Infinitesimal_Calculus

Infinitesimal Calculus. 1\. The infinitesimal calculus is the body of rules and processes by means of which continuously varying magnitudes are dealt with in mathematical analysis. The name “infinitesimal” has been applied to the calculus because most of the leading results were first obtained by means of arguments about “infinitely small” quantities; the “infinitely small” or “infinitesimal” quantities were vaguely conceived as being neither zero nor finite but in some intermediate, nascent or evanescent, state. There was no necessity for this confused conception, and it came to be understood that it can be dispensed with; but the calculus was not developed by its first founders in accordance with logical principles from precisely defined notions, and it gained adherents rather through the impressiveness and variety of the results that could be obtained by using it than through the cogency of the arguments by which it was established. A similar statement might be made in regard to other theories included in mathematical analysis, such, for instance, as the theory of infinite series. Many, perhaps all, of the mathematical and physical theories which have survived have had a similar history—a history which may be divided roughly into two periods: a period of construction, in which results are obtained from partially formed notions, and a period of criticism, in which the fundamental notions become progressively more and more precise, and are shown to be adequate bases for the constructions previously built upon them. These periods usually overlap. Critics of new theories are never lacking. On the other hand, as E. W. Hobson has well said, “pertinent criticism of fundamentals almost invariably gives rise to new construction.” In the history of the infinitesimal calculus the 17th and 18th centuries were mainly a period of construction, the 19th century mainly a period of criticism.

I. Nature of the Calculus.  
  
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Fig. 1.  
  
2\. The guise in which variable quantities presented themselves to the mathematicians of the 17th century was that of the lengths of variable lines. This method of representing variable quantities dates from the 14th century, Geometrical representation of Variable Quantities. when it was employed by Nicole Oresme, who studied and afterwards taught at the Collège de Navarre in Paris from 1348 to 1361. He represented one of two variable quantities, e.g. the time that has elapsed since some epoch, by a length, called the “longitude,” measured along a particular line; and he represented the other of the two quantities, e.g. the temperature at the instant, by a length, called the “latitude,” measured at right angles to this line. He recognized that the variation of the temperature with the time was represented by the line, straight or curved, which joined the ends of all the lines of “latitude.” Oresme’s longitude and latitude were what we should now call the abscissa and ordinate. The same method was used later by many writers, among whom Johannes Kepler and Galileo Galilei may be mentioned. In Galileo’s investigation of the motion of falling bodies (1638) the abscissa OA represents the time during which a body has been falling, and the ordinate AB represents the velocity acquired during that time (see fig. 1). The velocity being proportional to the time, the “curve” obtained is a straight line OB, and Galileo showed that the distance through which the body has fallen is represented by the area of the triangle OAB.

The most prominent problems in regard to a curve were the problem of finding the points at which the ordinate is a maximum or a minimum, the problem of drawing a tangent to the curve at an assigned point, and the problem of The problems of Maxima and Minima, Tangents, and Quadratures. determining the area of the curve. The relation of the problem of maxima and minima to the problem of tangents was understood in the sense that maxima or minima arise when a certain equation has equal roots, and, when this is the case, the curves by which the problem is to be solved touch each other. The reduction of problems of maxima and minima to problems of contact was known to Pappus. The problem of finding the area of a curve was usually presented in a particular form in which it is called the “problem of quadratures.” It was sought to determine the area contained between the curve, the axis of abscissae and two ordinates, of which one was regarded as fixed and the other as variable. Galileo’s investigation may serve as an example. In that example the fixed ordinate vanishes. From this investigation it may be seen that before the invention of the infinitesimal calculus the introduction of a curve into discussions of the course of any phenomenon, and the problem of quadratures for that curve, were not exclusively of geometrical import; the purpose for which the area of a curve was sought was often to find something which is not an area—for instance, a length, or a volume or a centre of gravity.

3\. The Greek geometers made little progress with the problem of tangents, but they devised methods for investigating the problem of quadratures. One of these methods was afterwards called the “method of exhaustions,” and Greek methods. the principle on which it is based was laid down in the lemma prefixed to the 12th book of Euclid’s Elements as follows: “If from the greater of two magnitudes there be taken more than its half, and from the remainder more than its half, and so on, there will at length remain a magnitude less than the smaller of the proposed magnitudes.” The method adopted by Archimedes was more general. It may be described as the enclosure of the magnitude to be evaluated between two others which can be brought by a definite process to differ from each other by less than any assigned magnitude. A simple example of its application is the 6th proposition of Archimedes’ treatise On the Sphere and Cylinder, in which it is proved that the area contained between a regular polygon inscribed in a circle and a similar polygon circumscribed to the same circle can be made less than any assigned area by increasing the number of sides of the polygon. The methods of Euclid and Archimedes were specimens of rigorous limiting processes (see Function). The new problems presented by the analytical geometry and natural philosophy of the 17th century led to new limiting processes.  
  
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Fig. 2.  
  
4\. In the problem of tangents the new process may be described as follows. Let P, P′ be two points of a curve (see fig. 2). Let x, y be the coordinates of P, and x + Δx, y + Δy those of P′. The symbol Δx means “the difference of two Differentiation. x’s” and there is a like meaning for the symbol Δy. The fraction Δy/Δx is the trigonometrical tangent of the angle which the secant PP′ makes with the axis of x. Now let Δx be continually diminished towards zero, so that P′ continually approaches P. If the curve has a tangent at P the secant PP′ approaches a limiting position (see § 33 below). When this is the case the fraction Δy/Δx tends to a limit, and this limit is the trigonometrical tangent of the angle which the tangent at P to the curve makes with the axis of x. The limit is denoted by

dy | .  
---|---  
dx  
  
If the equation of the curve is of the form y = ƒ(x) where ƒ is a functional symbol (see Function), then

Δy | = | ƒ(x + Δx) − ƒ(x) | ,  
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Δx | Δx  
  
and

dy | = lim.Δx = 0 | ƒ(x + Δx) − ƒ(x) | .  
---|---|---|---  
dx | Δx  
  
The limit expressed by the right-hand member of this defining equation is often written

ƒ′(x),

and is called the “derived function” of ƒ(x), sometimes the “derivative” or “derivate” of ƒ(x). When the function ƒ(x) is a rational integral function, the division by Δx can be performed, and the limit is found by substituting zero for Δx in the quotient. For example, if ƒ(x) = x2, we have

ƒ(x + Δx) − ƒ(x) | = | (x + Δx)2 − x2 | = | 2xΔx + (Δx)2 | ,  
---|---|---|---|---|---  
Δx | Δx | Δx  
  
and

ƒ′(x) = 2x.

The process of forming the derived function of a given function is called differentiation. The fraction Δy/Δx is called the “quotient of differences,” and its limit dy/dx is called the “differential coefficient of y with respect to x.” The rules for forming differential coefficients constitute the differential calculus.

The problem of tangents is solved at one stroke by the formation of the differential coefficient; and the problem of maxima and minima is solved, apart from the discrimination of maxima from minima and some further refinements, by equating the differential coefficient to zero (see Maxima and Minima).  
  
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Fig. 3.  
  
5\. The problem of quadratures leads to a type of limiting process which may be described as follows: Let y = ƒ(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 Integration. 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 + Δx that of Q, x′ that of P′, the limit in question might be written

lim. Σba ƒ(x′) Δx,

where the letters a, b written below and above the sign of summation Σ indicate the extreme values of x. This limit is called “the definite integral of ƒ(x) between the limits a and b,” and the notation for it is

∫ba ƒ(x) dx.

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.  
  
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Fig. 4.  
  
6\. The chief of these rules is obtained by regarding the extreme ordinate BD as variable. Let ξ now denote the abscissa of B. The area A of the figure ACDB is represented by the Theorem of Inversion. integral ∫ξa ƒ(x)dx, and it is a function of ξ. Let BD be displaced to B′D′ so that ξ becomes ξ \+ δξ (see fig. 4). The area of the figure ACD′B′ is represented by the integral ∫ξ+Δξa ƒ(x)dx, and the increment ΔA of the area is given by the formula

ΔA = ∫ξ+Δξξ ƒ(x) dx,

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 ΔA is intermediate between HΔξ and hΔξ, and the quotient of differences ΔA/Δξ is intermediate between H and h. If the function ƒ(x) is continuous at B (see Function), then, as Δξ is diminished without limit, H and h tend to BD, or ƒ(ξ), as a limit, and we have

dA | = ƒ(ξ).  
---|---  
dξ  
  
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 ƒ(x) as its differential coefficient. If ƒ(x) is continuous between a and b, we can prove that

A = ∫ba ƒ(x) dx = F(b) − F(a).

When we recognize a function F(x) which has the property expressed by the equation

dF(x) | = ƒ(x),  
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dx  
  
we are said to integrate the function ƒ(x), and F(x) is called the indefinite integral of ƒ(x) with respect to x, and is written

∫ ƒ(x) dx.

7\. In the process of § 4 the increment Δy is not in general equal to the product of the increment Δx and the derived Differentials. function ƒ′(x). In general we can write down an equation of the form

Δy = ƒ′(x) Δx + R,

in which R is different from zero when Δx is different from zero; and then we have not only

lim.Δx=0 R = 0,

but also

lim.Δx=0 | R | = 0.  
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Δx  
  
We may separate Δy into two parts: the part ƒ′(x)Δx and the part R. The part ƒ′(x)Δx alone is useful for forming the differential coefficient, and it is convenient to give it a name. It is called the differential of ƒ(x), and is written dƒ(x), or dy when y is written for ƒ(x). When this notation is adopted dx is written instead of Δx, and is called the “differential of x,” so that we have

dƒ(x) = ƒ′(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 ƒ(x), so that we have

dF(x) | = ƒ(x),  ∫ba ƒ(x) dx = F(b) − F(a).  
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dx  
  
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, ... xn−1 be the abscissae of the points in order. The integral is the limit of the sum

ƒ(a) (x1 − a) + ƒ(x1) (x2 − x1) + ... + ƒ(xr) (xr+1 − xr) \+ ... + ƒ(xn−1) (b − xn−1),

every term of which is a differential of the form ƒ(x)dx. Further the integral is equal to the sum of differences

{F(x1) − F(a)} + {F(x2) − F(x1)} + ... + {F(xr+1) − F(xr)} \+ ... + {F(b) − F(xn−1)},

for this sum is F(b) − F(a). Now the difference F(xr+1) − F(xr) is not equal to the differential ƒ(xr) (xr+1 − xr), 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.

8\. The notation of the infinitesimal calculus is intimately bound up with the notions of differentials and sums of elements. The letter Notation.  
  
Fundamental Artifice. “d” is the initial letter of the word differentia (difference) and the symbol ∫ 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).

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 Δx and Δy, by substituting x + Δx for x and y + Δy for y. If there is a differential coefficient of y with respect to x, then Δy can be expressed in the form φ.Δx + R, where lim.Δx=0 (R/Δx) = 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 φ·Δx, or φ.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

lim.MN = 0 | MN · QR | = 0.  
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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.

10\. This principle is usually expressed by introducing the notion of orders of small quantities. If x, y are two variable numbers which are Orders of small quantities. 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

lim.x=0 | x | = 0,  
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y  
  
or

lim.x=0 | y | = 0,  
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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.

11\. By the aid of this artifice, or directly by carrying out the appropriate limiting processes, we may obtain the Rules of Differentiation. 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 | .  
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dx | dx | dx | dx  
  
(iii.) For a product uv

d(uv) | = u | dv | \+ v | du | .  
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dx | dx | dx  
  
(iv.) For a quotient u/v

d(u/v) | = ( v | du | − u | dv | ) / v2.  
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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 | .  
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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  
---|---  
xn | nxn−1  
for all values of n  
logax | x-1 logae  
ax | ax logea  
sin x | cos x  
cos x | −sin x  
sin−1x | (1 − x2)−1/2  
tan−1x | (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.

12\. The processes of the integral calculus consist largely in transformations Indefinite Integrals. 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:—

ƒ(x) | ∫ƒ(x)dx  
---|---  
xn | xn+1 / (n + 1)  
for all values of n except −1  
1/x | loge x  
eax | a−1eax  
cos x | sin x  
sin x | −cos x  
(a2 − x2)−1/2 | sin−1 (x/a)  
1 / (a2 \+ x2) | (1/a) tan−1 (x/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.

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 Kepler’s methods of Integration. (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.  
  
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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 adφ, where a is the radius of the auxiliary circle, and φ 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

∫φ0 (a + ae cos φ) dφ,

where e is the eccentricity. Kepler evaluated the sum as proportional to φ \+ e sin φ.

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.

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 Logarithms. 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.

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, Cavalieri’s Indivisibles. 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

∫x0 xm dx = | xm+1 | , (m integral).  
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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.

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 Successors of Cavalieri. 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 ∫10 xmdx by the process of forming the sum

0m \+ 1m \+ 2m \+ ... (n − 1)m | ,  
---|---  
nm+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 xm by forming this sum was found also by Fermat (1636), who stated expressly that he Fermat’s method of Integration. 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.

The result that the problem of quadratures could be solved for any curve whose equation could be expressed in the form

y = xm (m ≠ −1),

or in the form

y = a1 xm1 \+ a2 xm2 \+ ... + an xmn,

where none of the indices is equal to −1, was used by John Various Integrations. 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.

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 Integration before the Integral Calculus. 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).

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 Fermat’s methods of Differentiation. 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.  
  
---  
Fig. 6.  
  
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.

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 φ(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.  
  
---  
Fig. 7.  
  
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 Barrow’s Differential Triangle. 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.”

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 = ƒ(x) is known, so that the area up to the ordinate x is given by F(x), the curve Barrow’s Inversion-theorem. 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.

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 Nature of the discovery called the Infinitesimal Calculus. 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.

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 Newton’s investigations. 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.

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 Newton’s method of Series. 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 = xm(m ± −1) in the form z = xm+1/(m + 1). In the proof he transformed this formula to the form zn = cn xp, 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

zn \+ nzn−1 ov + ... = cn (xp \+ pxp−1 o + ...),

from which he deduced the relation

nzn−1 v = cn pxp−1

by omitting the equal terms zn and cnxp and dividing the remaining terms by o, tacitly putting o = 0 after division. This relation is the same as v = xm. Newton pointed out that, conversely, from the relation v = xm the relation z = xm+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 axm; 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 = xn (a + bxm)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.

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 Newton’s method of Fluxions. 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 ẋo. 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 + ẋo and y +ẏo 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

dxm | = mxm−1, (m integral).  
---|---  
dx  
  
a result which Newton obtained by expanding (x = ẋo)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

∫ xm dx = | xm+1 | .  
---|---|---  
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.

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 Publication of the Fluxional Notation. of which ẋ, ẏ, ̇z, ... are the fluxions, so ẋ, ẏ, ̇z, ... can be treated as fluents of which the fluxions may be denoted by ẍ, ̈y, ̈z,... In like manner the fluxions of these may be denoted by ẍ, ̈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, ẋ, ẍ, ... 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.

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 Retarded Publication of the method of Fluxions. 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.”

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 Leibnitz’s course of discovery. 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

1⁄4π = 1 − 1⁄3 \+ 1⁄5 − 1⁄7 \+ ...

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 “∫ ydx,” but within a day or two he wrote “∫ y”. He regarded the symbol “∫” 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 “∫” represents “sum,” “d” represents “difference.” His notation appears to have been practically settled before the end of 1675, for in November he wrote ∫ ydy = ½ y2, just as we do now.

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 Correspondence of Newton and Leibnitz. 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.

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. Leibnitz’s Differential Calculus. 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(xm) = mxm−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.

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. Development of the Calculus. This was the first publication of the notation ∫ 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 “∫.” 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 (ax) of becoming infinitely great in comparison with any power (xn) 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.

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 Dispute concerning Priority. 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.”

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 British and Continental Schools of Mathematics. 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.

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, Oppositions to the calculus. 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 The “Analyst” controversy. 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 Cauchy’s method of limits. 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.

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, Arithmetical basis of modern analysis. 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.  
  
---  
Fig. 8.  
  
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 Geometrical limits. 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 δ, 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 δ. 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 Tangents. may have a tangent at P it is necessary that, if any angle α, however small, is specified, a distance δ can be found such that when P is between Q and Q′, and PQ and PQ′ are less than δ, the angle RPR′ is less than α 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 Δy/Δx has a limit when Δx 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 ƒ(x) is always defined by the formula

ƒ′(x) = | dƒ(x) | = lim.h=0 | ƒ(x + h) − ƒ(x) | .  
---|---|---|---|---  
dx | 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.

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 Progressive and Regressive Differential Coefficients. 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.

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 Areas. 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 ƒydx 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, ... Rn−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 \+ ... + Rn−1Q.

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 Lengths of Curves. 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 = √ { (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.

34\. In regard to differential coefficients it is an important theorem Constants of Integration. that, if the derived function ƒ′(x) vanishes at all points of an interval, the function ƒ(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.

35\. The differential coefficient dy/dx, or the derived function ƒ′(x), is itself a function of x, and its differential coefficient is denoted Higher Differential Coefficients. by ƒ″(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 ƒ(n)(x) or by dny/dxn. When the second differential coefficient exists, or the first is differentiable, we have the relation

ƒ″(x) = lim.h=0 | ƒ(x + h) − 2ƒ(x) + ƒ(x − h) | .  
---|---|---  
h2  
  
(i.)

The limit expressed by the right-hand member of this equation may exist in cases in which ƒ′(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 ƒ(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

ƒ(n)(x) = lim.h=0 h−n [ ƒ(x + nh) − nf {(x + (n − 1) h }

+ | n (n − 1) | ƒ {x + (n − 2) h } − ... + (−1)n ƒ(x) ].  
---|---|---  
2!  
  
(ii.)

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 ƒ′(x), ƒ″(x), ... ƒ(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

dn(uv) | = u | dnv | \+ n | du |   | dn−1v | + | n(n − 1) |   | d2u |   | dn−2v | \+ ... + | dnu | v,  
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---  
dxn | dxn | dx | dxn−1 | 1·2 | dx2 | dxn−2 | dxn  
  
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 = ƒ(x), the nth differential dny is defined by the equation

dny = ƒ(n) (x) (dx)n,

in which dx is the (arbitrary) differential of x.

When d/dx is regarded as a single symbol of operation the symbol ƒ ... dx represents the inverse operation. If the former is denoted by D, the latter may be denoted by D−1. Dn means that Symbols of operation. 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 Dn. 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.  
  
---  
Fig. 9.  
  
36\. After the formation of differential coefficients the most important theorem of the differential calculus is the theorem of intermediate value 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 = ƒ(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 ƒ′(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

ƒ(b) − ƒ(a) | = ƒ′(x1).  
---|---  
b − a  
  
(i.)

The value x1 can be expressed in the form a + θ(b − a) where θ is a number between 0 and 1.

A slightly more general theorem was given by Cauchy (1823) to the effect that, if ƒ′(x) and F′(x) are continuous between x = a and x = b, then there is a number θ between 0 and 1 which has the property expressed by the equation

F(b) − F(a) | = | F′ {a + θ(b − a) } | .  
---|---|---|---  
ƒ(b) − ƒ(a) | ƒ′ {a + θ(b − a) }  
  
The theorem expressed by the relation (i.) was first noted by Rolle (1690) for the case where ƒ(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

F(b) − F(a) = ∫ba F′(x) dx

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 ƒ(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 θ between 0 and 1 which has the property expressed by the equation

ƒ(b) = ƒ(a) + (b − a) ƒ′(a) + | (b − a)2 | ƒ″(a) + ... + | (b − a)n−1 | ƒ(n−1)(a)  
---|---|---|---|---  
2! | (n − 1)!  
+ | (b − a)n | ƒ(n) {a + θ (b − a) }.  
---|---|---  
n!  
  
(ii.)

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 Taylor’s Theorem. 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 ƒ(x + z) in terms of ƒ(x), ƒ′(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 − θ)n−1 ƒn {a + θ(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

ƒ(x) = | 1 | \+ Σn=∞n=1 | (−1)n |   | 1 | .  
---|---|---|---|---|---|---  
1 + x2 | n! |  1 + 32n x2  
  
(i.)

The sum of the series

ƒ(0) + xƒ′(0) + | x2 | ƒ″(0)+ ...  
---|---|---  
2!  
  
(ii.)

is the same as that of the series

e−1 − x2 e−32 \+ x4 e−34 − ...

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/√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.

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 Expansions in power series. regarded his result as a “quadrature of the hyperbola.” Newton (1669) obtained the expansion of sin−1x by expanding (l − x2)−1/2 by the binomial theorem and integrating the series term by term. James Gregory (1671) gave the series for tan−1x. Newton also obtained the series for sin x, cos x, and ex 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 | = 1 − | x | + | B1 | x2 − | B2 | x4 + | B3 | x6 − ...,  
---|---|---|---|---|---|---|---|---|---  
ex − 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

Bn = | (2n)! | ( | 1 | + | 1 | + | 1 | \+ ... ).  
---|---|---|---|---|---|---|---|---  
22n−1 π2n | 12n | 22n | 32n  
  
The function

xm − | m | xm−1 + | m·m − 1 | B1 xm−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 ≠ 0) = yƒ0(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,

y = | x | \+ Σn=∞n=2 [ | xn | · | dn−1 |   | 1 | ]y=0  
---|---|---|---|---|---|---|---|---  
ƒ0(0) | n! | dyn−1 | {ƒ0(y)}n  
  
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 + xƒ(y),

for which problem Lagrange (1770) obtained the formula

y = a + xƒ(a) + Σn=∞n=2 [ | n | · | dn−1 | {ƒ(a)}n ].  
---|---|---|---|---  
n! | dan−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).  
  
---  
Fig. 10.  
  
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 φ(x) and ψ(x) both vanish at x = a, the fraction φ(x)/ψ(x) may have a finite Indeterminate forms. 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 = φ(x)/ψ(x) near the ordinate x = a, when the curves y = φ(x) and y = ψ(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 φ, ψ, 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 =φ’(a) and QM : AM = ψ′(a) could be assumed to hold, and he arrived at the result that the “true value” of φ(a)/ψ(a) is φ′(a)/ψ′(a). It can be proved rigorously that, if ψ′(x) does not vanish at x = a, while φ(a) = 0 and ψ(a) = 0, then

lim.x=a | φ(x) | = | φ′(a) | .  
---|---|---|---|---  
ψ(x) | ψ′(a)  
  
It can be proved further if that φm(x) and ψn(x) are the differential coefficients of lowest order of φ(x) and ψ(x) which do not vanish at x = a, and if m = n, then

lim.x=a | φ(x) | = | φn(a) | .  
---|---|---|---|---  
ψ(x) | ψn(a)  
  
If m > n the limit is zero; but if m < n the function represented by the quotient φ(x)/ψ(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 × ∞, or 1∞, or ∞/∞, or ∞ − ∞. The most notable of the forms 1∞ is lim.x=0(1 + x)1/x, which is e. The case in which φ(x) and ψ(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 φ′(x) and ψ′(x) have determinate finite limits when x is increased indefinitely, while φ(x) and ψ(x) are determinately (positively or negatively) infinite, we have the result expressed by the equation

lim.x=∞ | φ(x) | = | lim.x=∞ φ′(x) | .  
---|---|---|---|---  
ψ(x) | lim.x=∞ ψ′(x)  
  
For the meaning of the statement that φ(x) and ψ(x) are determinately infinite reference may be made to the article Function. The evaluation of forms of the type ∞/∞ leads to a scale of increasing “infinities,” each being infinite in comparison with the preceding. Such a scale is

log x, ... x, x2, ... xn, ... ex, ... xx;

each of the limits expressed by such forms as lim.x=∞ φ(x)/ψ(x), where φ(x) precedes ψ(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 ∞/∞ 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 φ(x) becomes infinite at x = a, and another function ψ(x) also becomes infinite at x = a in such a way that φ(x)/ψ(x) has a finite limit C, we say that φ(x) and ψ(x) become “infinite of the same order.” We may write φ(x) = Cψ(x) + φ1(x), where lim.x=aφ1(x)/ψ(x) = 0, and thus φ1(x) is of a lower order than φ(x); it may be finite or infinite at x = a. If it is finite, we describe Cψ(x) as the “infinite part” of φ(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 ∞ − ∞.

For example lim.x=0{(ex − 1)−1 − x−1} is finite and equal to = ½, and the function (ex − 1)−1 − x−1 can be expanded in a power series in x.

39\. Functions of several variables. 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.

40\. Let u or ƒ (x, y) denote a function of two variables x and y. If we regard y as constant, u or ƒ 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 Partial differentiation. coefficient which is formed in this way is called the “partial differential coefficient” of u or ƒ with respect to x, and is denoted by ∂u/∂x or ∂ƒ/∂x. The symbol “∂” 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 (dƒ/dx) for the partial differential coefficient of ƒ 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 “(dƒ/dx)y” or “(∂ƒ/∂x)y”. This course is often adopted by writers on Thermodynamics. Sometimes the symbols d or ∂ are dropped, and the partial differential coefficient is denoted by ux or ƒx. As a definition of the partial differential coefficient we have the formula

∂ƒ | = lim.h=0 |  ƒ(x + h, y) − ƒ(x, y) | .  
---|---|---|---  
∂x | 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 = ƒ(x, y), the direction cosines of the normal to the surface at any point are in the ratios ∂ƒ/∂x : ∂ƒ/∂y : = 1. If f is a maximum or a minimum at (x, y), then ∂ƒ/∂x and ∂ƒ/∂y 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 + Δx, y + Δy, z + Δz), and let Δs 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

lim.Δs=0 | F(P′) = F(P) | ,  
---|---|---  
Δs  
  
in which Δs 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 ∂F/∂h, 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

∂F |  | ∂F |  | ∂F |  | ∂F | .  
---|---|---|---|---|---|---|---  
∂h | ∂x | ∂y | ∂z  
  
The operation of forming ∂F/∂h is called “differentiation with respect to an axis” or “vector differentiation.”

41\. The most important theorem in regard to partial differential coefficients is the theorem of the total differential. We may write down the equation

ƒ(a + h, b + k) − ƒ(a, b) = ƒ(a + h, b + k) − ƒ(a, b + k) \+ ƒ(a, b + k) − ƒ(a, b).

If Theorem of the Total Differential. ƒx is a continuous function of x when x lies between a and a + h and y = b + k, and if further ƒy is a continuous function of y when y lies between b and d + k, there exist values of θ and η which lie between 0 and 1 and have the properties expressed by the equations

ƒ(a + h, b + k) − ƒ(a, b + k) = hƒx(a + θh, b + k),  
ƒ(a, b + k) − ƒ(a, b) = kƒy(a, b + ηk).

Further, ƒx(a + θh, b + k) and ƒy(a, b + ηk) tend to the limits ƒx(a, b) and ƒy(a, b) when h and k tend to zero, provided the differential coefficients ƒx, ƒy, are continuous at the point (a, b). Hence in this case the above equation can be written

ƒ(a + h, b + k) − ƒ(a, b) = hƒx(a, b) + kƒy(a, b) + R,

where

lim.h=0, k=0 | R | = 0 and lim.h=0, k=0 | R | = 0.  
---|---|---|---|---  
h | k  
  
In accordance with the notation of differentials this equation gives

dƒ = | ∂f | dx + | ∂y | dy.  
---|---|---|---|---  
∂x | ∂y  
  
Just as in the case of functions of one variable, dx and dy are arbitrary finite differences, and dƒ is not the difference of two values of ƒ, 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 ƒ(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 | = − | ∂ƒ | / | ∂ƒ | .  
---|---|---|---|---|---  
dx | ∂x | ∂y  
  
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 ƒ(x, y) is a homogeneous function of degree n then

x | ∂f | \+ y | ∂f |  = nƒ(x, y).  
---|---|---|---|---  
∂x | ∂y  
  
The theorem is applicable to functions of any number of variables and is generally known as Euler’s theorem of homogeneous functions.

42\. Jacobians.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, ... un with respect to x1, x2, ... xn is the determinant

| ∂u1 |  | ∂u1 | ... | ∂u1  
---|---|---|---|---  
∂x1 | ∂x2 | ∂xn  
| ∂u2 |  | ∂u2 | ... | ∂u2  
---|---|---|---|---  
∂x1 | ∂x2 | ∂xn  
:  
| ∂un |  | ∂un | ... | ∂un  
---|---|---|---|---  
∂x1 | ∂x2 | ∂xn  
  
in which the constituents of the rth row are the n partial differential coefficients of ur, with respect to the n variables x. This determinant is expressed shortly by

∂(u1, u2, ..., un) | .  
---|---  
∂(x1, x2, ..., xn)  
  
Jacobians possess many properties analogous to those of ordinary differential coefficients, for example, the following:—

∂(u1, u2, ..., un) | × | ∂(x1, x2, ..., xn) | = 1, |   |    
---|---|---|---|---|---  
∂(x1, x2, ..., xn) | ∂(u1, u2, ..., un) |    
∂(u1, u2, ..., un) | × | ∂(y1, y2, ..., yn) | = | ∂(u1, u2, ..., un) | .  
∂(y1, y2, ..., yn) | ∂(x1, x2, ..., xn) | ∂(x1, x2, ..., xn)  
  
If n functions (u1, u2, ... un) of n variables (x1, x2, ..., xn) are not independent, but are connected by a relation ƒ(u1, u2, ... un) = 0, then

∂(u1, u2, ..., un) | = 0;  
---|---  
∂(x1, x2, ..., xn)  
  
and, conversely, when this condition is satisfied identically the functions u1, u2 ..., un are not independent.

43\. Partial differential coefficients of the second and higher Interchange of order of differentiations. orders can be formed in the same way as those of the first order. For example, when there are two variables x, y, the first partial derivatives ∂ƒ/∂x and ∂ƒ/∂y are functions of x and y, which we may seek to differentiate partially with respect to x or y. The most important theorem in relation to partial differential coefficients of orders higher than the first is the theorem that the values of such coefficients do not depend upon the order in which the differentiations are performed. For example, we have the equation

∂ | ( | ∂ƒ | ) = | ∂ | ( | ∂ƒ | ).  
---|---|---|---|---|---|---|---  
∂x | ∂y | ∂y | ∂x  
  
(i.)

This theorem is not true without limitation. The conditions for its validity have been investigated very completely by H. A. Schwarz (see his Ges. math. Abhandlungen, Bd. 2, Berlin, 1890, p. 275). It is a sufficient, though not a necessary, condition that all the differential coefficients concerned should be continuous functions of x, y. In consequence of the relation (i.) the differential coefficients expressed in the two members of this relation are written

∂2ƒ | or | ∂2ƒ | .  
---|---|---|---  
∂x∂y | ∂y∂x  
  
The differential coefficient

∂nƒ | ,  
---|---  
∂xp ∂yq ∂zr  
  
in which p + g + r = n, is formed by differentiating p times with respect to x, q times with respect to y, r times with respect to z, the differentiations being performed in any order. Abbreviated notations are sometimes used in such forms as

ƒ xp yq zr or  ƒ | (p, q, r) | .  
---|---|---  
x, y, z  
  
Differentials of higher orders are introduced by the defining equation

dnƒ = ( dx | ∂ | \+ dy | ∂ | ) | n | ƒ  
---|---|---|---|---|---|---  
∂x | ∂y |    
= (dx)n |  ∂nƒ | \+ n(dx)n−1 dy | ∂nƒ | \+ ...  
---|---|---|---|---  
∂xn | ∂xn−1 ∂y  
  
in which the expression (dx·∂/∂x + dy·∂/∂y)n is developed by the binomial theorem in the same way as if dx·∂/∂x and dy·∂/∂y were numbers, and (∂/∂x)r·(∂/∂y)n−r ƒ is replaced by ∂nƒ/∂xr ∂yn−r. When there are more than two variables the multinomial theorem must be used instead of the binomial theorem.

The problem of forming the second and higher differential coefficients of implicit functions can be solved at once by means of partial differential coefficients, for example, if ƒ(x, y) = 0 is the equation defining y as a function of x, we have

d2y | = ( | ∂ƒ | ) | −3 | { ( | ∂ƒ | )2 | ∂2ƒ | − 2 | ∂ƒ | · | ∂ƒ | · | ∂2ƒ | \+ ( | ∂ƒ | )2 | ∂2ƒ | }.  
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---  
dx2 | ∂y |   | ∂y | ∂x2 | ∂x | ∂y | ∂x∂y | ∂x | ∂y2  
  
The differential expression Xdx + Ydy, in which both X and Y are functions of the two variables x and y, is a total differential if there exists a function ƒ of x and y which is such that

∂ƒ/∂x = X,  ∂ƒ/∂y = Y.

When this is the case we have the relation

∂Y/∂x = ∂X/∂y.

(ii.)

Conversely, when this equation is satisfied there exists a function ƒ which is such that

dƒ= Xdx + Ydy.

The expression Xdx + Ydy in which X and Y are connected by the relation (ii.) is often described as a “perfect differential.” The theory of the perfect differential can be extended to functions of n variables, and in this case there are ½n(n − 1) such relations as (ii.).

In the case of a function of two variables x, y an abbreviated notation is often adopted for differential coefficients. The function being denoted by z, we write

p, q, r, s, t for | ∂z | , | ∂z | , | ∂2z | , | ∂2z | , | ∂2z | .  
---|---|---|---|---|---|---|---|---|---|---  
∂x | ∂y | ∂x2 | ∂x∂y | ∂y2  
  
Partial differential coefficients of the second order are important in geometry as expressing the curvature of surfaces. When a surface is given by an equation of the form z = ƒ(x, y), the lines of curvature are determined by the equation

{ (l + q2)s − pqt} (dy)2 \+ { (1 + q2)r − (1 + p2)t } dxdy − { (1 + p2)s − pqr} (dx)2 = 0,

and the principal radii of curvature are the values of R which satisfy the equation

R2(rt − s2) − R { (1 + q2)r − 2pqs + (1 + p2)t } √(1 + p2 \+ q2) \+ (1 + p2 \+ q2)2 = 0.

44\. Change of variables.The problem of change of variables was first considered by Brook Taylor in his Methodus incrementorum. In the case considered by Taylor y is expressed as a function of z, and z as a function of x, and it is desired to express the differential coefficients of y with respect to x without eliminating z. The result can be obtained at once by the rules for differentiating a product and a function of a function. We have

dy | = | dy | · | dz | , |   |   |   |   |   |   |   |   |   |    
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---  
dx | dz | dx |   |   |   |   |    
d2y | = | dy | · | d2z | + | d2y | · ( | dz | )2, |   |   |   |   |   |    
dx2 | dz | dx2 | dz2 | dx |   |   |    
d3y | = | dy | · | d3z | \+ 3 | d2y | · | dz | · | d2z | + | d3y | · ( | dz | )3,  
dx3 | dz | dx3 | dz2 | dx | dx2 | dz3 | dx  
. . . . . . . . . . . . . . . . . . . . .  
  
The introduction of partial differential coefficients enables us to deal with more general cases of change of variables than that considered above. If u, v are new variables, and x, y are connected with them by equations of the type

x = ƒ1(u, v),  y = ƒ2(u, v),

(i.)

while y is either an explicit or an implicit function of x, we have the problem of expressing the differential coefficients of various orders of y with respect to x in terms of the differential coefficients of v with respect to u. We have

dy | = ( | ∂ƒ2 | + | ∂ƒ2 |   | dv | ) / ( | ∂ƒ1 | + | ∂ƒ1 |   | dv | )  
---|---|---|---|---|---|---|---|---|---|---|---|---|---  
dx | ∂u | ∂v | du | ∂u | ∂v | du  
  
by the rule of the total differential. In the same way, by means of differentials of higher orders, we may express d2y/dx2, and so on.

Equations such as (i.) may be interpreted as effecting a transformation by which a point (u, v) is made to correspond to a point (x, y). The whole theory of transformations, and of functions, or differential expressions, which remain invariant under groups of transformations, has been studied exhaustively by Sophus Lie (see, in particular, his Theorie der Transformationsgruppen, Leipzig, 1888-1893). (See also Differential Equations and Groups).

A more general problem of change of variables is presented when it is desired to express the partial differential coefficients of a function V with respect to x, y, ... in terms of those with respect to u, v, ..., where u, v, ... are connected with x, y, ... by any functional relations. When there are two variables x, y, and u, v are given functions of x, y, we have

∂V | = | ∂V |   | ∂u | + | ∂V |   | ∂v | ,  
---|---|---|---|---|---|---|---|---|---  
∂x | ∂u | ∂x | ∂v | ∂x  
∂V | = | ∂V |   | ∂u | + | ∂V |   | ∂v | ,  
∂y | ∂u | ∂y | ∂v | ∂y  
  
and the differential coefficients of higher orders are to be formed by repeated applications of the rule for differentiating a product and the rules of the type

∂ | = | ∂u |   | ∂ | + | ∂v |   | ∂ | .  
---|---|---|---|---|---|---|---|---|---  
∂x | ∂x | ∂u | ∂x | ∂v  
  
When x, y are given functions of u, v, ... we have, instead of the above, such equations as

∂V | = | ∂V |   | ∂x | + | ∂V |   | ∂y | ;  
---|---|---|---|---|---|---|---|---|---  
∂u | ∂x | ∂u | ∂y | ∂u  
  
and ∂V/∂x, ∂V/∂y can be found by solving these equations, provided the Jacobian ∂(x, y)/∂(u, v) is not zero. The generalization of this method for the case of more than two variables need not detain us.

In cases like that here considered it is sometimes more convenient not to regard the equations connecting x, y with u, v as effecting a point transformation, but to consider the loci u = const., v = const. as two “families” of curves. Then in any region of the plane of (x, y) in which the Jacobian ∂(x, y)/∂(u, v) does not vanish or become infinite, any point (x, y) is uniquely determined by the values of u and v which belong to the curves of the two families that pass through the point. Such variables as u, v are then described as “curvilinear coordinates” of the point. This method is applicable to any number of variables. When the loci u = const., ... intersect each other at right angles, the variables are “orthogonal” curvilinear coordinates. Three-dimensional systems of such coordinates have important applications in mathematical physics. Reference may be made to G. Lamé, Leçons sur les coordonnées curvilignes (Paris, 1859), and to G. Darboux, Leçons sur les coordonnées curvilignes et systèmes orthogonaux (Paris, 1898).

When such a coordinate as u is connected with x and y by a functional relation of the form ƒ(x, y, u) = 0 the curves u = const. are a family of curves, and this family may be such that no two curves of the family have a common point. When this is not the case the points in which a curve ƒ(x, y, u) = 0 is intersected by a curve ƒ(x, y, u + Δu) = 0 tend to limiting positions as Δu is diminished indefinitely. The locus of these limiting positions is the “envelope” of the family, and in general it touches all the curves of the family. It is easy to see that, if u, v are the parameters of two families of curves which have envelopes, the Jacobian ∂(x, y)/∂(u, v) vanishes at all points on these envelopes. It is easy to see also that at any point where the reciprocal Jacobian ∂(u, v)/∂(x, y) vanishes, a curve of the family u touches a curve of the family v.

If three variables x, y, z are connected by a functional relation ƒ(x, y, z) = 0, one of them, z say, may be regarded as an implicit function of the other two, and the partial differential coefficients of z with respect to x and y can be formed by the rule of the total differential. We have

∂z | = − | ∂ƒ | / | ∂ƒ | ,  | ∂z | = − | ∂ƒ | / | ∂ƒ | ;  
---|---|---|---|---|---|---|---|---|---|---|---  
∂x | ∂x | ∂z | ∂y | ∂y | ∂z  
  
and there is no difficulty in proceeding to express the higher differential coefficients. There arises the problem of expressing the partial differential coefficients of x with respect to y and z in terms of those of z with respect to x and y. The problem is known as that of “changing the dependent variable.” It is solved by applying the rule of the total differential. Similar considerations are applicable to all cases in which n variables are connected by fewer than n equations.

45\. Extension of Taylor’s theorem.Taylor’s theorem can be extended to functions of several variables. In the case of two variables the general formula, with a remainder after n terms, can be written most simply in the form

ƒ(a + h, b + k) = ƒ(a, b) + dƒ(a, b) + | 1 | d2 ƒ(a, b) + ...  
---|---|---  
2!  
+ | 1 | dn−1 ƒ(a, b) + | 1 | dn ƒ(a + θh, b + θk),  
---|---|---|---|---  
(n − 1)! | n!  
  
in which

dr ƒ(a, b) = [ ( h | ∂ | \+ k | ∂ | )rƒ(x, y) ] |   | ,  
---|---|---|---|---|---|---  
∂x | ∂y | x=a, y=b  
  
and

dn ƒ(a + θh, b + θk) = [ ( h | ∂ | \+ k | ∂ | )nƒ(x, y) ] |   | .  
---|---|---|---|---|---|---  
∂x | ∂y | x=a+θh, y=b+θk  
  
The last expression is the remainder after n terms, and in it θ denotes some particular number between 0 and 1. The results for three or more variables can be written in the same form. The extension of Taylor’s theorem was given by Lagrange (1797); the form written above is due to Cauchy (1823). For the validity of the theorem in this form it is necessary that all the differential coefficients up to the nth should be continuous in a region bounded by x = a ± h, y = b ± k. When all the differential coefficients, no matter how high the order, are continuous in such a region, the theorem leads to an expansion of the function in a multiple power series. Such expansions are just as important in analysis, geometry and mechanics as expansions of functions of one variable. Among the problems which are solved by means of such expansions are the problem of maxima and minima for functions of more than one variable (see Maxima and Minima).

46\. Plane curves.In treatises on the differential calculus much space is usually devoted to the differential geometry of curves and surfaces. A few remarks and results relating to the differential geometry of plane curves are set down here.

(i.) If ψ denotes the angle which the radius vector drawn from the origin makes with the tangent to a curve at a point whose polar coordinates are r, θ and if p denotes the perpendicular from the origin to the tangent, then

cos ψ = dr/ds,  sin ψ = rdθ/ds = p/r,

where ds denotes the element of arc. The curve may be determined by an equation connecting p with r.

(ii.) The locus of the foot of the perpendicular let fall from the origin upon the tangent to a curve at a point is called the pedal of the curve with respect to the origin. The angle ψ for the pedal is the same as the angle ψ for the curve. Hence the (p, r) equation of the pedal can be deduced. If the pedal is regarded as the primary curve, the curve of which it is the pedal is the “negative pedal” of the primary. We may have pedals of pedals and so on, also negative pedals of negative pedals and so on. Negative pedals are usually determined as envelopes.

(iii.) If φ denotes the angle which the tangent at any point makes with a fixed line, we have

r2 = p2 \+ (dp/dφ)2.

(iv.) The “average curvature” of the arc Δs of a curve between two points is measured by the quotient

| Δφ  
---  
Δs  
  
where the upright lines denote, as usual, that the absolute value of the included expression is to be taken, and φ is the angle which the tangent makes with a fixed line, so that Δφ is the angle between the tangents (or normals) at the points. As one of the points moves up to coincidence with the other this average curvature tends to a limit which is the “curvature” of the curve at the point. It is denoted by

| dφ  
---  
ds  
  
Sometimes the upright lines are omitted and a rule of signs is given:—Let the arc s of the curve be measured from some point along the curve in a chosen sense, and let the normal be drawn towards that side to which the curve is concave; if the normal is directed towards the left of an observer looking along the tangent in the chosen sense of description the curvature is reckoned positive, in the contrary case negative. The differential dφ is often called the “angle of contingence.” In the 14th century the size of the angle between a curve and its tangent seems to have been seriously debated, and the name “angle of contingence” was then given to the supposed angle.

(v.) The curvature of a curve at a point is the same as that of a certain circle which touches the curve at the point, and the “radius of curvature” ρ is the radius of this circle. We have 1/ρ = |dφ/ds|. The centre of the circle is called the “centre of curvature”; it is the limiting position of the point of intersection of the normal at the point and the normal at a neighbouring point, when the second point moves up to coincidence with the first. If a circle is described to intersect the curve at the point P and at two other points, and one of these two points is moved up to coincidence with P, the circle touches the curve at the point P and meets it in another point; the centre of the circle is then on the normal. As the third point now moves up to coincidence with P, the centre of the circle moves to the centre of curvature. The circle is then said to “osculate” the curve, or to have “contact of the second order” with it at P.

(vi.) The following are formulae for the radius of curvature:—

1 | = | { 1 + ( | dy | )2 }−3/2 | d2y | |,  
---|---|---|---|---|---  
ρ | dx | dx2  
ρ = | r | dr | | = | p + | d2p | |.  
---|---|---|---|---  
dp | dφ2  
  
(vii.) The points at which the curvature vanishes are “points of inflection.” If P is a point of inflection and Q a neighbouring point, then, as Q moves up to coincidence with P, the distance from P to the point of intersection of the normals at P and Q becomes greater than any distance that can be assigned. The equation which gives the abscissae of the points in which a straight line meets the curve being expressed in the form ƒ(x) = 0, the function ƒ(x) has a factor (x − x0)3, where x0 is the abscissa of the point of inflection P, and the line is the tangent at P. When the factor (x − x0) occurs (n + 1) times in ƒ(x), the curve is said to have “contact of the nth order” with the line. There is an obvious modification when the line is parallel to the axis of y.

(viii.) The locus of the centres of curvature, or envelope of the normals, of a curve is called the “evolute.” A curve which has a given curve as evolute is called an “involute” of the given curve. All the involutes are “parallel” curves, that is to say, they are such that one is derived from another by marking off a constant distance along the normal. The involutes are “orthogonal trajectories” of the tangents to the common evolute.

(ix.) The equation of an algebraic curve of the nth degree can be expressed in the form u0 \+ u1 \+ u2 \+ ... + un = 0, where u0 is a constant, and ur is a homogeneous rational integral function of x, y of the rth degree. When the origin is on the curve, u0 vanishes, and u1 = 0 represents the tangent at the origin. If u1 also vanishes, the origin is a double point and u2 = o represents the tangents at the origin. If u2 has distinct factors, or is of the form a(y − p1x) (y − p2x), the value of y on either branch of the curve can be expressed (for points sufficiently near the origin) in a power series, which is either

p1x + ½ q1 x2 \+ ...,  or p2x + ½ q2 x2 \+ ...,

where q1, ... and q2, ... are determined without ambiguity. If p1 and p2 are real the two branches have radii of curvature ρ1, ρ2 determined by the formulae

1 | = | (1 + p12)−3/2 q1 |,  | 1 | = | (1 + p22)−3/2 q2 |.  
---|---|---|---  
ρ1 | ρ2  
  
When p1 and p2 are imaginary the origin is the real point of intersection of two imaginary branches. In the real figure of the curve it is an isolated point. If u2 is a square, a(y − px)2, the origin is a cusp, and in general there is not a series for y in integral powers of x, which is valid in the neighbourhood of the origin. The further investigation of cusps and multiple points belongs rather to analytical geometry and the theory of algebraic functions than to differential calculus.

(x.) When the equation of a curve is given in the form u0 \+ u1 \+ ... \+ un−1 \+ un = 0 where the notation is the same as that in (ix.), the factors of un determine the directions of the asymptotes. If these factors are all real and distinct, there is an asymptote corresponding to each factor. If un = L1L2 ... Ln, where L1, ... are linear in x, y, we may resolve un−1/un into partial fractions according to the formula

un−1 | = | A1 | + | A2 | \+ ... + | An | ,  
---|---|---|---|---|---|---|---  
un | L1 | L2 | Ln  
  
and then L1 \+ A1 = 0, L2 \+ A2 = 0, ... are the equations of the asymptotes. When a real factor of un is repeated we may have two parallel asymptotes or we may have a “parabolic asymptote.” Sometimes the parallel asymptotes coincide, as in the curve x2(x2 \+ y2 − a2) = a4, where x = 0 is the only real asymptote. The whole theory of asymptotes belongs properly to analytical geometry and the theory of algebraic functions.

47\. The formal definition of an integral, the theorem of the existence of the integral for certain classes of functions, a list of Integral calculus. classes of “integrable” functions, extensions of the notion of integration to functions which become infinite or indeterminate, and to cases in which the limits of integration become infinite, the definitions of multiple integrals, and the possibility of defining functions by means of definite integrals—all these matters have been considered in Function. The definition of integration has been explained in § 5 above, and the results of some of the simplest integrations have been given in § 12. A few theorems relating to integrations have been noted in §§ 34, 35, 36 above.

48\. Methods of integration.The chief methods for the evaluation of indefinite integrals are the method of integration by parts, and the introduction of new variables.

From the equation d(uv) = u dv + v du we deduce the equation

∫ u | dv | dx = uv − ∫ v | du | dx,  
---|---|---|---|---  
dx | dx  
  
or, as it may be written

∫ uw dx = u ∫ w dx − ∫ | du | { ∫ w dx } dx.  
---|---|---  
dx  
  
This is the rule of “integration by parts.”

As an example we have

∫ xeax dx = x | eax | − ∫ | eax | dx = ( | x | − | 1 | ) eax.  
---|---|---|---|---|---|---|---|---  
a | a | a | a2  
  
When we introduce a new variable z in place of x, by means of an equation giving x in terms of z, we express ƒ(x) in terms of z. Let φ(z) denote the function of z into which ƒ(x) is transformed. Then from the equation

dx = | dx | dz  
---|---|---  
dz  
  
we deduce the equation

∫ ƒ(x) dx = ∫ φ(z) | dx | dz.  
---|---|---  
dz  
  
As an example, in the integral

∫ √(1 − x2)dx

put x = sin z; the integral becomes

∫ cos z · cos z dz = ∫ ½ (1 + cos 2z)dz = ½ (z + ½ sin 2z) = ½ (z + sin z cos z).

49\. The indefinite integrals of certain classes of functions can be expressed by means of a finite number of operations of addition or multiplication in terms of the so-called “elementary” functions. The elementary functions are rational algebraic Integration in terms of elementary functions. functions, implicit algebraic functions, exponentials and logarithms, trigonometrical and inverse circular functions. The following are among the classes of functions whose integrals involve the elementary functions only: (i.) all rational functions; (ii.) all irrational functions of the form ƒ(x, y), where ƒ denotes a rational algebraic function of x and y, and y is connected with x by an algebraic equation of the second degree; (iii.) all rational functions of sin x and cos x; (iv.) all rational functions of ex; (v.) all rational integral functions of the variables x, eax, ebx, ... sin mx, cos mx, sin nx, cos nx, ... in which a, b, ... and m, n, ... are any constants. The integration of a rational function is generally effected by resolving the function into partial fractions, the function being first expressed as the quotient of two rational integral functions. Corresponding to any simple root of the denominator there is a logarithmic term in the integral. If any of the roots of the denominator are repeated there are rational algebraic terms in the integral. The operation of resolving a fraction into partial fractions requires a knowledge of the roots of the denominator, but the algebraic part of the integral can always be found without obtaining all the roots of the denominator. Reference may be made to C. Hermite, Cours d’analyse, Paris, 1873. The integration of other functions, which can be integrated in terms of the elementary functions, can usually be effected by transforming the functions into rational functions, possibly after preliminary integrations by parts. In the case of rational functions of x and a radical of the form √(ax2 \+ bx + c) the radical can be reduced by a linear substitution to one of the forms √(a2 − x2), √(x2 − a2), √(x2 \+ a2). The substitutions x = a sin θ, x = a sec θ, x = a tan θ are then effective in the three cases. By these substitutions the subject of integration becomes a rational function of sin θ and cos θ, and it can be reduced to a rational function of t by the substitution tan ½θ = t. There are many other substitutions by which such integrals can be determined. Sometimes we may have information as to the functional character of the integral without being able to determine it. For example, when the subject of integration is of the form (ax4 \+ bx3 \+ cx2 \+ dx + e)−1/2 the integral cannot be expressed explicitly in terms of elementary functions. Such integrals lead to new functions (see Function).

Methods of reduction and substitution for the evaluation of indefinite integrals occupy a considerable space in text-books of the integral calculus. In regard to the functional character of the integral reference may be made to G. H. Hardy’s tract, The Integration of Functions of a Single Variable (Cambridge, 1905), and to the memoirs there quoted. A few results are added here

(i.)

∫ (x2 \+ a) − ½ dx = log {x + (x2 \+ a)1/2 }.

(ii.)

∫ | dx  
---|---  
(x − p) √(ax2 \+ 2bx + c)  
  
can be evaluated by the substitution x − p = 1/z, and

∫ | dx  
---|---  
(x − p)n √(ax2 \+ 2bx + c)  
  
can be deduced by differentiating (n − 1) times with respect to p.

(iii.)

∫ | (Hx + K) dx  
---|---  
(αx2 \+ 2βx + γ) √(ax2 \+ 2bx + c)  
  
can be reduced by the substitution y2 = (ax2 \+ 2bx + c)/(αx2 \+ 2βx + γ) to the form

A ∫ | dy | \+ B ∫ | dy  
---|---|---|---  
√(λ1 − y2) | √(y2 − λ2)  
  
where A and B are constants, and λ1 and λ2 are the two values of λ for which (a − λα)x2 \+ 2(b − λβ)x + c − λγ is a perfect square (see A. G. Greenhill, A Chapter in the Integral Calculus, London, 1888).

(iv.) ƒxm (axn \+ b)p dx, in which m, n, p are rational, can be reduced, by putting axn = bt, to depend upon ƒtq (1 + t)p dt. If p is an integer and q a fraction r/s, we put t = us. If q is an integer and p = r/s we put 1 + t = us. If p + q is an integer and p = r/s we put 1 + t = tus. These integrals, called “binomial integrals,” were investigated by Newton (De quadratura curvarum).

(v.)

∫ | dx |  = log tan | x | ,  
---|---|---|---|---  
sin x | 2  
  
(vi.)

∫ | dx | = log (tan x + sec x).  
---|---|---  
cos x  
  
(vii.) ∫ eax sin (bx + α) dx = (a2 \+ b2)−1 eax {a sin (bx + α) − b cos (bx + α) }.

(viii.) ∫ sinm x cosn x dx can be reduced by differentiating a function of the form sinp x cosq x;

e.g. | d |   | sin x | = | 1 | + | q sin2 x | = | 1 − q | + | q | .  
---|---|---|---|---|---|---|---|---|---|---|---|---  
dx | cosq x | cosq−1 x | cosq+1 x | cosq−1 x | cosq+1 x  
  
Hence

∫ | dx | = | sin x | + | n − 2 | ∫ | dx | .  
---|---|---|---|---|---|---|---|---  
cosn x | (n − 1) cosn−1 x | n − 1 | cosn−2 x  
  
(ix.)

∫1/2π0 sin2n x dx = ∫1/2π0 cos2n x dx = | 1·3 ... (2n − 1) | · | π | , (n an integer).  
---|---|---|---|---  
2·4 ... 2n | 2  
  
(x.)

∫1/2π0 sin2n+1 x dx = ∫1/2π0 cos2n+1 x dx = | 2·4 ... 2n | , (n an integer).  
---|---|---  
3·5 ... (2n + 1)  
  
(xi.)

∫ | dx | can be reduced by one of the substitutions  
---|---|---  
(1 + e cos x)n  
cos φ = | e + cos x | ,  cosh u = | e + cos x | ,  
---|---|---|---|---  
1 + e cos x | 1 + e cos x  
  
of which the first or the second is to be employed according as e < or > 1.

50\. New transcendents.Among the integrals of transcendental functions which lead to new transcendental functions we may notice

∫x0 | dx | ,  or ∫log x−∞ | ez | dz,  
---|---|---|---|---  
log x | z  
  
called the “logarithmic integral,” and denoted by “Li x,” also the integrals

∫x0 | sin x | dx and ∫x∞ | cos x | dx,  
---|---|---|---|---  
x | x  
  
called the “sine integral” and the “cosine integral,” and denoted by “Si x” and “Ci x,” also the integral

∫x0 e−x2 dx

called the “error-function integral,” and denoted by “Erf x.” All these functions have been tabulated (see Tables, Mathematical).

51\. Eulerian integrals.New functions can be introduced also by means of the definite integrals of functions of two or more variables with respect to one of the variables, the limits of integration being fixed. Prominent among such functions are the Beta and Gamma functions expressed by the equations

B(l, m) = ∫10 xl−1 (1 − x)m−1 dx,

Γ(n) = ∫∞0 e−t tn−1 dt.

When n is a positive integer Γ(n + 1) = n!. The Beta function (or “Eulerian integral of the first kind”) is expressible in terms of Gamma functions (or “Eulerian integrals of the second kind”) by the formula

B(l, m) · Γ(l + m) = Γ(l) · Γ(m).

The Gamma function satisfies the difference equation

Γ(x + 1) = x Γ(x),

and also the equation

Γ(x) · Γ(1 − x) = π/sin (xπ),

with the particular result

Γ(½)= √π.

The number

− [ | d | { log Γ (1 + x) } ]x=0, or −Γ′(1),  
---|---|---  
dx  
  
is called “Euler’s constant,” and is equal to the limit

lim.n=∞ [ ( 1 + ½ + 1⁄3 \+ ... + | 1 | ) − log n ];  
---|---|---  
n  
  
its value to 15 decimal places is 0.577 215 664 901 532.

The function log Γ(1 + x) can be expanded in the series

log Γ (1 + x) = ½ log ( | xπ | ) − ½ log | 1 + x | \+ { 1 + Γ′;(1) } x  
---|---|---|---|---  
sin xπ | 1 − x  
  
− 1⁄3 (S3 − 1) x3 − 1⁄5 (S5 − 1) x5 − ...,

where

S2r+1 = 1 + | 1 | + | 1 | \+ ...,  
---|---|---|---|---  
22r+1 |  32r+1  
  
and the series for log Γ(1 + x) converges when x lies between −1 and 1.

52\. Definite integrals.Definite integrals can sometimes be evaluated when the limits of integration are some particular numbers, although the corresponding indefinite integrals cannot be found. For example, we have the result

∫10 (1 − x2)−1/2 log x dx = −½ π log 2,

although the indefinite integral of (1 − x2)−1/2 log x cannot be found. Numbers of definite integrals are expressible in terms of the transcendental functions mentioned in § 50 or in terms of Gamma functions. For the calculation of definite integrals we have the following methods:—

(i.) | Differentiation with respect to a parameter.  
---|---  
(ii.) | Integration with respect to a parameter.  
(iii.) | Expansion in infinite series and integration term by term.  
(iv.) | Contour integration.  
  
The first three methods involve an interchange of the order of two limiting operations, and they are valid only when the functions satisfy certain conditions of continuity, or, in case the limits of integration are infinite, when the functions tend to zero at infinite distances in a sufficiently high order (see Function). The method of contour integration involves the introduction of complex variables (see Function: § Complex Variables).

A few results are added

(i.)

∫∞0 | xa−1 | dx = | π | , (1 > a > 0),  
---|---|---|---|---  
1 + x | sin aπ  
  
(ii.)

∫∞0 | xa−1 − xb−1 | dx = π (cot aπ − cot bπ), (0 < a or b < 1),  
---|---|---  
1 − x  
  
(iii.)

∫∞0 | xa−1 log x | dx = | π2 | , (a > 1),  
---|---|---|---|---  
x − 1 | sin2 aπ  
  
(iv.)

∫∞0 x2 · cos 2x · e−x2 dx = −1⁄4 e−1 √π,

(v.)

∫10 | 1 − x2 |   | dx | = log tan | π | ,  
---|---|---|---|---|---|---  
1 + x4 | log x | 8  
  
(vi.)

∫∞0 | sin mx | dx = ½ ( | 1 | − | 1 | + | 1 | ),  
---|---|---|---|---|---|---|---|---  
e2πx − 1 | em | m | 2  
  
(vii.)

∫π0 log (1 − 2α cos x + α2) dx = 0 or 2π log α according as α < or > 1,

(viii.)

∫∞0 | sin x | dx = ½ π,  
---|---|---  
x  
  
(ix.)

∫∞0 | cos ax | dx = ½ πb−1 e−ab,  
---|---|---  
x2 \+ b2  
  
(x.)

∫∞0 | cos ax − cos bx | dx = ½ π (b − a),  
---|---|---  
x2  
  
(xi.)

∫∞0 | cos ax − cos bx | dx = log | b | ,  
---|---|---|---|---  
x | a  
  
(xii.)

∫∞0 | cos x − e−mx | dx = log m,  
---|---|---  
x  
  
(xiii.)

∫∞−∞ e−x2+2ax dx = √π · ea2,

(xiv.)

∫∞0 x−1/2 sin x dx = ∫∞0 x−1/2 cos x dx = √(½ π),

53\. Multiple Integrals.The meaning of integration of a function of n variables through a domain of the same number of dimensions is explained in the article Function. In the case of two variables x, y we integrate a function ƒ(x, y) over an area; in the case of three variables x, y, z we integrate a function ƒ(x, y, z) through a volume. The integral of a function ƒ(x, y) over an area in the plane of (x, y) is denoted by

∫∫ ƒ(x, y) dx dy.

The notation refers to a method of evaluating the integral. We may suppose the area divided into a very large number of very small rectangles by lines parallel to the axes. Then we multiply the value of ƒ at any point within a rectangle by the measure of the area of the rectangle, sum for all the rectangles, and pass to a limit by increasing the number of rectangles indefinitely and diminishing all their sides indefinitely. The process is usually effected by summing first for all the rectangles which lie in a strip between two lines parallel to one axis, say the axis of y, and afterwards for all the strips. This process is equivalent to integrating ƒ(x, y) with respect to y, keeping x constant, and taking certain functions of x as the limits of integration for y, and then integrating the result with respect to x between constant limits. The integral obtained in this way may be written in such a form as

∫ba dx { ∫ | ƒ2 (x) | ƒ(x, y)dy },  
---|---|---  
ƒ1 (x)  
  
and is called a “repeated integral.” The identification of a surface integral, such as ∫∫ ƒ(x, y)dxdy, with a repeated integral cannot always be made, but implies that the function satisfies certain conditions of continuity. In the same way volume integrals are usually evaluated by regarding them as repeated integrals, and a volume integral is written in the form

∫∫∫ ƒ(x, y, z) dx dy dz.

Integrals such as surface and volume integrals are usually called “multiple integrals.” Thus we have “double” integrals, “triple” integrals, and so on. In contradistinction to multiple integrals the ordinary integral of a function of one variable with respect to that variable is called a “simple integral.”

A more general type of surface integral may be defined by taking an arbitrary surface, with or without an edge. We suppose in the first place that the surface is closed, or has no edge. We may mark a large number of points on the surface, and Surface Integrals. draw the tangent planes at all these points. These tangent planes form a polyhedron having a large number of faces, one to each marked point; and we may choose the marked points so that all the linear dimensions of any face are less than some arbitrarily chosen length. We may devise a rule for increasing the number of marked points indefinitely and decreasing the lengths of all the edges of the polyhedra indefinitely. If the sum of the areas of the faces tends to a limit, this limit is the area of the surface. If we multiply the value of a function ƒ at a point of the surface by the measure of the area of the corresponding face of the polyhedron, sum for all the faces, and pass to a limit as before, the result is a surface integral, and is written

∫∫∫ ƒ dS.

The Line Integrals.extension to the case of an open surface bounded by an edge presents no difficulty. A line integral taken along a curve is defined in a similar way, and is written

∫ ƒ ds

where ds is the element of arc of the curve (§ 33). The direction cosines of the tangent of a curve are dx/ds, dy/ds, dz/ds, and line integrals usually present themselves in the form

∫ ( u | dx | \+ v | dy | \+ w | dz | ) ds or ∫s (u dx + v dy + w dz).  
---|---|---|---|---|---|---  
ds | ds | ds  
  
In like manner surface integrals usually present themselves in the form

∫∫ (lξ \+ mη \+ nζ) dS

where l, m, n are the direction cosines of the normal to the surface drawn in a specified sense.

The area of a bounded portion of the plane of (x, y) may be expressed either as

½ ∫ (x dy − y dx),

or as

∫∫ dx dy,

the former integral being a line integral taken round the boundary of the portion, and the latter a surface integral taken over the area within this boundary. In forming the line integral the boundary is supposed to be described in the positive sense, so that the included area is on the left hand.

53a. Theorems of Green and Stokes.We have two theorems of transformation connecting volume integrals with surface integrals and surface integrals with line integrals. The first theorem, called “Green’s theorem,” is expressed by the equation

∫∫∫ ( | ∂ξ | + | ∂η | + | ∂ζ | ) dx dy dz = ∫∫ (lξ \+ mη \+ nζ) dS,  
---|---|---|---|---|---|---  
∂x | ∂y | ∂z  
  
where the volume integral on the left is taken through the volume within a closed surface S, and the surface integral on the right is taken over S, and l, m, n denote the direction cosines of the normal to S drawn outwards. There is a corresponding theorem for a closed curve in two dimensions, viz.,

∫∫ ( | ∂ξ | + | ∂η | ) dx dy = ∫ ( ξ | dy | − η | dx | ) ds,  
---|---|---|---|---|---|---|---|---  
∂x  | ∂y | ds | ds  
  
the sense of description of s being the positive sense. This theorem is a particular case of a more general theorem called “Stokes’s theorem.” Let s denote the edge of an open surface S, and let S be covered with a network of curves so that the meshes of the network are nearly plane, then we can choose a sense of description of the edge of any mesh, and a corresponding sense for the normal to S at any point within the mesh, so that these senses are related like the directions of rotation and translation in a right-handed screw. This convention fixes the sense of the normal (l, m, n) at any point on S when the sense of description of s is chosen. If the axes of x, y, z are a right-handed system, we have Stokes’s theorem in the form

∫s (u dx + v dy + w dz) = ∫∫ { l ( | ∂w | − | ∂v | ) \+ m ( | ∂u | − | ∂w | ) \+ n ( | ∂v | − | ∂u | ) } dS,  
---|---|---|---|---|---|---|---|---|---|---|---|---  
∂y | ∂z | ∂z | ∂x | ∂x | ∂y  
  
where the integral on the left is taken round the curve s in the chosen sense. When the axes are left-handed, we may either reverse the sense of l, m, n and maintain the formula, or retain the sense of l, m, n and change the sign of the right-hand member of the equation. For the validity of the theorems of Green and Stokes it is in general necessary that the functions involved should satisfy certain conditions of continuity. For example, in Green’s theorem the differential coefficients ∂ξ/∂x, ∂η/∂y, ∂ζ/∂z must be continuous within S. Further, there are restrictions upon the nature of the curves or surfaces involved. For example, Green’s theorem, as here stated, applies only to simply-connected regions of space. The correction for multiply-connected regions is important in several physical theories.

54\. The process of changing the variables in a multiple integral, such as a surface or volume integral, is divisible into two stages. It Change of Variables in a Multiple Integral. is necessary in the first place to determine the differential element expressed by the product of the differentials of the first set of variables in terms of the differentials of the second set of variables. It is necessary in the second place to determine the limits of integration which must be employed when the integral in terms of the new variables is evaluated as a repeated integral. The first part of the problem is solved at once by the introduction of the Jacobian. If the variables of one set are denoted by x1, x2, ..., xn, and those of the other set by u1, u2, ..., un, we have the relation

dx1 dx2 ... dxn = | ∂ (x1, x2, ..., xn) | du1 du2 ... dun.  
---|---|---  
∂ (u1, u2, ..., un)  
  
In regard to the second stage of the process the limits of integration must be determined by the rule that the integration with respect to the second set of variables is to be taken through the same domain as the integration with respect to the first set.

For example, when we have to integrate a function ƒ(x, y) over the area within a circle given by x2 \+ y2 = a2, and we introduce polar coordinates so that x = r cos θ, y = r sin θ, we find that r is the value of the Jacobian, and that all points within or on the circle are given by a ≥ r ≥ 0, 2π ≥ θ ≥ 0, and we have

∫a−a dx ∫ | √(a2−x2) | ƒ(x, y) dy = ∫a0 dr ∫2π0 ƒ(r cos θ, r sin θ) r dθ.  
---|---|---  
−√(a2−x2)  
  
If we have to integrate over the area of a rectangle a ≥ x ≥ 0, b ≥ y ≥ 0, and we transform to polar coordinates, the integral becomes the sum of two integrals, as follows:—

∫a0 dx ∫b0 ƒ(x, y) dy = ∫0tan−1b/a dθ ∫0a sec θ ƒ(r cos θ, r sin θ) r dr

\+ ∫1/2πtan−1b/a dθ ∫0b cosec θ ƒ(r cos θ, r sin θ) r dr.

55\. A few additional results in relation to line integrals and multiple integrals are set down here.

(i.) Any simple integral can be regarded as a line-integral taken along a portion of the axis of x. When a change of Line Integrals and Multiple Integrals. variables is made, the limits of integration with respect to the new variable must be such that the domain of integration is the same as before. This condition may require the replacing of the original integral by the sum of two or more simple integrals.

(ii.) The line integral of a perfect differential of a one-valued function, taken along any closed curve, is zero.

(iii.) The area within any plane closed curve can be expressed by either of the formulae

∫ ½ r2 dθ or ∫ ½ p ds,

where r, θ are polar coordinates, and p is the perpendicular drawn from a fixed point to the tangent. The integrals are to be understood as line integrals taken along the curve. When the same integrals are taken between limits which correspond to two points of the curve, in the sense of line integrals along the arc between the points, they represent the area bounded by the arc and the terminal radii vectores.

(iv.) The volume enclosed by a surface which is generated by the revolution of a curve about the axis of x is expressed by the formula

π ∫ y2 dx,

and the area of the surface is expressed by the formula

2π ∫ y ds,

where ds is the differential element of arc of the curve. When the former integral is taken between assigned limits it represents the volume contained between the surface and two planes which cut the axis of x at right angles. The latter integral is to be understood as a line integral taken along the curve, and it represents the area of the portion of the curved surface which is contained between two planes at right angles to the axis of x.

(v.) When we use curvilinear coordinates ξ, η which are conjugate functions of x, y, that is to say are such that

∂ξ/∂x = ∂η/∂y and ∂ξ/∂y = −∂η/∂x,

the Jacobian ∂(ξ, η)/∂(x, v) can be expressed in the form

( | ∂ξ | )2 \+ ( | ∂η | )2,  
---|---|---|---|---  
∂x | ∂x  
  
and in a number of equivalent forms. The area of any portion of the plane is represented by the double integral

∫∫ J−1 dξ dη,

where J denotes the above Jacobian, and the integration is taken through a suitable domain. When the boundary consists of portions of curves for which ξ = const., or η = const., the above is generally the simplest way of evaluating it.

(vi.) The problem of “rectifying” a plane curve, or finding its length, is solved by evaluating the integral

∫ { 1 + ( | dy | )2 }1/2 dx,  
---|---|---  
dx  
  
or, in polar coordinates, by evaluating the integral

∫ { r2 \+ ( | dr | )2 }1/2 dθ.  
---|---|---  
dθ  
  
In both cases the integrals are line integrals taken along the curve.

(vii.) When we use curvilinear coordinates ξ, η as in (v.) above, the length of any portion of a curve ξ = const. is given by the integral

∫ J−1/2 dη

taken between appropriate limits for η. There is a similar formula for the arc of a curve η = const.

(viii.) The area of a surface z = ƒ(x, y) can be expressed by the formula

∫∫ { 1 + ( | ∂z | )2 \+ ( | ∂z | )2 }1/2 dx dy.  
---|---|---|---|---  
∂x | ∂y  
  
When the coordinates of the points of a surface are expressed as functions of two parameters u, v, the area is expressed by the formula

∫∫ [ { | ∂(y, z) | }2 \+ { | ∂(z, x) | }2 \+ { | ∂(x, y) | }2 ]1/2 du dv.  
---|---|---|---|---|---|---  
∂(u, v) | ∂(u, v) | ∂(u, v)  
  
When the surface is referred to three-dimensional polar coordinates r, θ, φ given by the equations

x = r sin θ cos φ, y = r sin θ sin φ, z = r cos θ,

and the equation of the surface is of the form r = ƒ(θ, φ), the area is expressed by the formula

∫∫ r [ { r2 \+ ( | ∂r | )2 } sin2 θ \+ ( | ∂r | )2 ]1/2 dθ dφ.  
---|---|---|---|---  
∂θ | ∂φ  
  
The surface integral of a function of (θ, φ) over the surface of a sphere r = const. can be expressed in the form

∫2π0 dφ ∫π0 F (θ,φ) r2 sin θ dθ.

In every case the domain of integration must be chosen so as to include the whole surface.

(ix.) In three-dimensional polar coordinates the Jacobian

∂(x, y, z) | = r2 sin θ.  
---|---  
∂(r, θ, φ)  
  
The volume integral of a function F (r, θ, φ) through the volume of a sphere r = a is

∫a0 dr ∫2π0 dφ ∫π0 F (r, θ, φ) r2 sin θ dθ.

(x.) Integrations of rational functions through the volume of an ellipsoid x2/a2 \+ y2/b2 \+ z2/c2 = 1 are often effected by means of a general theorem due to Lejeune Dirichlet (1839), which is as follows: when the domain of integration is that given by the inequality

( | x1 | )a1 \+ ( | x2 | )a2 \+ ... + ( | xn | )an ≤ 1,  
---|---|---|---|---|---|---  
a1 | a2 | an  
  
where the a’s and α’s are positive, the value of the integral

∫∫ ... x1 n1−1 · x2 n2−1 ... dx1 dx2 ...

is

a1n1 a2n2 ... |   | Γ (n1/α1) Γ (n2/α2) ... | .  
---|---|---|---  
α1 α2 ... |  Γ (1 + n1/α1 \+ n2/α2 \+ ... )  
  
If, however, the object aimed at is an integration through the volume of an ellipsoid it is simpler to reduce the domain of integration to that within a sphere of radius unity by the transformation x = aξ, y = bη, z = cζ, and then to perform the integration through the sphere by transforming to polar coordinates as in (ix).

56\. Methods of approximate integration began to be devised very early. Kepler’s practical measurement of the focal sectors Approximate and Mechanical Integration. of ellipses (1609) was an approximate integration, as also was the method for the quadrature of the hyperbola given by James Gregory in the appendix to his Exercitationes geometricae (1668). In Newton’s Methodus differentialis (1711) the subject was taken up systematically. Newton’s object was to effect the approximate quadrature of a given curve by making a curve of the type

y = a0 \+ a1 x + a2 x2 \+ ... + an xn

pass through the vertices of (n + 1) equidistant ordinates of the given curve, and by taking the area of the new curve so determined as an approximation to the area of the given curve. In 1743 Thomas Simpson in his Mathematical Dissertations published a very convenient rule, obtained by taking the vertices of three consecutive equidistant ordinates to be points on the same parabola. The distance between the extreme ordinates corresponding to the abscissae x = a and x = b is divided into 2n equal segments by ordinates y1, y2, ... y2n−1, and the extreme ordinates are denoted by y0, y2n. The vertices of the ordinates y0, y1, y2 lie on a parabola with its axis parallel to the axis of y, so do the vertices of the ordinates y2, y3, y4, and so on. The area is expressed approximately by the formula

{ (b − a)/6n } [y0 \+ y2n \+ 2 (y2 \+ y4 \+ ... + y2n−2) + 4 (y1 \+ y3 \+ ... + y2n−1) ],

which is known as Simpson’s rule. Since all simple integrals can be represented as areas such rules are applicable to approximate integration in general. For the recent developments reference may be made to the article by A. Voss in Ency. d. Math. Wiss., Bd. II., A. 2 (1899), and to a monograph by B. P. Moors, Valeur approximative d’une intégrale définie (Paris, 1905).

Many instruments have been devised for registering mechanically the areas of closed curves and the values of integrals. The best known are perhaps the “planimeter” of J. Amsler (1854) and the “integraph” of Abdank-Abakanowicz (1882).

Bibliography.—For historical questions relating to the subject the chief authority is M. Cantor, Geschichte d. Mathematik (3 Bde., Leipzig, 1894-1901). For particular matters, or special periods, the following may be mentioned: H. G. Zeuthen, Geschichte d. Math. im Altertum u. Mittelalter (Copenhagen, 1896) and Gesch. d. Math. im XVI. u. XVII. Jahrhundert (Leipzig, 1903); S. Horsley, Isaaci Newtoni opera quae exstant omnia (5 vols., London, 1779-1785); C. I. Gerhardt, Leibnizens math. Schriften (7 Bde., Leipzig, 1849-1863); Joh. Bernoulli, Opera omnia (4 Bde., Lausanne and Geneva, 1742). Other writings of importance in the history of the subject are cited in the course of the article. A list of some of the more important treatises on the differential and integral calculus is appended. The list has no pretensions to completeness; in particular, most of the recent books in which the subject is presented in an elementary way for beginners or engineers are omitted.—L. Euler, Institutiones calculi differentialis (Petrop., 1755) and Institutiones calculi integralis (3 Bde., Petrop., 1768-1770); J. L. Lagrange, Leçons sur le calcul des fonctions (Paris, 1806, Œuvres, t. x.), and Théorie des fonctions analytiques (Paris, 1797, 2nd ed., 1813, Œuvres, t. ix.); S. F. Lacroix, Traité de calcul diff. et de calcul int. (3 tt., Paris, 1808-1819). There have been numerous later editions; a translation by Herschel, Peacock and Babbage of an abbreviated edition of Lacroix’s treatise was published at Cambridge in 1816. G. Peacock, Examples of the Differential and Integral Calculus (Cambridge, 1820); A. L. Cauchy, Résumé des leçons ... sur le calcul infinitésimale (Paris, 1823), and Leçons sur le calcul différentiel (Paris, 1829; Œuvres, sér. 2, t. iv.); F. Minding, Handbuch d. Diff.-u. Int.-Rechnung (Berlin, 1836); F. Moigno, Leçons sur le calcul diff. (4 tt., Paris, 1840-1861); A. de Morgan, Diff. and Int. Calc. (London, 1842); D. Gregory, Examples on the Diff. and Int. Calc. (2 vols., Cambridge, 1841-1846); I. Todhunter, Treatise on the Diff. Calc. and Treatise on the Int. Calc. (London, 1852), numerous later editions; B. Price, Treatise on the Infinitesimal Calculus (2 vols., Oxford, 1854), numerous later editions; D. Bierens de Haan, Tables d’intégrales définies (Amsterdam, 1858); M. Stegemann, Grundriss d. Diff.- u. Int.-Rechnung (2 Bde., Hanover, 1862) numerous later editions; J. Bertrand, Traité de calc. diff. et int. (2 tt., Paris, 1864-1870); J. A. Serret, Cours de calc. diff. et int. (2 tt., Paris, 1868, 2nd ed., 1880, German edition by Harnack, Leipzig, 1884-1886, later German editions by Bohlmann, 1896, and Scheffers, 1906, incomplete); B. Williamson, Treatise on the Diff. Calc. (Dublin, 1872), and Treatise on the Int. Calc. (Dublin, 1874) numerous later editions of both; also the article “Infinitesimal Calculus” in the 9th ed. of the Ency. Brit.; C. Hermite, Cours d’analyse (Paris, 1873); O. Schlömilch, Compendium d. höheren Analysis (2 Bde., Leipzig, 1874) numerous later editions; J. Thomae, Einleitung in d. Theorie d. bestimmten Integrale (Halle, 1875); R. Lipschitz, Lehrbuch d. Analysis (2 Bde., Bonn, 1877, 1880); A. Harnack, Elemente d. Diff.- u. Int.-Rechnung (Leipzig, 1882, Eng. trans. by Cathcart, London, 1891); M. Pasch, Einleitung in d. Diff.- u. Int.-Rechnung (Leipzig, 1882); Genocchi and Peano, Calcolo differenziale (Turin, 1884, German edition by Bohlmann and Schepp, Leipzig, 1898, 1899); H. Laurent, Traité d’analyse (7 tt., Paris, 1885-1891); J. Edwards, Elementary Treatise on the Diff. Calc. (London, 1886), several later editions; A. G. Greenhill, Diff. and Int. Calc. (London, 1886, 2nd ed., 1891); É. Picard, Traité d’analyse (3 tt., Paris, 1891-1896); O. Stolz, Grundzüge d. Diff.- u. Int.-Rechnung (3 Bde., Leipzig, 1893-1899); C. Jordan, Cours d’analyse (3 tt., Paris, 1893-1896); L. Kronecker, Vorlesungen ü. d. Theorie d. einfachen u. vielfachen Integrale (Leipzig, 1894); J. Perry, The Calculus for Engineers (London, 1897); H. Lamb, An Elementary Course of Infinitesimal Calculus (Cambridge, 1897); G. A. Gibson, An Elementary Treatise on the Calculus (London, 1901); É. Goursat, Cours d’analyse mathématique (2 tt., Paris, 1902-1905); C.-J. de la Vallée Poussin, Cours d’analyse infinitésimale (2 tt., Louvain and Paris, 1903-1906); A. E. H. Love, Elements of the Diff. and Int. Calc. (Cambridge, 1909); W. H. Young, The Fundamental Theorems of the Diff. Calc. (Cambridge, 1910). A résumé of the infinitesimal calculus is given in the articles “Diff.- u. Int-Rechnung” by A. Voss, and “Bestimmte Integrale” by G. Brunel in Ency. d. math. Wiss. (Bde. ii. A. 2, and ii. A. 3, Leipzig, 1899, 1900). Many questions of principle are discussed exhaustively by E. W. Hobson, The Theory of Functions of a Real Variable (Cambridge, 1907).

(A. E. H. L.)

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