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INFINITESIMAL CALCULUS

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Originally appearing in Volume V14, Page 537 of the 1911 Encyclopedia Britannica.
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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 " in-finitely 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 the17th and 18th centuries were mainly a period of construction, the 19th century mainly a period of criticism. 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, y} metrkal when it was employed by Nicole Oresme, who studied represent-and afterwards taught at the College de Navarre in atton of Paris from 1348 to 1361. He represented one of two variabltttte variable quantities, e.g. the time that has elapsed Qnanes. 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. I). 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 probthe curve at an assigned point, and the problem of lems of determining the area of the curve. The relation of Maxima the problem of maxima and minima to the problem Mtalma. of tangents was understood in the sense that maxima Tangents, or minima arise when a certain equation has equal and Quad-roots, and, when this is the case, the curves by which ratures. 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 creek afterwards called the " method of exhaustions," and methods. the principle on which it is based was laid down in the lemma prefixed to the I2th 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+Ax, y+Ay those Differen- of' P'. The symbol Ax means " the difference of two tiado:a x's " and there is a like meaning for the symbol Ay. The fraction Ay/Ax is the trigonometrical tangent of the angle which the secant PP' makes with the axis of x. Now let Ax be continually diminished towards zero, so that P' continually approaches P. If the curve has a tangent at P the secant P P' approaches a limiting position (see § 33 below). When this is the case the fraction Ay/Ax 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 7x• If the equation of the curve is of the form y=f(x) where f is a functional symbol (see FUNCTION), then Ly —f(x+Ax)—f( .x Ax ' and dy= f(x+Ax)—f(x) dx hm. ox-Q Ax The limit expressed by the right-hand member of this defining equation is often written f'(x), and is called the " derived function " of f(x), sometimes the " derivative " or " derivate " of f(x). When the function f(x) is a rational integral function, the division by Ax can be performed, and the limit is found by substituting zero for Ax in the quotient. For example, if f(x) =x2, we have f(x+Ax) —f(x) _ (x+Ax)2—x2—2xAx+(Ax)2=2x+Ax, ax Ax Ax and f'(x) =2x. The process of forming the derived function of a given function is called differentiation. The fraction Ay/Ax is called the " quotient of differences," and its limit dy/dx is called the " differential co-efficient 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 y=f(x) be the equation of Integra- a curve, and let AC and BD be the ordinates of the points don. C and D (see fig. 3). Let a, b be the abscissae of these points. Let the segment AB be divided into a number of segments by means of intermediate points such as M, and let MN be one such segment. Let PM and QN be those ordinates of the curve which have M and N as their feet. On MN as base describe two rectangles, of which the heights are the greatest and least values of y which correspond to points s r D 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. A M N $ When the number of the points 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 what-ever 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'MVI' 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 theabscissa of P, x+Ax that of Q, x' that of P', the limit in question might be written lim. f (x')Ax, where the letters a, b written where the letters a, b written below and above the sign of summation indicate the extreme values of x. This limit is called " the definite integral of f (x) between the limits a and b," and the notation for it is rf(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 E now denote the abscissa of B. The area A of the figure ACDB is represented by the Theorem integral J f(x)dx, and it is a function of E. Let BD of Inver- be displaced to B'D' so that becomes E+ LIE (see ab a. fig. 4). The area of the figure ACD'B' is represented by the integral f+ f(x)dx, and the increment DA of the area is given by the formula AA=J +jf(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 AA is intermediate between HAE and hAl, and the quotient of differences AA/AE is intermediate between H and h. If the function f(x) is continuous at B (see FUNCTION), then, as A:: is diminished without limit, H and h tend to BD, or AO, as a limit, and we have aE =f(t). The introduction of the process of differentiation, together with the theorem here proved, placed the solution of the problem of quadratures on a new basis. It appears that we can always find the area A if we know a function F (x) which has f(x) as its differential coefficient. If f(x) is continuous between a and b, we can prove that dF(x) dx —Ax)' we are said to integrate the function f(x), and F(x) is called the indefinite integral of f(x) with respect to x, and is written (f(x)dx. 7. In the process of § 4 the/increment Ay is not in general equal to the product of the increment Ax and the derived function f'(x). In general we can write down an equation of the form Ay = f'(x)Ax+R, in which R is different from zero when Ax is different from zero; and then we have not only lim. ox- oR = o, but also oxco, R = o. We may separate Ay into two parts: the part f'(x) Ax and the part R. The part f'(x) Ax alone is useful for forming the differential coefficient, and it is convenient to give it a name. It is called the differential of f(x), and is written df(x), or dy when y is written for f(x). When this notation is adopted dx is written instead of Ax, and is called the " differential of x," so that we have df(x) =f'(x)dx. Thus the differential of an independent variable such as x is a finite difference; in other words it is any number we please. The differ- ential 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 D Ds A sa A =ff (x)dx = F (b) —F (a). When we recognize a function F(x) which has the property expressed by the equation Differentials. are either finite differences, or are so much of certain finite differences as are useful for forming differential coefficients. Again let F(x) be the indefinite integral of a continuous function f(x), so that we have dF(x) =f(x), f1(x)dx=F(b) -F (a). 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- I in number, so that the segment AB is divided into n segments. Let x3, x2, ... x„-1 be the abscissae of the points in order. The integral is the limit of the sum f(a) (xi-a) +f(xi) (x2-x,)+...+f(xr) (x,+r-xr) + ... +,f(xn-1)(b-x„-1), every term of which is a differential of the form f(x)dx. Further the integral is equal to the sum of differences IF(x,)-F(a)1+IF(x2)-F(x:)1+ ... +IF(xr+1)-F(xr)1 + ... +IF(b)-F(x„-,)I, for this sum is F(b)-F(a). Now the difference F(xr+I)-F(xr) is not equal to the differential 1(xr) (xr+r-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 " d " is the initial letter of the word differentia (difference) Notation. and the symbol " f " 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 Funds- equation containing x and y we can deduce a new equation, mental containing also ox and Ay, by substituting x+ax for x Artifice. and y+'y for y. If there is a differential coefficient of y with respect to x, then Ay can be expressed in the form O.:~x+R, where lim.y,=0(R/Ox) =o, 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 sp.Ox, or ~. dx, which is the differential dy. In the same way, in all applications of the integral calculus to geometry or mechanics we form the element of an integral in the same way as the element of area y. dx is formed. In fig. 3 of § 5 the element of area y. dx is the area of the rectangle RM. The actual area of the curvilinear figure PQNM is greater than the area of this rectangle by the area of the curvilinear figure PQR; but the excess is less than the area of the rectangle PRQS, which is measured by the product of the numerical measures of MN and QR, and we have MN . QR lim.MN-o MN =o. 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. to. 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 connected together by any relation, and if when x tends to small zero y also tends to zero, the fraction y/x may tend to a quantities. 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 im x 4 =0, or lim.x_Ox = o. 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. n. 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 beiclassified as " formal rules " and" particular results." The formal rules may be stated as follows: (i.) The differential coefficient of a constant is zero. (ii.) For a sum u+v+ ... +z, where u, v,...are functions of x, d(u+v+ ... +z) du dv dz dx -dx+dx+ • . • +•(iii.) For a product uv d(uv) dv + du dx -udx' (iv.) For a quotient u/v d(u/v) (du dv\ / 2 dx vdx-udx v . (v.) For a function of a function, that is to say, for a function expressed in terms of a variable z, which is itself expressed as function of x, dy_dy dz dx- dz dx In addition to these formal rules we have particular results as to the differentiation of simple functions. The most important results are written down in the following table Y dy dx xn nxn—t for all values of n logax x ' loge as a' loggia sin x cos x cos x -sin x sin ix (I —x2)-4 tan-'x (1 +x2)—' 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
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