calcul differ entiel et de calcul integral.
(i) Calcul differentiel. Paris : Gauthier-Villars et fils. 1864.
Scarce. About $48.00 (2) Calcul integral (Integrates de"fin-
ies et inde'finies). Paris, 1870. Scarce. About $24.00.

Boucharlat, J.-L. : Elements de calcul differentiel et de calcul
integral. 9. edition, revue et annotee par H. Laurent. Paris :
Gauthier-Villars et fils. 1891. 8 fr. $2.40.

Moigno : Lecons de calcul differentiel et de calcul integral, 2
vols., Paris, 1840-1844. Scarce. About $9.60.

Navier : Lecons d1 analyse de V Ecole Polytechnique . Paris, 1840.
2nd ed. 1856. Out of print. About $3.60.

An able and practical work. Very popular in ;ts day. The typical
course of the Ecole Polyteclmique , and the basis of several of the trea
tises that followed, including that of Sturm. Also much used in its
German translation.

Cournot : Theorie des fonctions et du calcul infinitesimal. 2
vols. Paris, 1841. 2nd ed. 1856-1858. Out of print, and
scarce. About $3.00.

The first edition (1841) was " cordially recommended to teachers
and students" by De Morgan. Cournot was especially strong on the
philosophical side. He examined the foundations of many sciences
and developed original views on the theory of knowledge, which are
little known but have been largely drawn from by other philosophers-

Cauchy, A.: (Euvres completes. Tome III: Cours d' analyse
de I' Ecole Polytechnique. Tome IV : Resume des lecons
donnees h I"1 Ecole Polytechnique sur le calcul infinitesimal.
Lecons sur le calcul differentiel. Tome V : Lecons sur les
applications du calcul infinitesimal h la geometric . Paris :
Gauthier-Villars et fils, 1885-1897. 25 fr. each. $9.50 each.

The works of Cauchy, as well as those of Lagrange, which follow,
are mentioned for their high historical and educational importance.

Lagrange, J. L. : (Euvres completes. Tome IX : Theorie des fonc
tions analytiques. Tome X. : Lecons sur le calcul des fonc-

140 BIBLIOGRAPHY.

tions. Paris : Gauthier-Villars et fils, 1881-1884. 18 fr. per
volume. $5.40 per volume.

"The same power of abstraction and facility of treatment which
signalise these works are nowhere to be met with in the prior or sub
sequent history of the subject. In addition, they are replete with the
profoundest aperfus into the history of the development of analytical
truths,— aperfus which could have come only from a man who com
bined superior creative endowment with exact and comprehensive
knowledge of the facts. In the remarks woven into the body of the
text will be found what is virtually a detailed history of the subject,
and one which is not to be had elsewhere, least of all in diffuse his
tories of mathematics. The student, thus, not only learns in these
works how to think, but also discovers how people actually have
thought, and what are the ways which human instinct and reason
have pursued in the different individuals who have participated in
the elaboration of the science." — (E. Diihring.)

Euler, L.:

The Latin treatises of Euler are also to be mentioned in this con
nexion, for the benefit of those who wish to pursue the history of the
text-book making of this subject to its fountain-head. They are the
Differential Calculus (St. Petersburg, 1755), the Integral Calculus (3
vols., St. Petersburg, 1768-1770;, and the Introduction to the Infinitesi
mal Analysis (2 vols., Lausanne, 1748). Of the last-mentioned work
an old French translation by Labey exists (Paris: Gauthier-Villars),
and a new German translation (of Vol. I. only) by Maser (Berlin :
Julius Springer, 1885). Of the first-mentioned treatises on the Cal
culus proper there exist two old German translations, which are not
difficult to obtain.

GERMAN.

Harnack, Dr. Axel: Elemente der Differential- und Tntegral-
rechnung. Zur Einftihrung in das Studium dargestellt. Leip
zig : Teubner, 1881. M. 7.60. Bound, $2.80. (English trans
lation. London: Williams & Norgate. 1891.)

Junker, Dr. Friedrich : Htihere Analysts. I. Differentialrech-
nung. Mit 63 Figuren. II. Integralrechnung. Leipzig :
G. J. Goschen'sche Verlagshandlung. 1898-1899. 80 pf. each.
30 cents each.

These books are marvellously cheap, and very concise. They
contain no examples. Pocket-size.

Autenheimer, F. : Element arbuch der Differential- und Integral
rechnung mit zahlreichen An-wendungen aus der Analysis,
Geometric, Mechanik, Physik etc. Fur hohere Lehranstalten

BIBLIOGRAPHY. 141

und den Selbstunterricht. 4te verbesserte Auflage. Weimar :
Bernhard Friedrich Voigt. 1895.

As indicated by its title, this book is specially rich in practical
applications.

Stegemann : Grundriss der Differential- und fntegralrechnung,
8te Auflage, herausgegeben von Kiepert. Hannover : Hel-
wing, 1897. Two volumes, 26 marks. Two volumes, bound,

$8.50.

This work was highly recommended by Prof. Felix Klein at the
Evanston Colloquium in 1893.

Schlomilch : Compendium der h'dheren Analysis. Fifth edition,
1881. Two volumes, $6.80.

Schlomilch's text-books have been very successful. The present
work was long the standard manual.

Stolz, Dr. Otto : Grundziige der Differential- und Integralrech-
nung. In 2 Theilen. I. Theil. Reelle Verelnderliche und
Functionen. (460 S.) 1893. M. 8. II. Complexe Verander-
liche und Functionen. (3388.) Leipzig: Teubner. 1896.
M. 8. Two volumes, $6.00.

A supplementary 3rd part entitled Die Lehrt von den Doppel-
integralen has just been published (1899). Based on the works of J.
Tannery, Peano, and Dini.

Lipschitz, R.: Lehrbuch der Analysis. 1877-1880. Two vol
umes, bound, $12.30.

Specially good on the theoretical side.

COLLECTIONS OF EXAMPLES AND ILLUSTRATIONS.

Byerly, W. E. : Problems in Differential Calculus. Supplemen
tary to a Treatise on Differential Calculus. Boston : Ginn &

Co. 75 cents.

Gregory : Examples on the Differential and Integral Calculus.
1841. Second edition. 1846. Out of print. About $6.40.

Frenet : Recueil d' exercises sur le calcul infinitesimal. 5. Edi
tion, augmentee d'un appendice, par H. Laurent. Paris :
Gauthier-Villars et fils. 1891. 8 fr. $2.40.

142 BIBLIOGRAPHY.

Tisserand, F.: Recueil complement air e d* exercises sur le calcul
infinitesimal. Second edition. Paris: Gauthier-Villarset fils.
1896.

Complementary to Frenet.

Laisant, C. A. : Recueil de problemes de matliematiques. Tome
VII. Calcul infinitesimal et calcul des fonctions. Mecanique.
Astronomic. (Announced for publication.) Paris : Gauthier-
Villars et fils.

Schlomilch, Dr. Oscar : Ucbungsbuch zum Studium der htiheren
Analysis. I. Theil. Aufgaben aus der Differentialrechnung.
4te Auflage. (336 S.) 1887. M. 6. II. Aufgaben aus der
Integralrechnung. 3 te Auflage. (3848.) Leipzig : Teubner,
1882. M. 7.60. Both volumes, bound, $7.60.

Sohncke, L. A. : Sammlung von Aufgaben aus der Differential-
und Integralrechnung. Herausgegeben von Heis. Two vol
umes, in — 8. Bound, $3.00.

Fuhrmann, Dr. Arwed : Anwendungen der Infinitesimalrech-
nung in den Naturzuissenschaften, im Hochbau und in der
Technik. Lehrbuch und Aufgabensammlung. In sechs Thei-
Isn, von denen jeder ein selbststandiges Ganzes bildet. Theil
I. Naturwissenschaftliche Anwendungen der Differentialrech
nung. Theil II. Naturwissenschaftliche Anwendungen der
Integralrechnung. Berlin : Verlag von Ernst & Korn. 1888-
1890. Vol. I., Cloth, $1.35. Vol. II., Cloth, $2.20.

INDEX.

Accelerated motion, 57, 60.

Accelerating force, 62.

Advice for studying the Calculus,

132, 133-
Angle, unit employed in measuring

an, 51.
Approximate solutions in the Integral

Calculus, 132, 133.
Arc and its chord, a continuously

decreasing, 7 et seq., 39 et seq.
Archimedes, 127.
Astronomical ephemeris, 76.

Calculus, notation of, 25, 79 et seq.

Circle, equation of, 31 et seq.

Circle cut by straight line, investi
gated, 31 et seq.

Coefficients, differential, 22 et seq.,
38, 55, 82, 88, 96, 100, 112.

Complete Differential Coefficients,
96.

Constants, 14. •

Contiguous values, 112.

Continuous quantities, 7 et seq., 53.

Co-ordinates, 30.

Curve, magnified, 40.

Curvilinear areas, determination of,
124 et seq.

Density, continuously varying, 130 et
seq.

Derivatives, 19, 21, 22.

Derived Functions, 19 et seq., 21.

Differences, arithmetical, 4; of incre
ments, 26; calculus of, 89.

Differential coefficients, 22 et seq..
38, 55, 82, as the index of the change

of a function, 112; of higher orders,
88.

Differentials, partial, 78 et seq.; total
78 et seq.

Differentiation, of the common func
tions, 85, 86; successive, 88 et seq.;
implicit, 94 et seq.; of complicatpd
functions, 100 et seq.

Direct function, 97.

Direction, 36.

Equality, 4.

Equations, solution of, 77.

Equidistant values, 104.

Euler, 27, 124.

Errors, in the valuation of quantities,

75, 84.
Explicit functions, 107.

Falling bodies, 56.

Finite differences, 88 et seq.

Fluxions, n, 60, 112.

Force, 61-63.

Functions, definition of, 14 et seq.;
derived, 19 et seq., 21; direct and
indirect, 97; implicit and explicit,
107, 108; inverse, 102 et seq.. of sev
eral variables,78 etseq.; recapitula
tion of results in the theory of, 74.

Generally, the word, 16.

Implicit, differentiation, 94 et seq.;

function, 107, 108.
Impulse, 60.
Increase without limit, 5 et seq., 65

et seq.
Increment, 16, 11,3.

i44

INDEX.

Independent variables, 106.

Indirect function, 97.

Indivisibles, method of, 127 et seq.;
notion of, in mechanics, 129 et seq.

Infinite, the word, 128.

Infinitely small, the notion of, 12, 38
et seq., 49, 59, 83.

Infinity, orders of, 42 et seq.

Integral Calculus, 73, 115 et seq., no
tation of, 119.

Integrals, definition of, 119 et seq.;
relations between differential co
efficients and, 121 ; indefinite, 122,
123.

Intersections, limit of, 46 et seq.

Inverse functions, 102 et seq.

Iron bar continually varying in dens
ity, weight of, 130 et seq.

Ladder against wall, 45 et seq.

Lagrange, 124.

Laplace, 124.

Leibnitz, n, 13, 38, 42, 48, 59, 60, 83,

123, 124, 128, 129.

Limit of intersections, 46 et seq.
Limits, 26 et seq.
Limiting ratios, 65 et seq., 81.
Logarithms, 20, 38, 86, 87, 112 et seq.

Magnified curve, 40.
Motion, accelerated, 60; simple har
monic, 57.

Newton, u, 60.

Notation, of the Differential Calcu
lus, as, 79 et seq.; of the Integral
Calculus, 119.

Orders, differential coefficients of

higher, 88.
Orders of infinity, 42 et seq.

Parabola, the, 30, 124 et seq., 127.
Partial, differentials, 78 et seq.; dif
ferential coefficients, 96.
Paint, the word, 129.

Points, the number of, in a straight

line, 129.
Polygon, 38.
Proportion, 2 et seq.

Quantities, continuous, 7 et seq., 53

Ratio, defined, 2 et seq.; of two in
crements, 87.

Ratios, limiting, 65 et seq., 81.

Rough methods of solution in the In
tegral Calculus, 132, 133.

Series, 15 et seq., 24 et seq.
Signs, 31 et seq.
Simple harmonic motion, 57.
Sines, 87.

Singular values, 16.
Small, has no precise meaning, 12.
Specific gravity, continuously vary
ing, 130 et seq.

Successive differentiation, 88 et seq.
Sun's longitude, 76.

Tangent, 37, 38, 40.

Taylor's Theorem, 15 et seq., 19 et

seq.

Time, idea of, 4, no et seq.
Total, differential coefficient, 100,

differentials, 78 et seq.; variations,

95-
Transit instrument, 84.

Uniformly accelerated, 57, 60.

Values, contiguous, 112; equidistant,
104.

Variables, independent and depen
dent, 14, 15, 106; functions of sev
eral, 78 et seq.

Variations, total, 95.

Velocity, linear, 53 et seq., uij §a-.
gular, 59.

Weight of an iron bar of which the
density varies from point to point,
130 et seq.

Portraits of
Eminent Mathematicians

Three portfolios edited by DAVID EUGENE SMITH, Ph. D.. Professor of
Mathematics in Teachers' College, Columbia University, New York City.

In response to a widespread demand from those interested in mathe
matics and the history of education. Professor Smith has edited three port
folios of the portraits of some of the most eminent of the world's contributors
to the mathematical sciences. Accompanying each portrait is a brief bio
graphical sketch, with occasional notes of interest concerning the artists
represented, The pictures are of a size that allows for framing (11x14), it
being the hope that a new interest in mathematics may be aroused through
the decoration of classrooms by the portraits of those who helped to create
the science.

"Dnff-fnlin l\T/\ 1 Twelve great mathematicians down to 1700 A. D.:
JrOrLIOllO 1NO. 1. Thalcs, Pythagoras. Euclid, Archimedes. Leon
ardo of Pisa, Cardan, Vieta, Napier, Descartes, Fermat, Newton, Leibniz.

T>r\»-ffr\1irv "Mn 9 The most eminent founders and promoters of
JTUl L1U11U 11U. 6. the infinitesimal calculus: Cavallieri. Johann
and Jakob Bernoulli, Pascal. L'Hopital. Barrow, Laplace, Lagrange, Euler
Gauss. Monge and Niccolo Tartaglia.

"M/\ ^ Eight portraits selected from the two former.

1NO. O. portfolios especially adapted for high schools
and academies, including portraits of TH ALES— with whom began the study
of scientific geometry; PYTHAGORAS — who proved the proposition of the
square on the hy pothenuse : EUCLID— whose Elements of Geometry form the
basis of all modern text books; ARCHIMEDES— whose treatment of the
circle, cone, cylinder and sphere influences our work today; DESCARTES —
to whom we are indebted for the graphic algebra in our high schools:
NEWTON — who generalized the binomial theorem and invented the calculus;
NAPIER — who invented logarithms and contributed to trigpnometry;
PASCAL — who discovered the "Mystic Hexagram" at the age of sixteen.

PRICES

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Portfolio 3 (8 portraits) on Japanese vellum, size 11x14, $3.50; single portraits, 50 cents.
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