1911 Encyclopædia Britannica/Infinitesimal Calculus/Nature of the Calculus

 Infinitesimal Calculus (§1) Infinitesimal CalculusI. Nature of the Calculus (§2-12) II. History (§13-22)

I. Nature of the Calculus.

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, when it was employed by Nicole Oresme, who studied Geometrical representation of Variable Quantities. 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.

 Fig. 1.

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 determining the area of the curve. The relation of The problems of Maxima and Minima, Tangents, and Quadratures. 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 the principle on which it is based was laid down in the Greek methods. 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.

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 x’s” and there is a like meaning for the symbol Δy.Differentiation.

 Fig. 2.

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

${\displaystyle {\frac {dy}{dx}}}$.

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

${\displaystyle {\frac {\Delta y}{\Delta x}}={\frac {f(x+\Delta x)-f(x)}{\Delta x}}}$.

and

${\displaystyle {\frac {dy}{dx}}=\lim _{\Delta x\to 0}{\frac {f(x+\Delta x)-f(x)}{\Delta x}}}$.

The limit expressed by the right-hand member of this defining equation is often written

${\displaystyle f^{\prime }(x)\,}$

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

${\displaystyle {\frac {f(x+\Delta x)-f(x)}{\Delta x}}={\frac {f(x+\Delta x)^{2}-x^{2}}{\Delta x}}={\frac {2x\Delta x+(\Delta x)^{2}}{\Delta x}}=2x+\Delta x}$

and

${\displaystyle f^{\prime }(x)=2x\,}$

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).

5. The problem of quadratures leads to a type of limiting process which may be described as follows: Let yx be the equation of a curve, and let AC and BD be the ordinates of the points C and D (see fig. 3). Let a, b be the abscissae of these points. Let the segment AB be divided into a numberIntegration. 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

 Fig. 3.

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

${\displaystyle lim.{\begin{matrix}\sum _{a}^{b}f(x^{\prime })\Delta x\end{matrix}}}$

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

${\displaystyle \int _{a}^{b}f(x)\,dx}$

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.

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 integral ${\displaystyle \int _{a}^{\xi }f(x)\,dx}$, and it is a function of ξ. Let BD be displaced to B′D′ so that becomes ${\displaystyle \xi +\Delta \xi }$ (seeTheorem of Inversion. fig. 4). The area of the figure ACD′B′ is represented by the integral ${\displaystyle \int _{a}^{\xi +\Delta \xi }f(x)\,dx}$ and the increment ΔA is given by the formula:

${\displaystyle \Delta A=\int _{\xi }^{\xi +\Delta \xi }f(x)\,dx}$

which represents the area BDD′B′.

 Fig. 4.

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:

${\displaystyle {\frac {d{\text{A}}}{d\xi }}=f(\xi )}$

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

${\displaystyle A=\int _{a}^{b}f(x)\,dx=F(b)-F(a)}$

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

${\displaystyle {\frac {dF(x)}{dx}}=f(x)}$,

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

${\displaystyle \int f(x)\,dx}$

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

${\displaystyle \Delta y=f^{\prime }(x)\Delta x+{\text{R}}}$,

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

${\displaystyle \lim _{\Delta x\to 0}{\text{R}}=0}$,

but also

${\displaystyle \lim _{\Delta x\to 0}{\frac {\text{R}}{\Delta x}}=0}$

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

${\displaystyle df(x)=f^{\prime }(x)dx.}$

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

${\displaystyle {\frac {dF(x)}{dx}}=f(x),\int _{a}^{b}f(x)\,dx=F(b)-F(a)}$

When the points M of the process explained in § 5 are inserted between the points whose abscissae are a and b, we may take them to be n − 1 in number, so that the segment AB is divided into n segments. Let x1, x2, ... xn−1 be the abscissae of the points in order. The integral is the limit of the sum

${\displaystyle f(a)(x_{1}-a)+f(x_{1})(x_{2}-x_{1})+\ldots +f(x_{r})(x_{r+1}-x_{r})+\ldots +f(x_{n-1})(b-x_{n-1})}$

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

${\displaystyle {F(x_{1})-F(a)}+{F(x_{2})-F(x_{1})}+\ldots +{F(x_{r+1})-F(x_{r})}+\ldots +{F(b)-F(x_{n-1})}}$

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+1xr), 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 “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 Notation.(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 Fundamental Artifice.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

${\displaystyle {\begin{matrix}\lim _{MN\to 0}{\frac {MN.QR}{MN}}\end{matrix}}=0.}$

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 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 Orders of small quantities.same order.” When this is not the case we may have either

${\displaystyle {\begin{matrix}\lim _{x\to 0}{\frac {x}{y}}\end{matrix}}=0,}$

or

${\displaystyle {\begin{matrix}\lim _{x\to 0}{\frac {y}{x}}\end{matrix}}=0.}$

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 by which differential coefficients are formed. These rules may be classified as “formal rules” and “particular results.” The formal rules may be stated as Rules of Differentiation. follows:—

(i.) The differential coefficient of a constant is zero
(ii.) For a sum u+v+ . . ., where u,v,... are functions of x,

${\displaystyle {\frac {d(u+v+\ldots +z)}{dx}}={\frac {du}{dx}}+{\frac {dv}{dx}}+\ldots +{\frac {dz}{dx}}}$

(iii.) For a product uv

${\displaystyle {\frac {d(uv)}{dx}}=u{\frac {dv}{dx}}+v{\frac {du}{dx}}.}$

(iv.) For a quotient u/v

${\displaystyle {\frac {d(u/v)}{dx}}=\left(v{\frac {du}{dx}}-u{\frac {dv}{dx}}\right){\Bigg /}v^{2}.}$

(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,

${\displaystyle {\frac {dy}{dx}}={\frac {dy}{dz}}\cdot {\frac {dz}{dx}}.}$

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:—

 ${\displaystyle y\,}$ ${\displaystyle {\frac {dy}{dx}}}$ ${\displaystyle x^{n}\,}$ ${\displaystyle {\begin{matrix}nx^{n-1}\\{\mbox{ for all values of }}n\end{matrix}}}$ ${\displaystyle \log _{a}x\,}$ ${\displaystyle x^{-1}log_{a}e\,}$ ${\displaystyle a^{x}}$ ${\displaystyle a^{x}log_{e}a\,}$ ${\displaystyle \sin {x}\,}$ ${\displaystyle \cos {x}\,}$ ${\displaystyle \cos {x}\,}$ ${\displaystyle -\sin {x}\,}$ ${\displaystyle \sin ^{-1}x\,}$ ${\displaystyle (1-x^{2})^{-{\tfrac {1}{2}}}\,}$ ${\displaystyle \tan ^{-1}x\,}$ ${\displaystyle (1+x^{2})^{-1}\,}$

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 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 Indefinite Integrals. have those in the following table:—

 ${\displaystyle f(x)\,}$ ${\displaystyle \int f(x)dx}$ ${\displaystyle x^{n}\,}$ ${\displaystyle {\begin{matrix}{\frac {x^{n+1}}{n+1}}\\{\mbox{ for all values of }}n{\mbox{ except }}-1\end{matrix}}}$ ${\displaystyle {\frac {1}{x}}\,}$ ${\displaystyle \log _{e}x\,}$ ${\displaystyle e^{ax}\,}$ ${\displaystyle a^{-1}e^{ax}\,}$ ${\displaystyle \cos {x}\,}$ ${\displaystyle \sin {x}\,}$ ${\displaystyle \sin {x}\,}$ ${\displaystyle -\cos {x}\,}$ ${\displaystyle (a^{2}-x^{2})^{-{\frac {1}{2}}}\,}$ ${\displaystyle \sin ^{-1}{\frac {x}{a}}\,}$ ${\displaystyle {\frac {1}{a^{2}+x^{2}}}}$ ${\displaystyle {\frac {1}{a}}\tan ^{-1}{\frac {x}{a}}}$

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).