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Originally appearing in Volume V14, Page 547 of the 1911 Encyclopedia Britannica.
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FUNCTION. 12. The processes of the integral calculus consist largely in trans-formations of the functions to be integrated into such fndef.+nite forms that they can be recognized as differential co- efficients lnteara/s. of functions which have previously been differ- entiated. Corresponding to the results in the table of § II we have those in the following table: f(x) ff(x)dx --- xa+u x° n+I for all values of n except - I x log rx ea: a-lea: cosx sin x sin x -cos x (¢2-x2)-z x sin la a tan ' x a'2+x2 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 6 d8 hPlnw) either Rules of Differentiation. y a II. History. 13. The new limiting processes which were introduced in the development of the higher analysis were in the first instance Kepler's related to problems of the integral calculus. Johannes methods Kepler in his Astronomia nova . . . de motibus stellae of Integra- Martis (16og) stated his laws of planetary motion, to lion. 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. 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 add), where a is the radius of the auxiliary circle, and 4 is the angle ACp, that is A 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 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 f 0(a+ae cos 4 )d4 , where e is the eccentricity. Kepler evaluated the sum as proportional to d)+e sin o. Kepler soon afterwards occupied himself with the volumes of solids. The vintage of the year 'fors 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 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 Cava- the summation of indivisibles. The notion of a lien's continuum, such as the area within a closed curve, Inds- es. as being made up of indivisible parts, " atoms " of visib 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 regardeda 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 Jy xm+I ax'"dx m+I,(m integral). 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 in communication were Bonaventura Cavaliers, 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 f px'"dx by the process of forming the sum 0 1 I" }2 { ... (n—1)'" 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 x°" by forming this sum was found also Fermat's by Fermat (1636), who stated expressly that he method of arrived at it by generalizing a method employed by Integra-don. 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 latte' extension and the proofs were given in his memoir, Proportions 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=x"'(m'p—I), or in the form y = alx'"l+a2x'"2+ ... +a"x", L ogarithms. Successors of Cavalieri. where none of the indices is equal to -1, was used by John Wallis in his Arithmetica infinitorum (1655) as well as by Fermat (1659). The case in which m=—1 was that of the various ordinary rectangular hyperbola; and Gregory of Integra- lions. 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= r/(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 Integra- been obtained, and applications of them had been tion before made to the quadrature of the circle and other conic theintegrafsections, and to various problems concerning the Calculus' 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 geometrie (16J9). 16. The problem of maxima and minima and the problem of tangents had also by the same time been effectively solved. Fermatas Oresme in the r4th century knew that at a point where methods of the ordinate of a curve is a maximum or a minimum Differen- its variation from point to point of the curve is slowest; nation. Ind Kepler in the Stereofnetria 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 Rene Descartes early in 1638 on receiving a copy of Descartes's Gecmetrie (1637), and with it he sent to Descartes an account of his methods for solving the problem of tangents and for determining centres of gravity. Fermat's method for maxima and minima is essentially our method. Expressed in a more modern notation, what he did was to begin by connecting the ordinate y and the abscissa x of a point of a curve by an equation which holds at all points of the curve, then to subtract the value of y in terms of x from the value obtained by substituting x+E for x, then to divide the difference by E, to put E=o 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 0(x, y, A, E) =o, the left-hand member of which is divisible by E. Omitting the factor E, and putting E =o 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 Herigone 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 Rene 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. 17. Among the comparatively few mathematicians who before 1665 could perform differentiations was Isaac Barrow. In his book entitled Lectiones opticae et geometricae, Barrow's written apparently in 1663, 1664, and published in Differ-1669, 167o, he gave a method of tangents like that entiai of Roberval and Torricelli, compounding two velocities Triangle. 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. T 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 some-times 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 =f(x) is known, so that the area up to the ordinate x is given by F(x), the curve Barrow's y = F(x) can be drawn, and Barrow showed that the theorsion. subtangent of this curve is measured by the ratio of theorem. 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 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 r x or M'M Flo. 6. M N FIG. 7. 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 Newton's original thinking on the subject dates from the year lavestiga- tions. 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 Newton's it might be made known. One way of making it known method of would have been to print it in the Philosophical Trans-Series actions actions of the Royal Society, but this course was not adopted. Collins made a copy of the tract and sent it to Lord Brouncker, but neither of them brought it before the Royal Society. The tract contains a general proof of Barrow's inversion-theorem which is the same in principle as that in § 6 above. In this proof and elsewhere in the tract a notation is introduced for the momentary increment (momentum) of the abscissa or area of a curve; this " moment " is evidently meant to represent a moment of time, the abscissa representing time, and it is effectively the same as our differential element—the thing that Fermat had denoted by E, and Barrow by e, in the case of the abscissa. Newton denoted the moment of the abscissa by o, that of the area z by ov. He used the letter v for the ordinate y, thus suggesting that his curve is a velocity-time graph such as Galileo had used. Newton gave the formula for the area of a curve v=x'"(m -1) in the form z=x"'+1/(m+1). In the proof he transformed this formula to the form z" =c"x5, where n and p are positive integers, substituted x+o for x and z+ov for z, and expanded by the binomial theorem for a positive integral exponent, thus obtaining the relation z"+nz"-lov+ . . . =c^(xp ~pxp 10 F ...), from which he deduced the relation nz"-lv = c^ pxr-1 by an infinite series, using for this purpose the binomial theorem for negative and fractional exponents, that is to say, the expansion of (i +x)" 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 en-closed by surfaces, and centres of gravity, can be formulated as problems of quadratures, and can thus be solved either in finite terms or by means of infinite series. In the Quadratura (1676) the method of integration which is founded upon the inversion-theorem was carried out systematically. Among other results there given is the quadrature of curves expressed by equations of the form y=x"(a+bx'")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 Newton's the generated quantity a " fluent " and its rate of genera- method of tion a " fluxion." The fluxion of a fluent x is represented Fluxions. by x, and its moment, or " infinitely " small increment accruing in an " infinitely " short time, is represented by to. The problems of the calculus are stated to be (i.) to find the velocity at any time when the distance traversed is given; (u.) 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+xo and y+90 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 f unctions 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 § together with the particular result which we write (Ix"' dx =mx"-1 (m integral). a result which Newton obtained by expanding (x+io)'" 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 f x"dx=m+1. Newton added applications of his methods to maxima and minima, tangents and curvature. In a letter to Collins of date 1672 Newton stated that he had certain methods, and he described certain results which he had found by using them. These methods and results are those which are to be found in the Methodus fluxionum; but the letter makes no mention of fluxions and fluents or of the characteristic notation. The rule for tangents is said in the letter to be analogous to de Sluse's, but to be applicable to equations that contain irrational terms. 22. Newton gave the fluxional notation also in the tract De Quadratura curvarum (1676), and he there added to it notation for the higher differential coefficients and for indefinite integrals, as we call them. Just as x, y, z, . . . are fluents of which .x, 9, z, . . are the fluxions, so z, y, z, . can be treated as fluents of which the fluxions may be denoted by x, 9, 2', . . . In like manner the fluxions of these may be denoted by y, y, z, . and so on. Again x, y, z, . . may be regarded as fluxions of which the fluents maybe denoted by x, y, i and these again as fluxions of other quantities denoted by x, y, z, . . . and so on. No use was made of the notation c, 3, . . . 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 o, , x. . occur, but not the symbols z, 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. Nature of the discovery called the tesimal Calculus. by omitting the equal terms z" and c"x5 and dividing the remaining terms by o, tacitly putting o =--o after division. This relation is the same as v=x'". Newton pointed out that, conversely, from the relation v=xm the relation z=x'n+1/(m-F1) follows. He applied his formula to the quadrature of curves whose ordinates can be expressed as the sum of a finite number of terms of the form ax'"; and gave examples of its application to curves in which the ordinate is expressed x,"+1 Publication of the Fluxional Notation. Retarded Publication of the method of Fluxions. 23. It is difficult to account for the fragmentary manner of publication of the Fluxional Calculus and for the long delays which took place. At the time (1671) when Newton composed the Methodus fluxionum he contemplated bringing out an edition of Gerhard Kinckhuysen's treatise on algebra and prefixing his tract to this treatise. In the same year his " Theory of Light and Colours " was published in the Philosophical Transactions, and the opposition which it excited led to the abandonment of which we write " ;f ydx," but within a day or two he wrote " f y." He regarded the symbol " f " 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 " f " represents " sum," " d" represents " difference." His notation appears to have been practically settled before the end of 1675, for in November he wrote f ydy = by', 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 cones. statement of the binomial theorem and many results pondence relating to series. He stated that by means of such ofNewseries he could find areas and lengths of curves, centres ton and of gravity and volumes and surfaces of solids, but, as Leibnitz. 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 Leibnitz's quantitates moratur, et singulare pro illis calculi genus. Differ- In this memoir the differential dx of a variable x, considered as the abscissa of a point of a curve, is said calculus. to be an arbitrary quantity, and the differential dy of a related variable y, considered as the ordinate of the point, is defined as a quantity which has to dx the ratio of the ordinate to the subtangent, and rules are given for operating with differentials. These are the rules for forming the differential of a constant, a sum (or difference), a product, a quotient, a power (or root). They are equivalent to our rules (i.)-(iv.) of § 11 and the particular result d(x'") =mx"' -1dx. 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 the project with regard to fluxions. In 168o Collins sought the assistance of the Royal Society for the publication of the tract, and this was granted in x682. 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 intro-duce 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 differentialis, 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 Leibnitz's on the occasion of his first visit to London, early in course of diseoYefy, 1693, 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' Geometrie. He also read Pascal's Lettres de Deltonville, Gregory of St Vincent's Opus geometricum, Cavalieri's Indivisibles and the Synopsis geometrica of Honore 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 demon-station 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 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 amnia y for that 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, Develop- in which Leibnitz described the method of using his ment new differential calculus for the problem of quadratures. of the This was the first publication of the notation fydx. calculus. 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 " f." In 1696 the marquis de 1'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 (az) of becoming infinitely great in comparison with any power (20°) when xis 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 Dispute the publication by Nicolas Fatio de Duillier of a con- tract in which he stated that Newton was not only the cerning first, but by many years the first inventor, and insinu-Priority. ated 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 Honore 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 Keil], 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 fluxion, 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 Ada 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 Band `'lttsl' Con-notation of Leibnitz was universally adopted. This tlnental severance did not at first prevent a great advance in schools of mathematics in Great Britain. So long as attention =tit: 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 fluxion' 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 Colon 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 Opposi- that of any other scientific man of the day, declared tion that the employment of differentials was unnecessary, to the and that Leibnitz's second differential was meaningless calculus. (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 "Ana- opinions of mathematicians in matters of faith are cyst"con- likely to be more trustworthy than those of divines, troversy. 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 Institutlones 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 Encyclopedic methodique (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 Reflexions sur la metaphysique du calcul infinitesimal (1799) and by Joseph Louis de la Grange (generally called Lagrange) in writings from 176o onwards. Lagrange proposed in his Theorie 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 Resume des Mons ... sur le calcul infinitesimal of Augustin Louis Cauchy. Since that time it has been understood that the use of thephrase " infinitely small " in any mathematical argument is a figurative mode of expression pointing to a limiting process. In the opinion of many eminent Canchy's method of mathematicians such modes of expression are limits. 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 Arithdone in order to formulate any problem analytically, metica/ even though he might not be absolutely clear as to the basis of proper interpretation of the symbols, or able to render modem analysis. 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 " arithmetizatien 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 Igth century was directed to discovering the most general conditions in which particular processes, frequently but not universally applicable, can be used without scruple. 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. 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 de- QeO-termined by the functional relations form a curve. The metrical 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 6, 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 o. 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 R R above is sometimes called a " Jordan curve." It by no means follows that every curve of this kind has a tan-gent. In order that the curve may have a tangent Tangents. at P it is necessary that, if any angle a, however o small, is specified, a distance b can be found such that when P is between Q and Q', and PQ and PQ' are less than 6, the angle RPR' is less than a 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 Ay/Ox has a limit when Ox tends to zero, y is a differentiable function of x, and the limit in question is the differential coefficient. The derived function, or differential coefficient, of a function f(x) is always defined by the formula f'(x) ddx) =lim.I,=of(x+ -f(x) Rules for the formation of differential coefficients in particular cases have been given in §r1 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 co-efficient 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 Progress values, and a different value when h approaches zero siveead, through negative values. The two limits are then called Regressive the " progressive " and ' regressive " differential co-Dltlerea- efficients. In applications to dynamics, when x denotes tial Co- a coordinate and t the time, dx/dt denotes a velocity. If e/fideats. the velocity is changed suddenly the progressive differ- ential 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 Ames. they must be so specified. Geometrical intuition may fail to answer the question of the existence or non-existence of the appropriate limits. In the last resort the definitions of many quantities of geometrical import must be analytical, not geometrical. As illustrations of this statement we may take the definitions of the areas and lengths of curves. We may not assume that every curve has an area or a length. To find out whether a curve has an area or not, we must ascertain whether the limit expressed by fydx 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 RI, R2, ... R,,_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 PRI+RIR2-1- ... +R"-IQ. 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 indc-Lengths finitely, this limit is the length of the arc PQ. The limit 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 cf 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 = 1(dx)2+(dy)'l, 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 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 that, if the derived function f'(x) vanishes at all points of an interval, Constants the function f(x) is constant in the interval. It follows oflnte- that, if two functions have the same derived function gration. 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 f'(x), is itself a function of x, and its differential coefficient is denoted Nigher by f"(x) or d2y/dx2. In the second of these notations Dighea- d/dx is regarded as the symbol of an operation, that of tiaiCe differentiation with respect to x, and the index 2 means eittdeats- that the operation is repeated. In like manner we may express the results of n successive differentiations by fl") (x) or by d"y/dx". When the second differential coefficient exists, or the first is differentiable, we have the relationexist in cases in which f'(x) does not exist or is not differentiable. The result that, when the limit here expressed can be shown to vanish at all points of an interval, then f(x) must be a linear f unction of x in the interval, is important. The relation (i.) is a particular case of the more general relation fit) (x) =lim.r,.noh- "[f(x+nh) - of {(x+(n - Ohl +n(2 -, 1), f {x+ (n -2)h} -...+ (- I)nf(x)], As in the case of relation (i.) the limit expressed by the right-hand member may exist although some or all of the derived functions f'(x), f"(x), ... f("-i)(x) do not exist. Corresponding to the rule iii. of § 11 we have the rule for forming the nth differential coefficient of a product in the form d"(uv) - d"v du d"-'v n(n - I) d'-u d"-2v d"u dx" dx" dx dx'^_I 1.2 dx2 dx"-2 dx" where the coefficients are those of the expansion of (1+x)" to powers of x (n being a positive integer). The rule is due to Leibnitz, (1695). Differentials of higher orders may be introduced in the same way as the differential of the first order. In general when y=f(x), the nth differential d"y is defined by the equation d"y=f(")(x) (dx)", in which dx is the (arbitrary) differential of x. When d/dx is regarded as a single symbol of operation the symbol f...dx represents the inverse operation. If the former is denoted by D, the latter may be denoted by D-i. D" means that symbols the operation D is to be performed n times in succession; oPopera-D-" that the operation of forming the indefinite integral tioa, is to be performed n times in succession. Leibnitz's course of thought (§ 24) naturally led him to inquire after an interpretation of D". 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. 36. After the formation of differential coefficients the most important theorem of the differential calculus is the theorem of inter-mediate value (" theorem of mean Theorem value," " theorem of finite incre- of Interments," " Rolle's theorem," are mediate other names for it). This theorem value. may be explained as follows: Let A, B be two points of a curve y=f(x) (fig. 9). Then there is a point P between A and B at which the tangent is parallel to the secant AB. This theorem is expressed analytically in the statement that if f'(x) is continuous between a and b, there is a value xi of x between a and b which has the property expressed by the equation f(b) -f(¢).= f (xi). (i.) b-a The value xi can be expressed in the form a+0(b-a) where 0 is a number between o and 1. A slightly more general theorem was given by Cauchy (1823) to the effect that, if f'(x) and F'(x) are continuous between x=a and x = b, then there is a number 0 between o and 1 which has the property expressed by the equation F(h) - F(a) =F'}a+0(b - a)} f(b)-f(a) f'{a+0(b-a)}- The theorem expressed by the relation (i.) was first noted by Rolle (1690) for the case where f(x) is a rational integral function which vanishes when x=a and also when x=b. The general theorem was given by Lagrange (1997). 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) =f y' (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 f(x) and all its differential coefficients up to the nth inclusive are continuous in the interval betweenx=a andx fib, then there is a number 0 between o and i which has the property expressed by the equation f(b)=f(a)+(b-a)f'(a)+(b ¢)f"(a) +... +(b-a)"i f(" "(a) 2 (n- I)! .+(bn) f"){a+0(b-a)l. (i.) 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 Taylor's point are known. The function is expressed by a termin- Theorem. ated series, and, when the remainder tends to zero as n increases, it may be transformed into an infinite series. The theorem The limit expressed by the right-hand member of this equation may f"(x) =lim.s..aL(x±II) -2f(x)+f (x-h) h2 (i.) (ii.) was first given by Brook Taylor in his Methodus Incrementorum (1717) To this problem is reducible that of expanding y in powers of x when as a corollary to a theorem concerning finite differences. Taylor I x and y are connected by an equation of the form save the expression fcr f(x+z) in terms of f(x), f'(x), . as an infinite series proceeding by powers of z. His notation was that appropriate to the method of fluxions which he used. This rule for expressing a function as an infinite series is known as Taylor's theorem. The relation (i.), in which the remainder after n terms is put in evidence, was first obtained by Lagrange (1797). Another form of the remainder was given by Cauchy (1823) viz., (n—a) ~(I —B)"-'f"(a+B(b—a)}. The conditions of validity of Taylor's expansion in an infinite series have been investigated very completely oy 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 f(x)=1+xz+n oo( n)" I+3znxz (i.) n=1 The sum of the series f(o)+af(o)+Iff"(o)+ ... (ii.) is the same as that of the series e '—x2e 32+x4e as—... It is easy to prove that this is less than e' when x lies between o and t, and also that f(x) is greater than e -I when x=1/d 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=o is often called Maclaurin's theorem, because it was first explicitly stated by Colin Maclaurin in his Treatise of Fluxions (1i42). Maclaurin like Taylor worked exclusively with the fluxional calculus. Examples of expansions in series had been known for some time. The series for log (i+x) was obtained by Nicolaus Mercator (1668) the Expas- division, nand i tegrating the series m e mdbyf term. He slims in regarded his result as a " quadrature of the hyperbola." power Newton (1669) obtained the expansion of sin 'x by ex- serlea, panding (1- x2)-i by the binomial theorem and integrating the series term by term. James Gregory (1671) gave the series for tan'x. 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 Con-,tectandi (1713). These numbers Bi, B2, . . . called Bernoulli's numbers, are the coefficients so denoted in the formula esx = 1 -2+Bix2—4.x4+-at —..., and they are connected with the sums of powers of the reciprocals of the natural numbers by equations of the type (2n)! I 1 f " 2.n—1,2n12n 22n 32n ~- " The function x"'— 22~ IBix"''2—.. . has been called Bernoulli's function of the nth 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 xis given in terms of y by means of a power series of the form x =y(Co+Ciy+C2y2+...) (Co +o) =yfo(y), say, there arises the problem of expressing y as a power series in x. This problem is that of reversion of series. It can be shown that provided the absolute value of x is not too great, n = co x d"-' 1 n=2 I. dyn—` ifo(y) ini b=oy=a+xf(y), for which problem Lagrange (1770) obtained the formula x" d"-i to y—a I xf(a) } n=2 Cn! . d¢"-Ltf(a)) ]. For the history of the problem and the generalizations of Lagrange's result reference may be made to O. Stolz, Grundzuge d. Duff. u. Int. Rechnung, T. 2 (Leipzig, 1896). 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 c(x) and ,/-(x) both vanish at x=a, the fraction 0(x)At,(x) may have a finite Indetere limit at a. This limit is described as the limit of an informs. " indeterminate form." Such indeterminate forms were forms. considered first by de 1'Hospital (1696) to whom the problem of evaluating the limit presented itself in the form of tracing the curve y=0(x)/¢(x) near the ordinate x=a, when the curves y=0(x) and y=il,(x) both cross the axis of x at the same point as this ordinate. In fig. to PA and QA represent short arcs of the curves 4,, ¢, chosen so that P and Q have the same abscissa. k' 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 1'Hospital treated PM and QM as " infinitesimal," so that the equations PM :AM =45'(a) and QM :AM =¢'(a) could FIG. to. be assumed to hold, and he arrived at the result that the " true value " of 4,(a)/O(a) is o'(a)l¢'(a). It can be proved rigorously that, if ¢'(x) does not vanish at x =a, while 4,(a) =o and ¢(a) =o, then rim 0(x) -4',(a) 4(x) ~(a) It can be proved further if that 4,"'(x) and st,"(x) are the differential coefficients of lowest order of 4,(x) and,y (x) which do not vanish at x =a, and if m =n, then Iim. 0(x)=~n(a) x=a4,(x) iG"(a). If m>n the limit is zero; but if m Differentials of higher orders are introduced by the defining equation dnf = (dxax+dyay) f anf = (dx)"axa+n(dx)n-idyaxn_iay+ .. . in which the expression (dx -+dya) n is developed by the binomial Y theorem in the same way as if dxax and dyay were numbers, and (a ) r (ay) n f is replaced by ax*af • 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 f (x, y) =o is the equation defining y as a function of x, we have f y_ f 3 2 _2 2 dx2 - ( ay) i \ay) axe ax ay axay+ \ax) aye 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 f of x and y which is such that of/ax=X, of/ay=Y. When this is the case we have the relation aY/ax = aX /ay. (ii.) Conversely, when this equation is satisfied there exists a function f which is such that df =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 ;n(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 az az s 02z p, q, r, s, t for ax' ay axe' axay' aye' 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 =f(x,y), the lines of curvature are determined by the equation 1(1 +q2)s -pqt) (dy)2+I (I +q2)r- (I +p2)t}dxdy -~(1+p2)s-pgr)(dx)2=o, and the principal radii of curvature are the values of R which satisfy the equation
End of Article: FUNCTION

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