which may be described as follows: Let y = [f](x) be the equation of a
curve, and let AC and BD be the ordinates of the points C and D (see
fig. 3). Let a, b be the abscissae of these points. Let the segment AB
be divided into a number of segments by means of intermediate points
such as M, and let MN be one such segment. Let PM and QN be those
ordinates of the curve which have M and N as their feet. On MN as base
describe two rectangles, of which the heights are the greatest and
least values of y which correspond to points on the arc PQ of the
curve. In fig. 3 these are the rectangles RM, SN. Let the sum of the
areas of such rectangles as RM be formed, and likewise the sum of the
areas of such rectangles as SN. When the number of the points such as
M is increased without limit, and the lengths of all the segments such
as MN are diminished without limit, these two sums of areas tend to
limits. When they tend to the same limit the curvilinear figure ACDB
has an area, and the limit is the measure of this area (see § 33
below). The limit in question is the same whatever law may be adopted
for inserting the points such as M between A and B, and for
diminishing the lengths of the segments such as MN. Further, if P´ is
any point on the arc PQ, and P´M´ is the ordinate of P´, we may
construct a rectangle of which the height is P´M´ and the base is MN,
and the limit of the sum of the areas of all such rectangles is the
area of the figure as before. If x is the abscissa of P, x + [Delta]x
that of Q, x´ that of P´, the limit in question might be written
_b
lim. \ [f](x´)[Delta]x,
/_a
where the letters a, b written below and above the sign of summation
[Sigma] indicate the extreme values of x. This limit is called "the
definite integral of [f](x) between the limits a and b," and the
notation for it is
_
/ b
| [f](x)dx.
_/ a
The germs of this method of formulating the problem of quadratures are
found in the writings of Archimedes. The method leads to a definition
of a definite integral, but the direct application of it to the
evaluation of integrals is in general difficult. Any process for
evaluating a definite integral is a process of integration, and the
rules for evaluating integrals constitute the _integral calculus_.
Theorem of Inversion.
6. The chief of these rules is obtained by regarding the extreme
ordinate BD as variable. Let [xi] now denote the abscissa of B. The
area A of the figure ACDB is represented by the integral [int] {a to
[xi]} [f](x)dx, and it is a function of [xi]. Let BD be displaced to
B´D´ so that [xi] becomes [xi] + [delta][xi] (see fig. 4). The area of
the figure ACD´B´ is represented by the integral [int] {a to [xi] +
[Delta][xi]} [f](x)dx, and the increment [Delta]A of the area is given
by the formula
_[xi]+[Delta][xi]
/
[Delta]A = | [f](x) dx,
_/ [xi]
which represents the area BDD´B´. This area is intermediate between
those of two rectangles, having as a common base the segment BB´, and
as heights the greatest and least ordinates of points on the arc DD´
of the curve. Let these heights be H and h. Then [Delta]A is
intermediate between H[Delta][xi] and h[Delta][xi], and the quotient
of differences [Delta]A/[Delta][xi] is intermediate between H and h.
If the function [f](x) is continuous at B (see Function), then, as
[Delta][xi] is diminished without limit, H and h tend to BD, or
[f]([xi]), as a limit, and we have
dA
----- = [f]([xi]).
d[xi]
[Illustration: FIG. 4.]
The introduction of the process of differentiation, together with the
theorem here proved, placed the solution of the problem of quadratures
on a new basis. It appears that we can always find the area A if we
know a function F(x) which has [f](x) as its differential coefficient.
If [f](x) is continuous between a and b, we can prove that
_
/ b
A = | [f](x) dx = F(b) - F(a).
_/ a
When we recognize a function F(x) which has the property expressed by
the equation
dF(x)
----- = [f](x),
dx
we are said to _integrate_ the function [f](x), and F(x) is called the
_indefinite integral_ of [f](x) _with respect to_ x, and is written
_
/
| [f](x)dx.
_/
Differentials.
7. In the process of § 4 the increment [Delta]y is not in general
equal to the product of the increment [Delta]x and the derived
function [f]´(x). In general we can write down an equation of the form
[Delta]y = [f]´(x)[Delta]x + R,
in which R is different from zero when [Delta]x is different from
zero; and then we have not only
lim. R = 0,
[Delta]x=0
but also
R
lim. -------- = 0.
[Delta]x=0 [Delta]x
We may separate [Delta]y into two parts: the part [f]´(x)[Delta]x and
the part R. The part [f]´(x)[Delta]x alone is useful for forming the
differential coefficient, and it is convenient to give it a name. It
is called the _differential_ of [f](x), and is written d[f](x), or dy
when y is written for [f](x). When this notation is adopted dx is
written instead of [Delta]x, and is called the "differential of x," so
that we have
d[f](x) = [f]´(x) dx.
Thus the differential of an independent variable such as x is a finite
difference; in other words it is any number we please. The
differential of a dependent variable such as y, or of a function of
the independent variable x, is the product of the differential of x
and the differential coefficient or derived function. It is important
to observe that the differential coefficient is not to be defined as
the ratio of differentials, but the ratio of differentials is to be
defined as the previously introduced differential coefficient. The
differentials are either finite differences, or are so much of
certain finite differences as are useful for forming differential
coefficients.
Again let F(x) be the indefinite integral of a continuous function
[f](x), so that we have
_
dF(x) / b
----- = [f](x), | [f](x) dx = F(b) - F(a).
dx _/a
When the points M of the process explained in § 5 are inserted between
the points whose abscissae are a and b, we may take them to be n - 1
in number, so that the segment AB is divided into n segments. Let x1,
x2, ... x_(n-1) be the abscissae of the points in order. The integral
is the limit of the sum
[f](a)(x1 - a) + [f](x1)(x2 - x1) + ... + [f](x_r) [x_(r+1) - x_r]
+ ... + [f] [x_(n-1)] [b - x_(n-1)],
every term of which is a differential of the form [f](x)dx. Further
the integral is equal to the sum of differences
{F(x1) - F(a)} + {F(x2) - F(x1)} + ... + {F[x_(r+1)] - F(x_r)}
+ ... + {F(b) - F[x(n-1)]},
for this sum is F(b) - F(a). Now the difference F(x_(r+1)) - F(x_r) is
_not_ equal to the differential [f](x_r) [x_(r+1) - x_r], but the sum
of the differences is equal to the _limit_ of the sum of these
differentials. The differential may be regarded as so much of the
difference as is required to form the integral. From this point of
view a differential is called a _differential element of an integral_,
and the integral is the limit of the sum of differential elements. In
like manner the differential element ydx of the area of a curve (§ 5)
is not the area of the portion contained between two ordinates,
however near together, but is so much of this area as need be retained
for the purpose of finding the area of the curve by the limiting
process described.
Notation.
8. The notation of the infinitesimal calculus is intimately bound up
with the notions of differentials and sums of elements. The letter "d"
is the initial letter of the word _differentia_ (difference) and the
symbol [int] is a conventionally written "S," the initial letter of
the word _summa_ (sum or whole). The notation was introduced by
Leibnitz (see §§ 25-27, below).
Fundamental Artifice.
9. The fundamental artifice of the calculus is the artifice of forming
differentials without first forming differential coefficients. From an
equation containing x and y we can deduce a new equation, containing
also [Delta]x and [Delta]y, by substituting x + [Delta]x for x and y +
[Delta]y for y. If there is a differential coefficient of y with
respect to x, then [Delta]y can be expressed in the form
[phi].[Delta]x + R, where lim.{[Delta]x = 0} (R/[Delta]x) = 0, as in §
7 above. The artifice consists in rejecting _ab initio_ all terms of
the equation which belong to R. We do not form R at all, but only
[phi].[Delta]x, or [phi].dx, which is the differential dy. In the same
way, in all applications of the integral calculus to geometry or
mechanics we form the _element_ of an integral in the same way as the
element of area y·dx is formed. In fig. 3 of § 5 the element of area
y·dx is the area of the rectangle RM. The actual area of the
curvilinear figure PQNM is greater than the area of this rectangle by
the area of the curvilinear figure PQR; but the excess is less than
the area of the rectangle PRQS, which is measured by the product of
the numerical measures of MN and QR, and we have
MN·QR
lim. ------ = 0.
MN=0 MN
Thus the artifice by which differential elements of integrals are
formed is in principle the same as that by which differentials are
formed without first forming differential coefficients.
Orders of small quantities.
10. This principle is usually expressed by introducing the notion of
orders of small quantities. If x, y are two variable numbers which are
connected together by any relation, and if when x tends to zero y also
tends to zero, the fraction y/x may tend to a finite limit. In this
case x and y are said to be "of the same order." When this is not the
case we may have either
x
lim. --- = 0,
x=0 y
or
y
lim. --- = 0,
x=0 x
In the former case y is said to be "of a lower order" than x; in the
latter case y is said to be "of a higher order" than x. In accordance
with this notion we may say that the fundamental artifice of the
infinitesimal calculus consists in the rejection of small quantities
of an unnecessarily high order. This artifice is now merely an
incident in the conduct of a limiting process, but in the 17th
century, when limiting processes other than the Greek methods for
quadratures were new, the introduction of the artifice was a great
advance.
Rules of Differentiation.
11. By the aid of this artifice, or directly by carrying out the
appropriate limiting processes, we may obtain the rules by which
differential coefficients are formed. These rules may be classified as
"formal rules" and "particular results." The formal rules may be
stated as follows:--
(i.) The differential coefficient of a _constant_ is zero. (ii.) For a
_sum_ u + v + ... + z, where u, v, ... are functions of x,
d(u + v + ... + z) du dv dz
----------------- = -- + -- + ... + --.
dx dx dx dx
(iii.) For a _product_ uv
d(uv) dv du
----- = u -- + v --.
dx dx dx
(iv.) For a _quotient_ u/v
d(u/v) / du dv\ /
------ = ( v -- - u -- ) / v².
dx \ dx dx/ /
(v.) For a _function of a function_, that is to say, for a function y
expressed in terms of a variable z, which is itself expressed as a
function of x,
dy dy dz
-- = -- · --.
dx dz dx
In addition to these formal rules we have particular results as to the
differentiation of simple functions. The most important results are
written down in the following table:--
+---------+---------------------+
| y | dy/dx |
+---------+---------------------+
| x^n | nx^(n-1) |
| | for all values of n |
+---------+---------------------+
| log_a x | x^-1 log_a e |
+---------+---------------------+
| a^x | a^x log_e a |
+---------+---------------------+
| sin x | cos x |
+---------+---------------------+
| cos x | -sin x |
+---------+---------------------+
| sin^-1 x| (1 - x²)^-½ |
+---------+---------------------+
| tan^-1 x| (1 + x²)^-1 |
+---------+---------------------+
Each of the formal rules, and each of the particular results in the
table, is a theorem of the differential calculus. All functions (or
rather expressions) which can be made up from those in the table by a
finite number of operations of addition, subtraction, multiplication
or division can be differentiated by the formal rules. All such
functions are called _explicit_ functions. In addition to these we
have _implicit_ functions, or such as are determined by an equation
containing two variables when the equation cannot be solved so as to
exhibit the one variable expressed in terms of the other. We have also
functions of several variables. Further, since the derived function of
a given function is itself a function, we may seek to differentiate
it, and thus there arise the second and higher differential
coefficients. We postpone for the present the problems of differential
calculus which arise from these considerations. Again, we may have
explicit functions which are expressed as the results of limiting
operations, or by the limits of the results obtained by performing an
infinite number of algebraic operations upon the simple functions. For
the problem of differentiating such functions reference may be made to
FUNCTION.
Indefinite Integrals.
12. The processes of the integral calculus consist largely in
transformations of the functions to be integrated into such forms that
they can be recognized as differential coefficients of functions which
have previously been differentiated. Corresponding to the results in
the table of § 11 we have those in the following table:--
+----------------+-------------------------------+
| [f](x) | [int][f](x)dx |
+----------------+-------------------------------+
| | x^(n+1) |
| x^n | ------- |
| | n + 1 |
| | for all values of n except -1 |
+----------------+-------------------------------+
| 1 | |
| --- | log_e x |
| x | |
+----------------+-------------------------------+
| e^(ax) | a^-1 e^(ax) |
+----------------+-------------------------------+
| cos x | sin x |
+----------------+-------------------------------+
| sin x | -cos x |
+----------------+-------------------------------+
| | x |
| (a² - x²)^-½ | sin^-1 --- |
| | a |
+----------------+-------------------------------+
| 1 | 1 x |
| ------- | --- tan^-1 --- |
| a² + x² | a a |
+----------------+-------------------------------+
The formal rules of § 11 give us means for the transformation of
integrals into recognizable forms. For example, the rule (ii.) for a
sum leads to the result that the integral of a sum of a finite number
of terms is the sum of the integrals of the several terms. The rule
(iii.) for a product leads to the method of integration by parts. The
rule (v.) for a function of a function leads to the method of
substitution (see § 48 below.)
II. _History._
Kepler's methods of Integration.
13. The new limiting processes which were introduced in the development
of the higher analysis were in the first instance related to problems of
the integral calculus. Johannes Kepler in his _Astronomia nova ... de
motibus stellae Martis_ (1609) stated his laws of planetary motion, to
the effect that the orbits of the planets are ellipses with the sun at a
focus, and that the radii vectores drawn from the sun to the planets
describe equal areas in equal times. From these statements it is to be
concluded that Kepler could measure the areas of focal sectors of an
ellipse. When he made out these laws there was no method of evaluating
areas except the Greek methods. These methods would have sufficed for
the purpose, but Kepler invented his own method. He regarded the area as
measured by the "sum of the radii" drawn from the focus, and he verified
his laws of planetary motion by actually measuring a large number of
radii of the orbit, spaced according to a rule, and adding their
lengths.
[Illustration: FIG. 5.]
He had observed that the focal radius vector SP (fig. 5) is equal to
the perpendicular SZ drawn from S to the tangent at p to the auxiliary
circle, and he had further established the theorem which we should now
express in the form--the differential element of the area ASp as Sp
turns about S, is equal to the product of SZ and the differential
ad[phi], where a is the radius of the auxiliary circle, and [phi] is
the angle ACp, that is the eccentric angle of P on the ellipse. The
area ASP bears to the area ASp the ratio of the minor to the major
axis, a result known to Archimedes. Thus Kepler's radii are spaced
according to the rule that the eccentric angles of their ends are
equidifferent, and his "sum of radii" is proportional to the
expression which we should now write
_
/ [phi]
| (a + ae cos [phi]) d[phi],
_/ 0
where e is the eccentricity. Kepler evaluated the sum as proportional
to [phi] + e sin [phi].
Kepler soon afterwards occupied himself with the volumes of solids. The
vintage of the year 1612 was extraordinarily abundant, and the question
of the cubic content of wine casks was brought under his notice. This
fact accounts for the title of his work, _Nova stereometria doliorum;
accessit stereometriae Archimedeae supplementum_ (1615). In this
treatise he regarded solid bodies as being made up, as it were
(_veluti_), of "infinitely" many "infinitely" small cones or
"infinitely" thin disks, and he used the notion of summing the areas of
the disks in the way he had previously used the notion of summing the
focal radii of an ellipse.
Logarithms.
14. In connexion with the early history of the calculus it must not be
forgotten that the method by which logarithms were invented (1614) was
effectively a method of infinitesimals. Natural logarithms were not
invented as the indices of a certain base, and the notation e for the
base was first introduced by Euler more than a century after the
invention. Logarithms were introduced as numbers which increase in
arithmetic progression when other related numbers increase in geometric
progression. The two sets of numbers were supposed to increase together,
one at a uniform rate, the other at a variable rate, and the increments
were regarded for purposes of calculation as very small and as accruing
discontinuously.
Cavalieri's Indivisibles.
15. Kepler's methods of integration, for such they must be called, were
the origin of Bonaventura Cavalieri's theory of the summation of
indivisibles. The notion of a continuum, such as the area within a
closed curve, as being made up of indivisible parts, "atoms" of area, if
the expression may be allowed, is traceable to the speculations of early
Greek philosophers; and although the nature of continuity was better
understood by Aristotle and many other ancient writers yet the unsound
atomic conception was revived in the 13th century and has not yet been
finally uprooted. It is possible to contend that Cavalieri did not
himself hold the unsound doctrine, but his writing on this point is
rather obscure. In his treatise _Geometria indivisibilibus continuorum
nova quadam ratione promota_ (1635) he regarded a plane figure as
generated by a line moving so as to be always parallel to a fixed line,
and a solid figure as generated by a plane moving so as to be always
parallel to a fixed plane; and he compared the areas of two plane
figures, or the volumes of two solids, by determining the ratios of the
sums of all the indivisibles of which they are supposed to be made up,
these indivisibles being segments of parallel lines equally spaced in
the case of plane figures, and areas marked out upon parallel planes
equally spaced in the case of solids. By this method Cavalieri was able
to effect numerous integrations relating to the areas of portions of
conic sections and the volumes generated by the revolution of these
portions about various axes. At a later date, and partly in answer to an
attack made upon him by Paul Guldin, Cavalieri published a treatise
entitled _Exercitationes geometricae sex_ (1647), in which he adapted
his method to the determination of centres of gravity, in particular for
solids of variable density.
Among the results which he obtained is that which we should now write
_
/ x x^(m+1)
| x^m dx = -------, (m integral).
_/ 0 m + 1
He regarded the problem thus solved as that of determining the sum of
the mth powers of all the lines drawn across a parallelogram parallel
to one of its sides.
Successors of Cavalieri.
Fermat's method of Integration.
At this period scientific investigators communicated their results to
one another through one or more intermediate persons. Such
intermediaries were Pierre de Carcavy and Pater Marin Mersenne; and
among the writers thus in communication were Bonaventura Cavalieri,
Christiaan Huygens, Galileo Galilei, Giles Personnier de Roberval,
Pierre de Fermat, Evangelista Torricelli, and a little later Blaise
Pascal; but the letters of Carcavy or Mersenne would probably come into
the hands of any man who was likely to be interested in the matters
discussed. It often happened that, when some new method was invented, or
some new result obtained, the method or result was quickly known to a
wide circle, although it might not be printed until after the lapse of a
long time. When Cavalieri was printing his two treatises there was much
discussion of the problem of quadratures. Roberval (1634) regarded an
area as made up of "infinitely" many "infinitely" narrow strips, each of
which may be considered to be a rectangle, and he had similar ideas in
regard to lengths and volumes. He knew how to approximate to the
quantity which we express by [int] (0 to 1) x^m dx by the process of
forming the sum
0^m + 1^m + 2^m + ... (n - 1)^m
-------------------------------,
n^(m+1)
and he claimed to be able to prove that this sum tends to 1/(m + 1), as
n increases for all positive integral values of m. The method of
integrating x^m by forming this sum was found also by Fermat (1636), who
stated expressly that he arrived at it by generalizing a method employed
by Archimedes (for the cases m = 1 and m = 2) in his books on _Conoids
and Spheroids_ and on _Spirals_ (see T. L. Heath, _The Works of
Archimedes_, Cambridge, 1897). Fermat extended the result to the case
where m is fractional (1644), and to the case where m is negative. This
latter extension and the proofs were given in his memoir, _Proportionis
geometricae in quadrandis parabolis et hyperbolis usus_, which appears
to have received a final form before 1659, although not published until
1679. Fermat did not use fractional or negative indices, but he regarded
his problems as the quadratures of parabolas and hyperbolas of various
orders. His method was to divide the interval of integration into parts
by means of intermediate points the abscissae of which are in geometric
progression. In the process of § 5 above, the points M must be chosen
according to this rule. This restrictive condition being understood, we
may say that Fermat's formulation of the problem of quadratures is the
same as our definition of a definite integral.
Various Integrations.
The result that the problem of quadratures could be solved for any curve
whose equation could be expressed in the form
y = x^m (m [Not Equal] -1),
or in the form
y = a1 x^m1 + a2 x^m2 + ... + a_n x^m_n,
where none of the indices is equal to - 1, was used by John Wallis in
his _Arithmetica infinitorum_ (1655) as well as by Fermat (1659). The
case in which m = - 1 was that of the ordinary rectangular hyperbola;
and Gregory of St Vincent in his _Opus geometricum quadraturae circuli
et sectionum coni_ (1647) had proved by the method of exhaustions that
the area contained between the curve, one asymptote, and two ordinates
parallel to the other asymptote, increases in arithmetic progression as
the distance between the ordinates (the one nearer to the centre being
kept fixed) increases in geometric progression. Fermat described his
method of integration as a logarithmic method, and thus it is clear that
the relation between the quadrature of the hyperbola and logarithms was
understood although it was not expressed analytically. It was not very
long before the relation was used for the calculation of logarithms by
Nicolaus Mercator in his _Logarithmotechnia_ (1668). He began by writing
the equation of the curve in the form y = 1/(1 + x), expanded this
expression in powers of x by the method of division, and integrated it
term by term in accordance with the well-understood rule for finding the
quadrature of a curve given by such an equation as that written at the
foot of p. 325.
Integration before the Integral Calculus.
By the middle of the 17th century many mathematicians could perform
integrations. Very many particular results had been obtained, and
applications of them had been made to the quadrature of the circle and
other conic sections, and to various problems concerning the lengths of
curves, the areas they enclose, the volumes and superficial areas of
solids, and centres of gravity. A systematic account of the methods then
in use was given, along with much that was original on his part, by
Blaise Pascal in his _Lettres de Amos Dettonville sur quelques-unes de
ses inventions en géométrie_ (1659).
Fermat's methods of Differentiation.
16. The problem of maxima and minima and the problem of tangents had
also by the same time been effectively solved. Oresme in the 14th
century knew that at a point where the ordinate of a curve is a maximum
or a minimum its variation from point to point of the curve is slowest;
and Kepler in the _Stereometria doliorum_ remarked that at the places
where the ordinate passes from a smaller value to the greatest value and
then again to a smaller value, its variation becomes insensible. Fermat
in 1629 was in possession of a method which he then communicated to one
Despagnet of Bordeaux, and which he referred to in a letter to Roberval
of 1636. He communicated it to René Descartes early in 1638 on receiving
a copy of Descartes's _Géométrie_ (1637), and with it he sent to
Descartes an account of his methods for solving the problem of tangents
and for determining centres of gravity.
Fermat's method for maxima and minima is essentially our method.
Expressed in a more modern notation, what he did was to begin by
connecting the ordinate y and the abscissa x of a point of a curve by
an equation which holds at all points of the curve, then to subtract
the value of y in terms of x from the value obtained by substituting x
+ E for x, then to divide the difference by E, to put E = 0 in the
quotient, and to equate the quotient to zero. Thus he differentiated
with respect to x and equated the differential coefficient to zero.
[Illustration: FIG. 6.]
Fermat's method for solving the problem of tangents may be explained
as follows:--Let (x, y) be the coordinates of a point P of a curve,
(x´, y´), those of a neighbouring point P´ on the tangent at P, and
let MM´ = E (fig. 6).
From the similarity of the triangles P´TM´, PTM we have
y´:A - E = y:A,
where A denotes the subtangent TM. The point P´ being near the curve,
we may substitute in the equation of the curve x - E for x and (yA -
yE)/A for y. The equation of the curve is approximately satisfied. If
it is taken to be satisfied exactly, the result is an equation of the
form [phi](x, y, A, E) = 0, the left-hand member of which is divisible
by E. Omitting the factor E, and putting E = 0 in the remaining
factor, we have an equation which gives A. In this problem of tangents
also Fermat found the required result by a process equivalent to
differentiation.
Fermat gave several examples of the application of his method; among
them was one in which he showed that he could differentiate very
complicated irrational functions. For such functions his method was to
begin by obtaining a rational equation. In rationalizing equations
Fermat, in other writings, used the device of introducing new variables,
but he did not use this device to simplify the process of
differentiation. Some of his results were published by Pierre Hérigone
in his _Supplementum cursus mathematici_ (1642). His communication to
Descartes was not published in full until after his death (Fermat,
_Opera varia_, 1679). Methods similar to Fermat's were devised by René
de Sluse (1652) for tangents, and by Johannes Hudde (1658) for maxima
and minima. Other methods for the solution of the problem of tangents
were devised by Roberval and Torricelli, and published almost
simultaneously in 1644. These methods were founded upon the composition
of motions, the theory of which had been taught by Galileo (1638), and,
less completely, by Roberval (1636). Roberval and Torricelli could
construct the tangents of many curves, but they did not arrive at
Fermat's artifice. This artifice is that which we have noted in § 10 as
the fundamental artifice of the infinitesimal calculus.
[Illustration: FIG. 7.]
Barrow's Differential Triangle.
17. Among the comparatively few mathematicians who before 1665 could
perform differentiations was Isaac Barrow. In his book entitled
_Lectiones opticae et geometricae_, written apparently in 1663, 1664,
and published in 1669, 1670, he gave a method of tangents like that of
Roberval and Torricelli, compounding two velocities in the directions of
the axes of x and y to obtain a resultant along the tangent to a curve.
In an appendix to this book he gave another method which differs from
Fermat's in the introduction of a differential equivalent to our dy as
well as dx. Two neighbouring ordinates PM and QN of a curve (fig. 7) are
regarded as containing an indefinitely small (_indefinite parvum_) arc,
and PR is drawn parallel to the axis of x. The tangent PT at P is
regarded as identical with the secant PQ, and the position of the
tangent is determined by the similarity of the triangles PTM, PQR. The
increments QR, PR of the ordinate and abscissa are denoted by a and e;
and the ratio of a to e is determined by substituting x + e for x and y
+ a for y in the equation of the curve, rejecting all terms which are of
order higher than the first in a and e, and omitting the terms which do
not contain a or e. This process is equivalent to differentiation.
Barrow appears to have invented it himself, but to have put it into his
book at Newton's request. The triangle PQR is sometimes called "Barrow's
differential triangle."
Barrow's Inversion-theorem.
The reciprocal relation between differentiation and integration (§ 6)
was first observed explicitly by Barrow in the book cited above. If
the quadrature of a curve y = f(x) is known, so that the area up to
the ordinate x is given by F(x), the curve y = F(x) can be drawn, and
Barrow showed that the subtangent of this curve is measured by the
ratio of its ordinate to the ordinate of the original curve. The curve
y = F(x) is often called the "quadratrix" of the original curve; and
the result has been called "Barrow's inversion-theorem." He did not
use it as we do for the determination of quadratures, or indefinite
integrals, but for the solution of problems of the kind which were
then called "inverse problems of tangents." In these problems it was
sought to determine a curve from some property of its tangent, e.g.
the property that the subtangent is proportional to the square of the
abscissa. Such problems are now classed under "differential
equations." When Barrow wrote, quadratures were familiar and
differentiation unfamiliar, just as hyperbolas were trusted while
logarithms were strange. The functional notation was not invented till
long afterwards (see FUNCTION), and the want of it is felt in reading
all the mathematics of the 17th century.
Nature of the discovery called the Infinitesimal Calculus.
18. The great secret which afterwards came to be called the
"infinitesimal calculus" was almost discovered by Fermat, and still more
nearly by Barrow. Barrow went farther than Fermat in the theory of
differentiation, though not in the practice, for he compared two
increments; he went farther in the theory of integration, for he
obtained the inversion-theorem. The great discovery seems to consist
partly in the recognition of the fact that differentiation, known to be
a useful process, could always be performed, at least for the functions
then known, and partly in the recognition of the fact that the
inversion-theorem could be applied to problems of quadrature. By these
steps the problem of tangents could be solved once for all, and the
operation of integration, as we call it, could be rendered systematic. A
further step was necessary in order that the discovery, once made,
should become accessible to mathematicians in general; and this step was
the introduction of a suitable notation. The definite abandonment of the
old tentative methods of integration in favour of the method in which
this operation is regarded as the inverse of differentiation was
especially the work of Isaac Newton; the precise formulation of simple
rules for the process of differentiation in each special case, and the
introduction of the notation which has proved to be the best, were
especially the work of Gottfried Wilhelm Leibnitz. This statement
remains true although Newton invented a systematic notation, and
practised differentiation by rules equivalent to those of Leibnitz,
before Leibnitz had begun to work upon the subject, and Leibnitz
effected integrations by the method of recognizing differential
coefficients before he had had any opportunity of becoming acquainted
with Newton's methods.
Newton's investigations.
19. Newton was Barrow's pupil, and he knew to start with in 1664 all
that Barrow knew, and that was practically all that was known about the
subject at that time. His original thinking on the subject dates from
the year of the great plague (1665-1666), and it issued in the invention
of the "Calculus of Fluxions," the principles and methods of which were
developed by him in three tracts entitled _De analysi per aequationes
numero terminorum infinitas, Methodus fluxionum et serierum infinitarum,
and De quadratura curvarum_. None of these was published until long
after they were written. The _Analysis per aequationes_ was composed in
1666, but not printed until 1711, when it was published by William
Jones. The _Methodus fluxionum_ was composed in 1671 but not printed
till 1736, nine years after Newton's death, when an English translation
was published by John Colson. In Horsley's edition of Newton's works it
bears the title _Geometria analytica_. The _Quadratura_ appears to have
been composed in 1676, but was first printed in 1704 as an appendix to
Newton's _Opticks_.
Newton's method of Series.
20. The tract _De Analysi per aequationes ..._ was sent by Newton to
Barrow, who sent it to John Collins with a request that it might be
made known. One way of making it known would have been to print it in
the _Philosophical Transactions_ of the Royal Society, but this course
was not adopted. Collins made a copy of the tract and sent it to Lord
Brouncker, but neither of them brought it before the Royal Society.
The tract contains a general proof of Barrow's inversion-theorem which
is the same in principle as that in § 6 above. In this proof and
elsewhere in the tract a notation is introduced for the momentary
increment (_momentum_) of the abscissa or area of a curve; this
"moment" is evidently meant to represent a moment of time, the
abscissa representing time, and it is effectively the same as our
differential element--the thing that Fermat had denoted by E, and
Barrow by e, in the case of the abscissa. Newton denoted the moment of
the abscissa by o, that of the area z by ov. He used the letter v for
the ordinate y, thus suggesting that his curve is a velocity-time
graph such as Galileo had used. Newton gave the formula for the area
of a curve v = x^m (m ± -1) in the form z = x^(m+1)/(m + 1). In the
proof he transformed this formula to the form z^n = c^n x^p, where n
and p are positive integers, substituted x + o for x and z + ov for z,
and expanded by the binomial theorem for a positive integral exponent,
thus obtaining the relation
z^n + nz^(n-1) ov + ... = c^n (x_p + px^(p-1)o + ...),
from which he deduced the relation
nz_(n-1)v = c^n px^(p-1)
by omitting the equal terms z^n and c^n·x^p and dividing the remaining
terms by o, tacitly putting o = 0 after division. This relation is the
same as v = x^m. Newton pointed out that, conversely, from the
relation v = x^m the relation z = x^(m+1) / (m + 1) follows. He
applied his formula to the quadrature of curves whose ordinates can be
expressed as the sum of a finite number of terms of the form ax^m; and
gave examples of its application to curves in which the ordinate is
expressed by an infinite series, using for this purpose the binomial
theorem for negative and fractional exponents, that is to say, the
expansion of (1 + x)^n in an infinite series of powers of x. This
theorem he had discovered; but he did not in this tract state it in a
general form or give any proof of it. He pointed out, however, how it
may be used for the solution of equations by means of infinite series.
He observed also that all questions concerning lengths of curves,
volumes enclosed by surfaces, and centres of gravity, can be
formulated as problems of quadratures, and can thus be solved either
in finite terms or by means of infinite series. In the _Quadratura_
(1676) the method of integration which is founded upon the
inversion-theorem was carried out systematically. Among other results
there given is the quadrature of curves expressed by equations of the
form y = x^n·(a + bx^m)^p; this has passed into text-books under the
title "integration of binomial differentials" (see § 49). Newton
announced the result in letters to Collins and Oldenburg of 1676.
Newton's method of Fluxions.
21. In the _Methodus fluxionum_ (1671) Newton introduced his
characteristic notation. He regarded variable quantities as generated
by the motion of a point, or line, or plane, and called the generated
quantity a "fluent" and its rate of generation a "fluxion." The
fluxion of a fluent x is represented by x, and its moment, or
"infinitely" small increment accruing in an "infinitely" short time,
is represented by [.x]o. The problems of the calculus are stated to be
(i.) to find the velocity at any time when the distance traversed is
given; (ii.) to find the distance traversed when the velocity is
given. The first of these leads to differentiation. In any rational
equation containing x and y the expressions x + [.x]o and y +[.y]o are
to be substituted for x and y, the resulting equation is to be divided
by o, and afterwards o is to be omitted. In the case of irrational
functions, or rational functions which are not integral, new variables
are introduced in such a way as to make the equations contain rational
integral terms only. Thus Newton's rules of differentiation would be
in our notation the rules (i.), (ii.), (v.) of § 11, together with the
particular result which we write
dx^m
---- = mx^(m-1), (m integral).
dx
a result which Newton obtained by expanding (x = [.x]o)^m by the
binomial theorem. The second problem is the problem of integration,
and Newton's method for solving it was the method of series founded
upon the particular result which we write
_
/ x^(m+1)
| x^m dx = -------.
_/ m + 1
Newton added applications of his methods to maxima and minima,
tangents and curvature. In a letter to Collins of date 1672 Newton
stated that he had certain methods, and he described certain results
which he had found by using them. These methods and results are those
which are to be found in the _Methodus fluxionum_; but the letter
makes no mention of fluxions and fluents or of the characteristic
notation. The rule for tangents is said in the letter to be analogous
to de Sluse's, but to be applicable to equations that contain
irrational terms.
Publication of the Fluxional Notation.
22. Newton gave the fluxional notation also in the tract De
_Quadratura curvarum_ (1676), and he there added to it notation for
the higher differential coefficients and for indefinite integrals, as
we call them. Just as x, y, z, ... are fluents of which [.x], [.y],
[.z], ... are the fluxions, so [.x], [.y], [.z], ... can be treated as
fluents of which the fluxions may be denoted by [:x], [:y], [:z],...
In like manner the fluxions of these may be denoted by [:x], [:y],
[:z], ... and so on. Again x, y, z, ... may be regarded as fluxions of
which the fluents may be denoted by ['x], ['y], ['z], ... and these
again as fluxions of other quantities denoted by ["x], ["y], ["z], ...
and so on. No use was made of the notation ['x], ["x], ... in the
course of the tract. The first publication of the fluxional notation
was made by Wallis in the second edition of his _Algebra_ (1693) in
the form of extracts from communications made to him by Newton in
1692. In this account of the method the symbols 0, [.x], [:x], ...
occur, but not the symbols ['x], ["x], .... Wallis's treatise also
contains Newton's formulation of the problems of the calculus in the
words _Data aequatione fluentes quotcumque quantitates involvente
fluxiones invenire et vice versa_ ("an equation containing any number
of fluent quantities being given, to find their fluxions and vice
versa"). In the _Philosophiae naturalis principia mathematica_ (1687),
commonly called the "Principia," the words "fluxion" and "moment"
occur in a lemma in the second book; but the notation which is
characteristic of the calculus of fluxions is nowhere used.
Retarded Publication of the method of Fluxions.
23. It is difficult to account for the fragmentary manner of publication
of the Fluxional Calculus and for the long delays which took place. At
the time (1671) when Newton composed the _Methodus fluxionum_ he
contemplated bringing out an edition of Gerhard Kinckhuysen's treatise
on algebra and prefixing his tract to this treatise. In the same year
his "Theory of Light and Colours" was published in the _Philosophical
Transactions_, and the opposition which it excited led to the
abandonment of the project with regard to fluxions. In 1680 Collins
sought the assistance of the Royal Society for the publication of the
tract, and this was granted in 1682. Yet it remained unpublished. The
reason is unknown; but it is known that about 1679, 1680, Newton took up
again the studies in natural philosophy which he had intermitted for
several years, and that in 1684 he wrote the tract _De motu_ which was
in some sense a first draft of the _Principia_, and it may be
conjectured that the fluxions were held over until the _Principia_
should be finished. There is also reason to think that Newton had become
dissatisfied with the arguments about infinitesimals on which his
calculus was based. In the preface to the _De quadratura curvarum_
(1704), in which he describes this tract as something which he once
wrote ("_olim scripsi_") he says that there is no necessity to introduce
into the method of fluxions any argument about infinitely small
quantities; and in the _Principia_ (1687) he adopted instead of the
method of fluxions a new method, that of "Prime and Ultimate Ratios." By
the aid of this method it is possible, as Newton knew, and as was
afterwards seen by others, to found the calculus of fluxions on an
irreproachable method of limits. For the purpose of explaining his
discoveries in dynamics and astronomy Newton used the method of limits
only, without the notation of fluxions, and he presented all his results
and demonstrations in a geometrical form. There is no doubt that he
arrived at most of his theorems in the first instance by using the
method of fluxions. Further evidence of Newton's dissatisfaction with
arguments about infinitely small quantities is furnished by his tract
_Methodus diferentialis_, published in 1711 by William Jones, in which
he laid the foundations of the "Calculus of Finite Differences."
Leibnitz's course of discovery.
24. Leibnitz, unlike Newton, was practically a self-taught
mathematician. He seems to have been first attracted to mathematics as a
means of symbolical expression, and on the occasion of his first visit
to London, early in 1673, he learnt about the doctrine of infinite
series which James Gregory, Nicolaus Mercator, Lord Brouncker and
others, besides Newton, had used in their investigations. It appears
that he did not on this occasion become acquainted with Collins, or see
Newton's _Analysis per aequationes_, but he purchased Barrow's
_Lectiones_. On returning to Paris he made the acquaintance of Huygens,
who recommended him to read Descartes' _Géométrie_. He also read
Pascal's _Lettres de Dettonville_, Gregory of St Vincent's _Opus
geometricum_, Cavalieri's _Indivisibles_ and the _Synopsis geometrica_
of Honoré Fabri, a book which is practically a commentary on Cavalieri;
it would never have had any importance but for the influence which it
had on Leibnitz's thinking at this critical period. In August of this
year (1673) he was at work upon the problem of tangents, and he appears
to have made out the nature of the solution--the method involved in
Barrow's differential triangle--for himself by the aid of a diagram
drawn by Pascal in a demonstration of the formula for the area of a
spherical surface. He saw that the problem of the relation between the
differences of neighbouring ordinates and the ordinates themselves was
the important problem, and then that the solution of this problem was to
be effected by quadratures. Unlike Newton, who arrived at
differentiation and tangents through integration and areas, Leibnitz
proceeded from tangents to quadratures. When he turned his attention to
quadratures and indivisibles, and realized the nature of the process of
finding areas by summing "infinitesimal" rectangles, he proposed to
replace the rectangles by triangles having a common vertex, and obtained
by this method the result which we write
1 1 1 1
--- [pi] = 1 - --- + --- - --- + ...
4 3 5 7
In 1674 he sent an account of his method, called "transmutation," along
with this result to Huygens, and early in 1675 he sent it to Henry
Oldenburg, secretary of the Royal Society, with inquiries as to Newton's
discoveries in regard to quadratures. In October of 1675 he had begun to
devise a symbolical notation for quadratures, starting from Cavalieri's
indivisibles. At first he proposed to use the word _omnia_ as an
abbreviation for Cavalieri's "sum of all the lines," thus writing
_omnia_ y for that which we write "[int] ydx," but within a day or two
he wrote "[int] y". He regarded the symbol "[int]" as representing an
operation which raises the dimensions of the subject of operation--a
line becoming an area by the operation--and he devised his symbol "d" to
represent the inverse operation, by which the dimensions are diminished.
He observed that, whereas "[int]" represents "sum," "d" represents
"difference." His notation appears to have been practically settled
before the end of 1675, for in November he wrote [int] y dy = ½y², just
as we do now.
Correspondence of Newton and Leibnitz.
25. In July of 1676 Leibnitz received an answer to his inquiry in regard
to Newton's methods in a letter written by Newton to Oldenburg. In this
letter Newton gave a general statement of the binomial theorem and many
results relating to series. He stated that by means of such series he
could find areas and lengths of curves, centres of gravity and volumes
and surfaces of solids, but, as this would take too long to describe, he
would illustrate it by examples. He gave no proofs. Leibnitz replied in
August, stating some results which he had obtained, and which, as it
seemed, could not be obtained easily by the method of series, and he
asked for further information. Newton replied in a long letter to
Oldenburg of the 24th of October 1676. In this letter he gave a much
fuller account of his binomial theorem and indicated a method of proof.
Further he gave a number of results relating to quadratures; they were
afterwards printed in the tract _De quadratura curvarum_. He gave many
other results relating to the computation of natural logarithms and
other calculations in which series could be used. He gave a general
statement, similar to that in the letter to Collins, as to the kind of
problems relating to tangents, maxima and minima, &c., which he could
solve by his method, but he concealed his formulation of the calculus in
an anagram of transposed letters. The solution of the anagram was given
eleven years later in the _Principia_ in the words we have quoted from
Wallis's _Algebra_. In neither of the letters to Oldenburg does the
characteristic notation of the fluxional calculus occur, and the words
"fluxion" and "fluent" occur only in anagrams of transposed letters. The
letter of October 1676 was not despatched until May 1677, and Leibnitz
answered it in June of that year. In October 1676 Leibnitz was in
London, where he made the acquaintance of Collins and read the _Analysis
per aequationes_, and it seems to have been supposed afterwards that he
then read Newton's letter of October 1676, but he left London before
Oldenburg received this letter. In his answer of June 1677 Leibnitz gave
Newton a candid account of his differential calculus, nearly in the form
in which he afterwards published it, and explained how he used it for
quadratures and inverse problems of tangents. Newton never replied.
Leibnitz's Differential Calculus.
26. In the _Acta eruditorum_ of 1684 Leibnitz published a short memoir
entitled _Nova methodus pro maximis et minimis, itemque tangentibus,
quae nec fractas nec irrationales quantitates moratur, et singulare pro
illis calculi genus_. In this memoir the differential dx of a variable
x, considered as the abscissa of a point of a curve, is said to be an
arbitrary quantity, and the differential dy of a related variable y,
considered as the ordinate of the point, is defined as a quantity which
has to dx the ratio of the ordinate to the subtangent, and rules are
given for operating with differentials. These are the rules for forming
the differential of a constant, a sum (or difference), a product, a
quotient, a power (or root). They are equivalent to our rules (i.)-(iv.)
of § 11 and the particular result
d(x^m) = mx^(m-1) dx.
The rule for a function of a function is not stated explicitly but is
illustrated by examples in which new variables are introduced, in much
the same way as in Newton's _Methodus fluxionum_. In connexion with the
problem of maxima and minima, it is noted that the differential of y is
positive or negative according as y increases or decreases when x
increases, and the discrimination of maxima from minima depends upon the
sign of ddy, the differential of dy. In connexion with the problem of
tangents the differentials are said to be proportional to the momentary
increments of the abscissa and ordinate. A tangent is defined as a line
joining two "infinitely" near points of a curve, and the "infinitely"
small distances (e.g., the distance between the feet of the ordinates of
such points) are said to be expressible by means of the differentials
(e.g., dx). The method is illustrated by a few examples, and one example
is given of its application to "inverse problems of tangents." Barrow's
inversion-theorem and its application to quadratures are not mentioned.
No proofs are given, but it is stated that they can be obtained easily
by any one versed in such matters. The new methods in regard to
differentiation which were contained in this memoir were the use of the
second differential for the discrimination of maxima and minima, and the
introduction of new variables for the purpose of differentiating
complicated expressions. A greater novelty was the use of a letter (d),
not as a symbol for a number or magnitude, but as a symbol of operation.
None of these novelties account for the far-reaching effect which this
memoir has had upon the development of mathematical analysis. This
effect was a consequence of the simplicity and directness with which the
rules of differentiation were stated. Whatever indistinctness might be
felt to attach to the symbols, the processes for solving problems of
tangents and of maxima and minima were reduced once for all to a
definite routine.
Development of the Calculus.
27. This memoir was followed in 1686 by a second, entitled _De Geometria
recondita et analysi indivisibilium atque infinitorum_, in which
Leibnitz described the method of using his new differential calculus for
the problem of quadratures. This was the first publication of the
notation [int] ydx. The new method was called _calculus summatorius_.
The brothers Jacob (James) and Johann (John) Bernoulli were able by 1690
to begin to make substantial contributions to the development of the new
calculus, and Leibnitz adopted their word "integral" in 1695, they at
the same time adopting his symbol "[int]." In 1696 the marquis de
l'Hospital published the first treatise on the differential calculus
with the title _Analyse des infiniment petits pour l'intelligence des
lignes courbes_. The few references to fluxions in Newton's _Principia_
(1687) must have been quite unintelligible to the mathematicians of the
time, and the publication of the fluxional notation and calculus by
Wallis in 1693 was too late to be effective. Fluxions had been
supplanted before they were introduced.
The differential calculus and the integral calculus were rapidly
developed in the writings of Leibnitz and the Bernoullis. Leibnitz
(1695) was the first to differentiate a logarithm and an exponential,
and John Bernoulli was the first to recognize the property possessed by
an exponential (a^x) of becoming infinitely great in comparison with any
power (x^n) when x is increased indefinitely. Roger Cotes (1722) was the
first to differentiate a trigonometrical function. A great development
of infinitesimal methods took place through the founding in 1696-1697 of
the "Calculus of Variations" by the brothers Bernoulli.
Dispute concerning Priority.
28. The famous dispute as to the priority of Newton and Leibnitz in the
invention of the calculus began in 1699 through the publication by
Nicolas Fatio de Duillier of a tract in which he stated that Newton was
not only the first, but by many years the first inventor, and insinuated
that Leibnitz had stolen it. Leibnitz in his reply (_Acta Eruditorum_,
1700) cited Newton's letters and the testimony which Newton had rendered
to him in the _Principia_ as proofs of his independent authorship of the
method. Leibnitz was especially hurt at what he understood to be an
endorsement of Duillier's attack by the Royal Society, but it was
explained to him that the apparent approval was an accident. The dispute
was ended for a time. On the publication of Newton's tract _De
quadratura curvarum_, an anonymous review of it, written, as has since
been proved, by Leibnitz, appeared in the _Acta Eruditorum_, 1705. The
anonymous reviewer said: "Instead of the Leibnitzian differences Newton
uses and always has used fluxions ... just as Honoré Fabri in his
_Synopsis Geometrica_ substituted steps of movements for the method of
Cavalieri." This passage, when it became known in England, was
understood not merely as belittling Newton by comparing him with the
obscure Fabri, but also as implying that he had stolen his calculus of
fluxions from Leibnitz. Great indignation was aroused; and John Keill
took occasion, in a memoir on central forces which was printed in the
_Philosophical Transactions_ for 1708, to affirm that Newton was without
doubt the first inventor of the calculus, and that Leibnitz had merely
changed the name and mode of notation. The memoir was published in 1710.
Leibnitz wrote in 1711 to the secretary of the Royal Society (Hans
Sloane) requiring Keill to retract his accusation. Leibnitz's letter was
read at a meeting of the Royal Society, of which Newton was then
president, and Newton made to the society a statement of the course of
his invention of the fluxional calculus with the dates of particular
discoveries. Keill was requested by the society "to draw up an account
of the matter under dispute and set it in a just light." In his report
Keill referred to Newton's letters of 1676, and said that Newton had
there given so many indications of his method that it could have been
understood by a person of ordinary intelligence. Leibnitz wrote to
Sloane asking the society to stop these unjust attacks of Keill,
asserting that in the review in the _Acta Eruditorum_ no one had been
injured but each had received his due, submitting the matter to the
equity of the Royal Society, and stating that he was persuaded that
Newton himself would do him justice. A committee was appointed by the
society to examine the documents and furnish a report. Their report,
presented in April 1712, concluded as follows:
"The _differential method_ is one and the same with the _method of
fluxions_, excepting the name and mode of notation; Mr Leibnitz
calling those quantities _differences_ which Mr Newton calls _moments_
or _fluxions_, and marking them with the letter d, a mark not used by
Mr Newton. And therefore we take the proper question to be, not who
invented this or that method, but who was the first inventor of the
method; and we believe that those who have reputed Mr Leibnitz the
first inventor, knew little or nothing of his correspondence with Mr
Collins and Mr Oldenburg long before; nor of Mr Newton's having that
method above fifteen years before Mr. Leibnitz began to publish it in
the _Acta Eruditorum_ of Leipzig. For which reasons we reckon Mr
Newton the first inventor, and are of opinion that Mr Keill, in
asserting the same, has been no ways injurious to Mr Leibnitz."
The report with the letters and other documents was printed (1712) under
the title _Commercium Epistolicum D. Johannis Collins et aliorum de
analysi promota, jussu Societatis Regiae in lucem editum_, not at first
for publication. An account of the contents of the _Commercium
Epistolicum_ was printed in the _Philosophical Transactions_ for 1715. A
second edition of the _Commercium Epistolicum_ was published in 1722.
The dispute was continued for many years after the death of Leibnitz in
1716. To translate the words of Moritz Cantor, it "redounded to the
discredit of all concerned."
British and Continental Schools of Mathematics.
29. One lamentable consequence of the dispute was a severance of British
methods from continental ones. In Great Britain it became a point of
honour to use fluxions and other Newtonian methods, while on the
continent the notation of Leibnitz was universally adopted. This
severance did not at first prevent a great advance in mathematics in
Great Britain. So long as attention was directed to problems in which
there is but one independent variable (the time, or the abscissa of a
point of a curve), and all the other variables depend upon this one, the
fluxional notation could be used as well as the differential and
integral notation, though perhaps not quite so easily. Up to about the
middle of the 18th century important discoveries continued to be made by
the use of the method of fluxions. It was the introduction of partial
differentiation by Leonhard Euler (1734) and Alexis Claude Clairaut
(1739), and the developments which followed upon the systematic use of
partial differential coefficients, which led to Great Britain being left
behind; and it was not until after the reintroduction of continental
methods into England by Sir John Herschel, George Peacock and Charles
Babbage in 1815 that British mathematics began to flourish again. The
exclusion of continental mathematics from Great Britain was not
accompanied by any exclusion of British mathematics from the continent.
The discoveries of Brook Taylor and Colin Maclaurin were absorbed into
the rapidly growing continental analysis, and the more precise
conceptions reached through a critical scrutiny of the true nature of
Newton's fluxions and moments stimulated a like scrutiny of the basis of
the method of differentials.
Oppositions to the calculus.
The "Analyst" controversy.
Cauchy's method of limits.
30. This method had met with opposition from the first. Christiaan
Huygens, whose opinion carried more weight than that of any other
scientific man of the day, declared that the employment of differentials
was unnecessary, and that Leibnitz's second differential was meaningless
(1691). A Dutch physician named Bernhard Nieuwentijt attacked the method
on account of the use of quantities which are at one stage of the
process treated as somethings and at a later stage as nothings, and he
was especially severe in commenting upon the second and higher
differentials (1694, 1695). Other attacks were made by Michel Rolle
(1701), but they were directed rather against matters of detail than
against the general principles. The fact is that, although Leibnitz in
his answers to Nieuwentijt (1695), and to Rolle (1702), indicated that
the processes of the calculus could be justified by the methods of the
ancient geometry, he never expressed himself very clearly on the subject
of differentials, and he conveyed, probably without intending it, the
impression that the calculus leads to correct results by compensation of
errors. In England the method of fluxions had to face similar attacks.
George Berkeley, bishop and philosopher, wrote in 1734 a tract entitled
_The Analyst; or a Discourse addressed to an Infidel Mathematician_, in
which he proposed to destroy the presumption that the opinions of
mathematicians in matters of faith are likely to be more trustworthy
than those of divines, by contending that in the much vaunted fluxional
calculus there are mysteries which are accepted unquestioningly by the
mathematicians, but are incapable of logical demonstration. Berkeley's
criticism was levelled against all infinitesimals, that is to say, all
quantities vaguely conceived as in some intermediate state between
nullity and finiteness, as he took Newton's moments to be conceived. The
tract occasioned a controversy which had the important consequence of
making it plain that all arguments about infinitesimals must be given
up, and the calculus must be founded on the method of limits. During the
controversy Benjamin Robins gave an exceedingly clear explanation of
Newton's theories of fluxions and of prime and ultimate ratios regarded
as theories of limits. In this explanation he pointed out that Newton's
_moment_ (Leibnitz's "differential") is to be regarded as so much of the
actual difference between two neighbouring values of a variable as is
needful for the formation of the fluxion (or differential coefficient)
(see G. A. Gibson, "The Analyst Controversy," _Proc. Math. Soc._,
Edinburgh, xvii., 1899). Colin Maclaurin published in 1742 a _Treatise
of Fluxions_, in which he reduced the whole theory to a theory of
limits, and demonstrated it by the method of Archimedes. This notion was
gradually transferred to the continental mathematicians. Leonhard Euler
in his _Institutiones Calculi differentialis_ (1755) was reduced to the
position of one who asserts that all differentials are zero, but, as the
product of zero and any finite quantity is zero, the ratio of two zeros
can be a finite quantity which it is the business of the calculus to
determine. Jean le Rond d'Alembert in the _Encyclopédie méthodique_
(1755, 2nd ed. 1784) declared that differentials were unnecessary, and
that Leibnitz's calculus was a calculus of mutually compensating errors,
while Newton's method was entirely rigorous. D'Alembert's opinion of
Leibnitz's calculus was expressed also by Lazare N. M. Carnot in his
_Réflexions sur la métaphysique du calcul infinitésimal_ (1799) and by
Joseph Louis de la Grange (generally called Lagrange) in writings from
1760 onwards. Lagrange proposed in his _Théorie des fonctions
analytiques_ (1797) to found the whole of the calculus on the theory of
series. It was not until 1823 that a treatise on the differential
calculus founded upon the method of limits was published. The treatise
was the _Résumé des leçons ... sur le calcul infinitésimal_ of Augustin
Louis Cauchy. Since that time it has been understood that the use of the
phrase "infinitely small" in any mathematical argument is a figurative
mode of expression pointing to a limiting process. In the opinion of
many eminent mathematicians such modes of expression are confusing to
students, but in treatises on the calculus the traditional modes of
expression are still largely adopted.
Arithmetical basis of modern analysis.
31. Defective modes of expression did not hinder constructive work. It
was the great merit of Leibnitz's symbolism that a mathematician who
used it knew what was to be done in order to formulate any problem
analytically, even though he might not be absolutely clear as to the
proper interpretation of the symbols, or able to render a satisfactory
account of them. While new and varied results were promptly obtained by
using them, a long time elapsed before the theory of them was placed on
a sound basis. Even after Cauchy had formulated his theory much remained
to be done, both in the rapidly growing department of complex variables,
and in the regions opened up by the theory of expansions in
trigonometric series. In both directions it was seen that rigorous
demonstration demanded greater precision in regard to fundamental
notions, and the requirement of precision led to a gradual shifting of
the basis of analysis from geometrical intuition to arithmetical law. A
sketch of the outcome of this movement--the "arithmetization of
analysis," as it has been called--will be found in FUNCTION. Its general
tendency has been to show that many theories and processes, at first
accepted as of general validity, are liable to exceptions, and much of
the work of the analysts of the latter half of the 19th century was
directed to discovering the most general conditions in which particular
processes, frequently but not universally applicable, can be used
without scruple.
III. _Outlines of the Infinitesimal Calculus._
32. The general notions of functionality, limits and continuity are
explained in the article FUNCTION. Illustrations of the more immediate
ways in which these notions present themselves in the development of the
differential and integral calculus will be useful in what follows.
[Illustration: FIG. 8.]
Geometrical limits.
Tangents.
33. Let y be given as a function of x, or, more generally, let x and y
be given as functions of a variable t. The first of these cases is
included in the second by putting x = t. If certain conditions are
satisfied the aggregate of the points determined by the functional
relations form a curve. The first condition is that the aggregate of
the values of t to which values of x and y correspond must be
continuous, or, in other words, that these values must consist of all
real numbers, or of all those real numbers which lie between assigned
extreme numbers. When this condition is satisfied the points are
"ordered," and their order is determined by the order of the numbers
t, supposed to be arranged in order of increasing or decreasing
magnitude; also there are two senses of description of the curve,
according as t is taken to increase or to diminish. The second
condition is that the aggregate of the points which are determined by
the functional relations must be "continuous." This condition means
that, if any point P determined by a value of t is taken, and any
distance [delta], however small, is chosen, it is possible to find two
points Q, Q´ of the aggregate which are such that (i.) P is between Q
and Q´, (ii.) if R, R´ are any points between Q and Q´ the distance
RR´ is less than [delta]. The meaning of the word "between" in this
statement is fixed by the ordering of the points. Sometimes additional
conditions are imposed upon the functional relations before they are
regarded as defining a curve. An aggregate of points which satisfies
the two conditions stated above is sometimes called a "Jordan curve."
It by no means follows that every curve of this kind has a tangent. In
order that the curve may have a tangent at P it is necessary that, if
any angle [alpha], however small, is specified, a distance [delta] can
be found such that when P is between Q and Q´, and PQ and PQ´ are less
than [delta], the angle RPR´ is less than [alpha] for all pairs of
points R, R´ which are between P and Q, or between P and Q´ (fig. 8).
When this condition is satisfied y is a function of x which has a
differential coefficient. The only way of finding out whether this
condition is satisfied or not is to attempt to form the differential
coefficient. If the quotient of differences [Delta]y/[Delta]x has a
limit when [Delta]x tends to zero, y is a differentiable function of
x, and the limit in question is the differential coefficient. The
derived function, or differential coefficient, of a function [f](x) is
always defined by the formula
d[f](x) [f](x + h) - [f](x)
[f]´(x) = ------- = lim. -------------------.
dx h=0 h
Rules for the formation of differential coefficients in particular
cases have been given in § 11 above. The definition of a differential
coefficient, and the rules of differentiation are quite independent of
any geometrical interpretation, such as that concerning tangents to a
curve, and the tangent to a curve is properly defined by means of the
differential coefficient of a function, not the differential
coefficient by means of the tangent.
Progressive and Regressive Differential Coefficients.
It may happen that the limit employed in defining the differential
coefficient has one value when h approaches zero through positive
values, and a different value when h approaches zero through negative
values. The two limits are then called the "progressive" and
"regressive" differential coefficients. In applications to dynamics,
when x denotes a coordinate and t the time, dx/dt denotes a velocity.
If the velocity is changed suddenly the progressive differential
coefficient measures the velocity just after the change, and the
regressive differential coefficient measures the velocity just before
the change. Variable velocities are properly defined by means of
differential coefficients.
Areas.
Lengths of Curves.
All geometrical limits may be specified in terms similar to those
employed in specifying the tangent to a curve; in difficult cases they
must be so specified. Geometrical intuition may fail to answer the
question of the existence or non-existence of the appropriate limits.
In the last resort the definitions of many quantities of geometrical
import must be analytical, not geometrical. As illustrations of this
statement we may take the definitions of the areas and lengths of
curves. We may not assume that every curve has an area or a length. To
find out whether a curve has an area or not, we must ascertain whether
the limit expressed by [f]ydx exists. When the limit exists the curve
has an area. The definition of the integral is quite independent of
any geometrical interpretation. The length of a curve again is defined
by means of a limiting process. Let P, Q be two points of a curve, and
R1, R2, ... R_(n-1) a set of intermediate points of the curve,
supposed to be described in the sense in which Q comes after P. The
points R are supposed to be reached successively in the order of the
suffixes when the curve is described in this sense. We form a sum of
lengths of chords
PR1 + R1R2 + ... + R_(n-1)Q.
If this sum has a limit when the number of the points R is increased
indefinitely and the lengths of all the chords are diminished
indefinitely, this limit is the length of the arc PQ. The limit is the
same whatever law may be adopted for inserting the intermediate points
R and diminishing the lengths of the chords. It appears from this
statement that the differential element of the arc of a curve is the
length of the chord joining two neighbouring points. In accordance
with the fundamental artifice for forming differentials (§§ 9, 10),
the differential element of arc ds may be expressed by the formula
ds = [root] {(dx)² + (dy)²},
of which the right-hand member is really the measure of the distance
between two neighbouring points on the tangent. The square root must
be taken to be positive. We may describe this differential element as
being so much of the actual arc between two neighbouring points as
need be retained for the purpose of forming the integral expression
for an arc. This is a description, not a definition, because the
length of the short arc itself is only definable by means of the
integral expression. Similar considerations to those used in defining
the areas of plane figures and the lengths of plane curves are
applicable to the formation of expressions for differential elements
of volume or of the areas of curved surfaces.
Constants of Integration.
34. In regard to differential coefficients it is an important theorem
that, if the derived function [f]´(x) vanishes at all points of an
interval, the function [f](x) is constant in the interval. It follows
that, if two functions have the same derived function they can only
differ by a constant. Conversely, indefinite integrals are
indeterminate to the extent of an additive constant.
Higher Differential Coefficients.
35. The differential coefficient dy/dx, or the derived function
[f]´(x), is itself a function of x, and its differential coefficient
is denoted by [f]´´(x) or d²y/dx². In the second of these notations
d/dx is regarded as the symbol of an operation, that of
differentiation with respect to x, and the index 2 means that the
operation is repeated. In like manner we may express the results of n
successive differentiations by [f]^(n)(x) or by d^n·y/dx^n. When the
second differential coefficient exists, or the first is
differentiable, we have the relation
[f](x + h) - 2[f](x) + [f](x - h)
[f]´´(x) = lim. --------------------------------- (i.)
h=0 h²
The limit expressed by the right-hand member of this equation may
exist in cases in which [f]´(x) does not exist or is not
differentiable. The result that, when the limit here expressed can be
shown to vanish at all points of an interval, then [f](x) must be a
linear function of x in the interval, is important.
The relation (i.) is a particular case of the more general relation
[f]^(n)(x) = lim.(h=0) h^-n [[f](x + nh) -n[f] {(x + (n - 1)h}
n(n - 1)
+ -------- [f]{x + (n - 2)h} - ... +(-1)^n [f](x)]. (ii.)
2!
As in the case of relation (i.) the limit expressed by the right-hand
member may exist although some or all of the derived functions
[f]´(x), [f]´´(x), ... [f]^(n-1)(x) do not exist.
Corresponding to the rule iii. of § 11 we have the rule for forming
the nth differential coefficient of a product in the form
d^n(uv) d^n v du d^(n-1)v n(n - 1) d²u d^(n-2)v d^n u
------- = u ----- + n -- -------- + -------- ---- -------- + ... + ----- v,
dx^n dx^n dx dx^(n-1) 1.2 dx² dx^(n-2) dx^n
where the coefficients are those of the expansion of (1 + x)^n in
powers of x (n being a positive integer). The rule is due to Leibnitz,
(1695).
_Differentials of higher orders_ may be introduced in the same way as
the differential of the first order. In general when y = [f](x), the
nth differential d^n·y is defined by the equation
d^n·y = [f]^n(x)(dx)^n,
in which dx is the (arbitrary) differential of x.
Symbols of operation.
When d/dx is regarded as a single symbol of operation the symbol [f]
... dx represents the inverse operation. If the former is denoted by
D, the latter may be denoted by D^-1. D^n means that the operation D
is to be performed n times in succession; D^-n that the operation of
forming the indefinite integral is to be performed n times in
succession. Leibnitz's course of thought (§ 24) naturally led him to
inquire after an interpretation of D^n. where n is not an integer. For
an account of the researches to which this inquiry gave rise,
reference may be made to the article by A. Voss in _Ency. d. math.
Wiss._ Bd. ii. A, 2 (Leipzig, 1889). The matter is referred to as
"fractional" or "generalized" differentiation.
[Illustration: FIG. 9.]
Theorem of Intermediate Value.
36. After the formation of differential coefficients the most
important theorem of the differential calculus is the _theorem of
intermediate value_ ("theorem of mean value," "theorem of finite
increments," "Rolle's theorem," are other names for it). This theorem
may be explained as follows: Let A, B be two points of a curve y =
[f](x) (fig. 9). Then there is a point P between A and B at which the
tangent is parallel to the secant AB. This theorem is expressed
analytically in the statement that if [f]´(x) is continuous between a
and b, there is a value x1 of x between a and b which has the property
expressed by the equation
[f](b) - [f](a)
--------------- = [f]´(x1). (i.)
b - a
The value x1 can be expressed in the form a + [theta](b - a) where
[theta] is a number between 0 and 1.
A slightly more general theorem was given by Cauchy (1823) to the
effect that, if [f]´(x) and F´(x) are continuous between x = a and x =
b, then there is a number [theta] between 0 and 1 which has the
property expressed by the equation
F(b) - F(a) F´{a + [theta](b - a)}
--------------- = ------------------------.
[f](b) - [f](a) [f]´{a + [theta](b - a)}
The theorem expressed by the relation (i.) was first noted by Rolle
(1690) for the case where [f](x) is a rational integral function which
vanishes when x = a and also when x = b. The general theorem was given
by Lagrange (1797). Its fundamental importance was first recognized by
Cauchy (1823). It may be observed here that the theorem of integral
calculus expressed by the equation
_
/ b
F(b) - F(a) = | F´(x) dx
_/ a
follows at once from the definition of an integral and the theorem of
intermediate value.
The theorem of intermediate value may be generalized in the statement
that, if [f](x) and all its differential coefficients up to the nth
inclusive are continuous in the interval between x = a and x = b, then
there is a number [theta] between 0 and 1 which has the property
expressed by the equation
(b - a)² (b - a)^(n-1)
[f](b) = [f](a) + (b - a)[f]´(a) + -------- [f]´´ (a) + ... + ------------- [f]^(n-1)(a)
2! (n - 1)!
(b - a)^n
+ --------- [f]^(n) {a + [theta](b - a)}. (i.)
n!
Taylor's Theorem.
37. This theorem provides a means for computing the values of a
function at points near to an assigned point when the value of the
function and its differential coefficients at the assigned point are
known. The function is expressed by a terminated series, and, when the
remainder tends to zero as n increases, it may be transformed into an
infinite series. The theorem was first given by Brook Taylor in his
_Methodus Incrementorum_ (1717) as a corollary to a theorem concerning
finite differences. Taylor gave the expression for [f](x + z) in terms
of [f](x), [f]´(x), ... as an infinite series proceeding by powers of
z. His notation was that appropriate to the method of fluxions which
he used. This rule for expressing a function as an infinite series is
known as Taylor's theorem. The relation (i.), in which the remainder
after n terms is put in evidence, was first obtained by Lagrange
(1797). Another form of the remainder was given by Cauchy (1823) viz.,
(b - a)^n
--------- (1 - [theta])^(n-1) [f]^n {a + [theta](b - a)}.
(n - 1)!
The conditions of validity of Taylor's expansion in an infinite series
have been investigated very completely by A. Pringsheim (_Math. Ann._
Bd. xliv., 1894). It is not sufficient that the function and all its
differential coefficients should be finite at x = a; there must be a
_neighbourhood_ of a within which Cauchy's form of the remainder tends
to zero as n increases (cf. FUNCTION).
An example of the necessity of this condition is afforded by the
function f(x) which is given by the equation
__ n = [oo]
1 \ (-1)^n 1
[f](x) = ------ + ) ------ ------------ (i.)
1 + x² /__ n = 1 n! 1 + 3^(2n)x²
The sum of the series
x²
[f](0) + x[f]´(0) + -- [f]´´(0) + ... (ii.)
2!
is the same as that of the series
e^-1 - x² e^-3² + x^4 e^(-3^4) - ...
It is easy to prove that this is less than e^-1 when x lies between 0
and 1, and also that f(x) is greater than e^-l when x = 1/[root]3.
Hence the sum of the series (i.) is not equal to the sum of the series
(ii.).
The particular case of Taylor's theorem in which a = 0 is often called
Maclaurin's theorem, because it was first explicitly stated by Colin
Maclaurin in his _Treatise of Fluxions_ (1742). Maclaurin like Taylor
worked exclusively with the fluxional calculus.
Expansions in power series.
Examples of expansions in series had been known for some time. The
series for log (1 + x) was obtained by Nicolaus Mercator (1668) by
expanding (1 + x)^-1 by the method of algebraic division, and
integrating the series term by term. He regarded his result as a
"quadrature of the hyperbola." Newton (1669) obtained the expansion of
sin^-1 x by expanding (l - x²)^-½ by the binomial theorem and
integrating the series term by term. James Gregory (1671) gave the
series for tan^-1 x. Newton also obtained the series for sin x, cos x,
and e^x by reversion of series (1669). The symbol e for the base of
the Napierian logarithms was introduced by Euler (1739). All these
series can be obtained at once by Taylor's theorem. James Gregory
found also the first few terms of the series for tan x and sec x; the
terms of these series may be found successively by Taylor's theorem,
but the numerical coefficient of the general term cannot be obtained
in this way.
Taylor's theorem for the expansion of a function in a power series was
the basis of Lagrange's theory of functions, and it is fundamental
also in the theory of analytic functions of a complex variable as
developed later by Karl Weierstrass. It has also numerous applications
to problems of maxima and minima and to analytical geometry. These
matters are treated in the appropriate articles.
The forms of the coefficients in the series for tan x and sec x can be
expressed most simply in terms of a set of numbers introduced by James
Bernoulli in his treatise on probability entitled _Ars Conjectandi_
(1713). These numbers B1, B2, ... called Bernoulli's numbers, are the
coefficients so denoted in the formula
x x B1 B2 B3
------- = 1 - --- + -- x² - -- x^4 + -- x^6 - ...,
e^x - 1 2 2! 4! 6!
and they are connected with the sums of powers of the reciprocals of
the natural numbers by equations of the type
(2n)! / 1 1 1 \
B_n = ------------------ ( ------ + ------ + ------ + ... ).
2^(2n-1) [pi]^(2n) \ 1^(2n) 2^(2n) 3^(2n) /
The function
m m·m - 1
x^m - --- x^(m-1) + ------- B1 x^(m-2) - ...
2 2!
has been called Bernoulli's function of the mth order by J. L. Raabe
(Crelle's _J. f. Math._ Bd. xlii., 1851). Bernoulli's numbers and
functions are of especial importance in the calculus of finite
differences (see the article by D. Seliwanoff in _Ency. d. math.
Wiss._ Bd. i., E., 1901).
When x is given in terms of y by means of a power series of the form
x = y(C0 + C1y + C2y² + ...) (C0 [not eq.] 0) = y [f]0(y), say,
there arises the problem of expressing y as a power series in x. This
problem is that of _reversion of series_. It can be shown that
provided the absolute value of x is not too great,
__n=[oo] _ _
x \ | x^n d^(n-1) 1 |
y = ------ + ) | --- -------- ----------- |
[f](0) /__n=2 |_ n! dy^(n-1) {[f]0(y)}^n _| y=0
To this problem is reducible that of expanding y in powers of x when x
and y are connected by an equation of the form
y = a + x[f](y),
for which problem Lagrange (1770) obtained the formula
__n=[oo] _ _
\ | x^n d^(n-1) |
y = a + x[f](a) + ) | --- · -------- {[f](a)}^n |.
/__n=2 |_ n! da^(n-1) _|
For the history of the problem and the generalizations of Lagrange's
result reference may be made to O. Stolz, _Grundzüge d. Diff. u. Int.
Rechnung_, T. 2 (Leipzig, 1896).
[Illustration: FIG. 10.]
Indeterminate forms.
38. An important application of the theorem of intermediate value and
its generalization can be made to the problem of evaluating certain
limits. If two functions [phi](x) and [psi](x) both vanish at x = a,
the fraction [phi](x)/[psi](x) may have a finite limit at a. This
limit is described as the limit of an "indeterminate form." Such
indeterminate forms were considered first by de l'Hospital (1696) to
whom the problem of evaluating the limit presented itself in the form
of tracing the curve y = [phi](x)/[psi](x) near the ordinate x = a,
when the curves y = [phi](x) and y = [psi](x) both cross the axis of x
at the same point as this ordinate. In fig. 10 PA and QA represent
short arcs of the curves [phi], [psi], chosen so that P and Q have the
same abscissa. The value of the ordinate of the corresponding point R
of the compound curve is given by the ratio of the ordinates PM, QM.
De l'Hospital treated PM and QM as "infinitesimal," so that the
equations PM : AM =[phi]´(a) and QM : AM = [psi]´(a) could be assumed
to hold, and he arrived at the result that the "true value" of
[phi](a)/[psi](a) is [phi]´(a)/[psi]´(a). It can be proved rigorously
that, if [psi]´(x) does not vanish at x = a, while [phi](a) = 0 and
[psi](a) = 0, then
[phi](x) [phi]´(a)
lim. -------- = ---------.
x=a [psi](x) [psi]´(a)
It can be proved further if that [phi]^m (x) and [psi]^n (x) are the
differential coefficients of lowest order of [phi](x) and [psi](x)
which do not vanish at x = a, and if m = n, then
[phi](x) [phi]^n(a)
lim. -------- = ----------.
x=a [psi](x) [psi]^n(a)
If m > n the limit is zero; but if m < n the function represented by
the quotient [phi](x)/[psi](x) "becomes infinite" at x = a. If the
value of the function at x = a is not assigned by the definition of
the function, the function does not exist at x = a, and the meaning of
the statement that it "becomes infinite" is that it has no finite
limit. The statement does not mean that the function has a value which
we call infinity. There is no such value (see FUNCTION).
Such indeterminate forms as that described above are said to be of the
form 0/0. Other indeterminate forms are presented in the form 0 ×
[oo], or 1^[oo], or [oo]/[oo], or [oo] - [oo]. The most notable of the
forms 1^[oo] is lim.(x=0) (1 + x)^(1/x), which is e. The case in which
[phi](x) and [psi](x) both tend to become infinite at x = a is
reducible to the case in which both the functions tend to become
infinite when x is increased indefinitely. If [phi]´(x) and [psi]´(x)
have determinate finite limits when x is increased indefinitely, while
[phi](x) and [psi](x) are determinately (positively or negatively)
infinite, we have the result expressed by the equation
[phi](x) lim.x=[oo] [psi]´(x)
lim. -------- = --------------------.
x=[oo] [psi](x) lim.x=[oo] [psi](x)
For the meaning of the statement that [phi](x) and [psi](x) are
determinately infinite reference may be made to the article FUNCTION.
The evaluation of forms of the type [oo]/[oo] leads to a scale of
increasing "infinities," each being infinite in comparison with the
preceding. Such a scale is
log x,...x, x²,...x^n,...e^x,...x^x;
each of the limits expressed by such forms as lim.x=[oo]
[phi](x)/[psi](x), where [phi](x) precedes [psi](x) in the scale, is
zero. The construction of such scales, along with the problem of
constructing a complete scale was discussed in numerous writings by
Paul du Bois-Reymond (see in particular, _Math. Ann._ Bd. xi., 1877).
For the general problem of indeterminate forms reference may be made
to the article by A. Pringsheim in _Ency. d. math. Wiss._ Bd. ii., A.
1 (1899). Forms of the type 0/0 presented themselves to early writers
on analytical geometry in connexion with the determination of the
tangents at a double point of a curve; forms of the type [oo]/[oo]
presented themselves in like manner in connexion with the
determination of asymptotes of curves. The evaluation of limits has
innumerable applications in all parts of analysis. Cauchy's _Analyse
algébrique_ (1821) was an epoch-making treatise on limits.
If a function [phi](x) becomes infinite at x = a, and another function
[psi](x) also becomes infinite at x = a in such a way that
[phi](x)/[psi](x) has a finite limit C, we say that [phi](x) and
[psi](x) become "infinite of the same order." We may write [phi](x) =
C[psi](x) + [phi]1(x), where lim. x=a [phi]1(x)/[psi](x) = 0, and thus
[phi]1(x) is of a lower order than [phi](x); it may be finite or
infinite at x = a. If it is finite, we describe C[psi](x) as the
"infinite part" of [phi](x). The resolution of a function which
becomes infinite into an infinite part and a finite part can often be
effected by taking the infinite part to be infinite of the same order
as one of the functions in the scale written above, or in some more
comprehensive scale. This resolution is the inverse of the process of
evaluating an indeterminate form of the type [oo] - [oo].
For example lim.x=0 {(e^x - 1)^-1 - x^-1} is finite and equal to =
½, and the function (e^x - 1)^-1 - x^-1 can be expanded in a power
series in x.
Functions of several variables.
39. The nature of a function of two or more variables, and the meaning
to be attached to continuity and limits in respect of such functions,
have been explained under FUNCTION. The theorems of differential
calculus which relate to such functions are in general the same
whether the number of variables is two or any greater number, and it
will generally be convenient to state the theorems for two variables.
Partial differentiation.
40. Let u or [f](x, y) denote a function of two variables x and y. If
we regard y as constant, u or f becomes a function of one variable x,
and we may seek to differentiate it with respect to x. If the function
of x is differentiable, the differential coefficient which is formed
in this way is called the "partial differential coefficient" of u or f
with respect to x, and is denoted by ðu/ðx or ð[f]/ðx. The symbol "ð"
was appropriated for partial differentiation by C. G. J. Jacobi
(1841). It had before been written indifferently with "d" as a symbol
of differentiation. Euler had written (df/dx) for the partial
differential coefficient of f with respect to x. Sometimes it is
desirable to put in evidence the variable which is treated as
constant, and then the partial differential coefficient is written
"(df/dx)_y" or "(ð[f]/ðx)_y". This course is often adopted by writers
on Thermodynamics. Sometimes the symbols d or ð are dropped, and the
partial differential coefficient is denoted by u_x or [f]_x. As a
definition of the partial differential coefficient we have the formula
ð[f] [f](x + h, y) - f(x, y)
---- = lim. -----------------------.
ðx h=0 h
In the same way we may form the partial differential coefficient with
respect to y by treating x as a constant.
The introduction of partial differential coefficients enables us to
solve at once for a surface a problem analogous to the problem of
tangents for a curve; and it also enables us to take the first step in
the solution of the problem of maxima and minima for a function of
several variables. If the equation of a surface is expressed in the
form z = [f](x, y), the direction cosines of the normal to the surface
at any point are in the ratios ð[f]/ðx : ð[f]/ðy : = 1. If f is a maximum
or a minimum at (x, y), then ð[f]/ðx and ð[f]/ðy vanish at that point.
In applications of the differential calculus to mathematical physics
we are in general concerned with functions of three variables x, y, z,
which represent the coordinates of a point; and then considerable
importance attaches to partial differential coefficients which are
formed by a particular rule. Let F(x, y, z) be the function, P a point
(x, y, z), P´ a neighbouring point (x + [Delta]x, y + [Delta]y, z +
[Delta]z), and let [Delta]s be the length of PP´. The value of F(x, y,
z) at P may be denoted shortly by F(P). A limit of the same nature as
a partial differential coefficient is expressed by the formula
F(P´) = F(P)
lim. ------------,
[Delta]s=0 [Delta]s
in which [Delta]s is diminished indefinitely by bringing P´ up to P,
and P´ is supposed to approach P along a straight line, for example,
the tangent to a curve or the normal to a surface. The limit in
question is denoted by ðF/ðh, in which it is understood that h
indicates a direction, that of PP´. If l, m, n are the direction
cosines of the limiting direction of the line PP´, supposed drawn from
P to P´, then
ðF ðF ðF ðF
-- = l -- + m -- + n --.
ðh ðx ðy ðz
The operation of forming ðF/ðh is called "differentiation with respect
to an axis" or "vector differentiation."
Theorem of the Total Differential.
41. The most important theorem in regard to partial differential
coefficients is the _theorem of the total differential_. We may write
down the equation
[f](a + h, b + k) - [f](a, b) = [f](a + h, b + k) - [f](a, b + k)
+ [f](a, b + k) - [f](a, b).
If [f]x is a continuous function of x when x lies between a and a + h
and y = b + k, and if further [f]y is a continuous function of y when
y lies between b and d + k, there exist values of [Theta] and [eta]
which lie between 0 and 1 and have the properties expressed by the
equations
[f](a + h, b + k) - [f](a, b + k) = h[f]_x (a + [Theta]h, b + k),
[f](a, b + k) - [f](a, b) = k[f]_y (a, b + [eta]k).
Further, [f]x(a + [Theta]h, b + k) and [f]_y (a, b + [eta]k) tend to
the limits [f]_x (a, b) and [f]_y (a, b) when h and k tend to zero,
provided the differential coefficients [f]_x, [f]_y, are continuous
at the point (a, b). Hence in this case the above equation can be
written
[f](a + h, b + k) - [f](a, b) = h[f]_x (a, b) + k[f]_y (a, b) + R,
where
R R
lim. --- = 0 and lim. --- = 0.
h=0, k=0 h h=0, k=0 k
In accordance with the notation of differentials this equation gives
ð[f] ðy
d[f] = ---- dx + -- dy.
ðx ðy
Just as in the case of functions of one variable, dx and dy are
arbitrary finite differences, and d[f] is not the difference of two
values of [f], but is so much of this difference as need be retained
for the purpose of forming differential coefficients.
The theorem of the total differential is immediately applicable to the
differentiation of _implicit functions_. When y is a function of x
which is given by an equation of the form [f](x, y) = 0, and it is
either impossible or inconvenient to solve this equation so as to
express y as an explicit function of x, the differential coefficient
dy/dx can be formed without solving the equation. We have at once
dy ð[f] / ðf
-- = - ---- / --.
dx ðx / ðy
This rule was known, in all essentials, to Fermat and de Sluse before
the invention of the algorithm, of the differential calculus.
An important theorem, first proved by Euler, is immediately deducible
from the theorem of the total differential. If [f](x, y) is a
homogeneous function of degree n then
ð[f] ð[f]
x ---- + y ---- = n[f](x, y).
ðx ðy
The theorem is applicable to functions of any number of variables and
is generally known as _Euler's theorem of homogeneous functions_.
Jacobians.
42. Many problems in which partial differential coefficients occur are
simplified by the introduction of certain determinants called
"Jacobians" or "functional determinants." They were introduced into
Analysis by C. G. J. Jacobi (_J. f. Math._, Crelle, Bd. 22, 1841, p.
319). The Jacobian of u1, u2, ... u_n with respect to x1, x2, ... x_n
is the determinant
| ðu1 ðu1 ðu1 |
| --- --- ... ---- |
| ðx1 ðx2 ðx_n |
| |
| ðu2 ðu2 ðu2 |
| --- --- ... ---- |
| ðx1 ðx2 ðx_n |
| . |
| . |
| . |
| ðu_n ðu_n ðu_n |
| ---- ---- ... ----- |
| ðx1 ðx2 ðx_n |
in which the constituents of the rth row are the n partial
differential coefficients of u_r, with respect to the n variables x.
This determinant is expressed shortly by
ð(u1, u2, ..., u_n)
-------------------.
ð(x1, x2, ..., x_n)
Jacobians possess many properties analogous to those of ordinary
differential coefficients, for example, the following:--
ð(u1, u2, ..., u_n) ð(x1, x2, ..., x_n)
------------------- × ------------------- = 1,
ð(x1, x2, ..., x_n) ð(u1, u2, ..., u_n)
ð(u1, u2, ..., u_n) ð(y1, y2, ..., y_n) ð(u1, u2, ..., u_n)
------------------- × ------------------- = -------------------.
ð(y1, y2, ..., y_n) ð(x1, x2, ..., x_n) ð(x1, x2, ..., x_n)
If n functions (u1, u2, ... u_n) of n variables (x1, x2, ..., x_n) are
not independent, but are connected by a relation [f](u1, u2, ... u_n)
= 0, then
ð(u1, u2, ..., u_n)
------------------- = 0;
ð(x1, x2, ..., x_n)
and, conversely, when this condition is satisfied identically the
functions u1, u2 ..., u_n are not independent.
Interchange of order of differentiations.