Part 16

He had observed that the focal radius vector SP (fig. 5) is equal to the perpendicular SZ drawn from S to the tangent at p to the auxiliary circle, and he had further established the theorem which we should now express in the form--the differential element of the area ASp as Sp turns about S, is equal to the product of SZ and the differential ad[phi], where a is the radius of the auxiliary circle, and [phi] is the angle ACp, that is the eccentric angle of P on the ellipse. The area ASP bears to the area ASp the ratio of the minor to the major axis, a result known to Archimedes. Thus Kepler's radii are spaced according to the rule that the eccentric angles of their ends are equidifferent, and his "sum of radii" is proportional to the expression which we should now write _ / [phi] | (a + ae cos [phi]) d[phi], _/ 0

where e is the eccentricity. Kepler evaluated the sum as proportional to [phi] + e sin [phi].

Kepler soon afterwards occupied himself with the volumes of solids. The vintage of the year 1612 was extraordinarily abundant, and the question of the cubic content of wine casks was brought under his notice. This fact accounts for the title of his work, _Nova stereometria doliorum; accessit stereometriae Archimedeae supplementum_ (1615). In this treatise he regarded solid bodies as being made up, as it were (_veluti_), of "infinitely" many "infinitely" small cones or "infinitely" thin disks, and he used the notion of summing the areas of the disks in the way he had previously used the notion of summing the focal radii of an ellipse.

Logarithms.

14. In connexion with the early history of the calculus it must not be forgotten that the method by which logarithms were invented (1614) was effectively a method of infinitesimals. Natural logarithms were not invented as the indices of a certain base, and the notation e for the base was first introduced by Euler more than a century after the invention. Logarithms were introduced as numbers which increase in arithmetic progression when other related numbers increase in geometric progression. The two sets of numbers were supposed to increase together, one at a uniform rate, the other at a variable rate, and the increments were regarded for purposes of calculation as very small and as accruing discontinuously.

Cavalieri's Indivisibles.

15. Kepler's methods of integration, for such they must be called, were the origin of Bonaventura Cavalieri's theory of the summation of indivisibles. The notion of a continuum, such as the area within a closed curve, as being made up of indivisible parts, "atoms" of area, if the expression may be allowed, is traceable to the speculations of early Greek philosophers; and although the nature of continuity was better understood by Aristotle and many other ancient writers yet the unsound atomic conception was revived in the 13th century and has not yet been finally uprooted. It is possible to contend that Cavalieri did not himself hold the unsound doctrine, but his writing on this point is rather obscure. In his treatise _Geometria indivisibilibus continuorum nova quadam ratione promota_ (1635) he regarded a plane figure as generated by a line moving so as to be always parallel to a fixed line, and a solid figure as generated by a plane moving so as to be always parallel to a fixed plane; and he compared the areas of two plane figures, or the volumes of two solids, by determining the ratios of the sums of all the indivisibles of which they are supposed to be made up, these indivisibles being segments of parallel lines equally spaced in the case of plane figures, and areas marked out upon parallel planes equally spaced in the case of solids. By this method Cavalieri was able to effect numerous integrations relating to the areas of portions of conic sections and the volumes generated by the revolution of these portions about various axes. At a later date, and partly in answer to an attack made upon him by Paul Guldin, Cavalieri published a treatise entitled _Exercitationes geometricae sex_ (1647), in which he adapted his method to the determination of centres of gravity, in particular for solids of variable density.

Among the results which he obtained is that which we should now write _ / x x^(m+1) | x^m dx = -------, (m integral). _/ 0 m + 1

He regarded the problem thus solved as that of determining the sum of the mth powers of all the lines drawn across a parallelogram parallel to one of its sides.

Successors of Cavalieri.

Fermat's method of Integration.

At this period scientific investigators communicated their results to one another through one or more intermediate persons. Such intermediaries were Pierre de Carcavy and Pater Marin Mersenne; and among the writers thus in communication were Bonaventura Cavalieri, Christiaan Huygens, Galileo Galilei, Giles Personnier de Roberval, Pierre de Fermat, Evangelista Torricelli, and a little later Blaise Pascal; but the letters of Carcavy or Mersenne would probably come into the hands of any man who was likely to be interested in the matters discussed. It often happened that, when some new method was invented, or some new result obtained, the method or result was quickly known to a wide circle, although it might not be printed until after the lapse of a long time. When Cavalieri was printing his two treatises there was much discussion of the problem of quadratures. Roberval (1634) regarded an area as made up of "infinitely" many "infinitely" narrow strips, each of which may be considered to be a rectangle, and he had similar ideas in regard to lengths and volumes. He knew how to approximate to the quantity which we express by [int] (0 to 1) x^m dx by the process of forming the sum

0^m + 1^m + 2^m + ... (n - 1)^m -------------------------------, n^(m+1)

and he claimed to be able to prove that this sum tends to 1/(m + 1), as n increases for all positive integral values of m. The method of integrating x^m by forming this sum was found also by Fermat (1636), who stated expressly that he arrived at it by generalizing a method employed by Archimedes (for the cases m = 1 and m = 2) in his books on _Conoids and Spheroids_ and on _Spirals_ (see T. L. Heath, _The Works of Archimedes_, Cambridge, 1897). Fermat extended the result to the case where m is fractional (1644), and to the case where m is negative. This latter extension and the proofs were given in his memoir, _Proportionis geometricae in quadrandis parabolis et hyperbolis usus_, which appears to have received a final form before 1659, although not published until 1679. Fermat did not use fractional or negative indices, but he regarded his problems as the quadratures of parabolas and hyperbolas of various orders. His method was to divide the interval of integration into parts by means of intermediate points the abscissae of which are in geometric progression. In the process of § 5 above, the points M must be chosen according to this rule. This restrictive condition being understood, we may say that Fermat's formulation of the problem of quadratures is the same as our definition of a definite integral.

Various Integrations.

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

y = x^m (m [Not Equal] -1),

or in the form

y = a1 x^m1 + a2 x^m2 + ... + a_n x^m_n,

where none of the indices is equal to - 1, was used by John Wallis in his _Arithmetica infinitorum_ (1655) as well as by Fermat (1659). The case in which m = - 1 was that of the ordinary rectangular hyperbola; and Gregory of St Vincent in his _Opus geometricum quadraturae circuli et sectionum coni_ (1647) had proved by the method of exhaustions that the area contained between the curve, one asymptote, and two ordinates parallel to the other asymptote, increases in arithmetic progression as the distance between the ordinates (the one nearer to the centre being kept fixed) increases in geometric progression. Fermat described his method of integration as a logarithmic method, and thus it is clear that the relation between the quadrature of the hyperbola and logarithms was understood although it was not expressed analytically. It was not very long before the relation was used for the calculation of logarithms by Nicolaus Mercator in his _Logarithmotechnia_ (1668). He began by writing the equation of the curve in the form y = 1/(1 + x), expanded this expression in powers of x by the method of division, and integrated it term by term in accordance with the well-understood rule for finding the quadrature of a curve given by such an equation as that written at the foot of p. 325.

Integration before the Integral Calculus.

By the middle of the 17th century many mathematicians could perform integrations. Very many particular results had been obtained, and applications of them had been made to the quadrature of the circle and other conic sections, and to various problems concerning the lengths of curves, the areas they enclose, the volumes and superficial areas of solids, and centres of gravity. A systematic account of the methods then in use was given, along with much that was original on his part, by Blaise Pascal in his _Lettres de Amos Dettonville sur quelques-unes de ses inventions en géométrie_ (1659).

Fermat's methods of Differentiation.

16. The problem of maxima and minima and the problem of tangents had also by the same time been effectively solved. Oresme in the 14th century knew that at a point where the ordinate of a curve is a maximum or a minimum its variation from point to point of the curve is slowest; and Kepler in the _Stereometria doliorum_ remarked that at the places where the ordinate passes from a smaller value to the greatest value and then again to a smaller value, its variation becomes insensible. Fermat in 1629 was in possession of a method which he then communicated to one Despagnet of Bordeaux, and which he referred to in a letter to Roberval of 1636. He communicated it to René Descartes early in 1638 on receiving a copy of Descartes's _Géométrie_ (1637), and with it he sent to Descartes an account of his methods for solving the problem of tangents and for determining centres of gravity.

Fermat's method for maxima and minima is essentially our method. Expressed in a more modern notation, what he did was to begin by connecting the ordinate y and the abscissa x of a point of a curve by an equation which holds at all points of the curve, then to subtract the value of y in terms of x from the value obtained by substituting x + E for x, then to divide the difference by E, to put E = 0 in the quotient, and to equate the quotient to zero. Thus he differentiated with respect to x and equated the differential coefficient to zero.

[Illustration: FIG. 6.]

Fermat's method for solving the problem of tangents may be explained as follows:--Let (x, y) be the coordinates of a point P of a curve, (x´, y´), those of a neighbouring point P´ on the tangent at P, and let MM´ = E (fig. 6).

From the similarity of the triangles P´TM´, PTM we have

y´:A - E = y:A,

where A denotes the subtangent TM. The point P´ being near the curve, we may substitute in the equation of the curve x - E for x and (yA - yE)/A for y. The equation of the curve is approximately satisfied. If it is taken to be satisfied exactly, the result is an equation of the form [phi](x, y, A, E) = 0, the left-hand member of which is divisible by E. Omitting the factor E, and putting E = 0 in the remaining factor, we have an equation which gives A. In this problem of tangents also Fermat found the required result by a process equivalent to differentiation.

Fermat gave several examples of the application of his method; among them was one in which he showed that he could differentiate very complicated irrational functions. For such functions his method was to begin by obtaining a rational equation. In rationalizing equations Fermat, in other writings, used the device of introducing new variables, but he did not use this device to simplify the process of differentiation. Some of his results were published by Pierre Hérigone in his _Supplementum cursus mathematici_ (1642). His communication to Descartes was not published in full until after his death (Fermat, _Opera varia_, 1679). Methods similar to Fermat's were devised by René de Sluse (1652) for tangents, and by Johannes Hudde (1658) for maxima and minima. Other methods for the solution of the problem of tangents were devised by Roberval and Torricelli, and published almost simultaneously in 1644. These methods were founded upon the composition of motions, the theory of which had been taught by Galileo (1638), and, less completely, by Roberval (1636). Roberval and Torricelli could construct the tangents of many curves, but they did not arrive at Fermat's artifice. This artifice is that which we have noted in § 10 as the fundamental artifice of the infinitesimal calculus.

[Illustration: FIG. 7.]

Barrow's Differential Triangle.

17. Among the comparatively few mathematicians who before 1665 could perform differentiations was Isaac Barrow. In his book entitled _Lectiones opticae et geometricae_, written apparently in 1663, 1664, and published in 1669, 1670, he gave a method of tangents like that of Roberval and Torricelli, compounding two velocities in the directions of the axes of x and y to obtain a resultant along the tangent to a curve. In an appendix to this book he gave another method which differs from Fermat's in the introduction of a differential equivalent to our dy as well as dx. Two neighbouring ordinates PM and QN of a curve (fig. 7) are regarded as containing an indefinitely small (_indefinite parvum_) arc, and PR is drawn parallel to the axis of x. The tangent PT at P is regarded as identical with the secant PQ, and the position of the tangent is determined by the similarity of the triangles PTM, PQR. The increments QR, PR of the ordinate and abscissa are denoted by a and e; and the ratio of a to e is determined by substituting x + e for x and y + a for y in the equation of the curve, rejecting all terms which are of order higher than the first in a and e, and omitting the terms which do not contain a or e. This process is equivalent to differentiation. Barrow appears to have invented it himself, but to have put it into his book at Newton's request. The triangle PQR is sometimes called "Barrow's differential triangle."

Barrow's Inversion-theorem.

The reciprocal relation between differentiation and integration (§ 6) was first observed explicitly by Barrow in the book cited above. If the quadrature of a curve y = f(x) is known, so that the area up to the ordinate x is given by F(x), the curve y = F(x) can be drawn, and Barrow showed that the subtangent of this curve is measured by the ratio of its ordinate to the ordinate of the original curve. The curve y = F(x) is often called the "quadratrix" of the original curve; and the result has been called "Barrow's inversion-theorem." He did not use it as we do for the determination of quadratures, or indefinite integrals, but for the solution of problems of the kind which were then called "inverse problems of tangents." In these problems it was sought to determine a curve from some property of its tangent, e.g. the property that the subtangent is proportional to the square of the abscissa. Such problems are now classed under "differential equations." When Barrow wrote, quadratures were familiar and differentiation unfamiliar, just as hyperbolas were trusted while logarithms were strange. The functional notation was not invented till long afterwards (see FUNCTION), and the want of it is felt in reading all the mathematics of the 17th century.

Nature of the discovery called the Infinitesimal Calculus.

18. The great secret which afterwards came to be called the "infinitesimal calculus" was almost discovered by Fermat, and still more nearly by Barrow. Barrow went farther than Fermat in the theory of differentiation, though not in the practice, for he compared two increments; he went farther in the theory of integration, for he obtained the inversion-theorem. The great discovery seems to consist

## partly in the recognition of the fact that differentiation, known to be

a useful process, could always be performed, at least for the functions then known, and partly in the recognition of the fact that the inversion-theorem could be applied to problems of quadrature. By these steps the problem of tangents could be solved once for all, and the operation of integration, as we call it, could be rendered systematic. A further step was necessary in order that the discovery, once made, should become accessible to mathematicians in general; and this step was the introduction of a suitable notation. The definite abandonment of the old tentative methods of integration in favour of the method in which this operation is regarded as the inverse of differentiation was especially the work of Isaac Newton; the precise formulation of simple rules for the process of differentiation in each special case, and the introduction of the notation which has proved to be the best, were especially the work of Gottfried Wilhelm Leibnitz. This statement remains true although Newton invented a systematic notation, and practised differentiation by rules equivalent to those of Leibnitz, before Leibnitz had begun to work upon the subject, and Leibnitz effected integrations by the method of recognizing differential coefficients before he had had any opportunity of becoming acquainted with Newton's methods.

Newton's investigations.

19. Newton was Barrow's pupil, and he knew to start with in 1664 all that Barrow knew, and that was practically all that was known about the subject at that time. His original thinking on the subject dates from the year of the great plague (1665-1666), and it issued in the invention of the "Calculus of Fluxions," the principles and methods of which were developed by him in three tracts entitled _De analysi per aequationes numero terminorum infinitas, Methodus fluxionum et serierum infinitarum, and De quadratura curvarum_. None of these was published until long after they were written. The _Analysis per aequationes_ was composed in 1666, but not printed until 1711, when it was published by William Jones. The _Methodus fluxionum_ was composed in 1671 but not printed till 1736, nine years after Newton's death, when an English translation was published by John Colson. In Horsley's edition of Newton's works it bears the title _Geometria analytica_. The _Quadratura_ appears to have been composed in 1676, but was first printed in 1704 as an appendix to Newton's _Opticks_.

Newton's method of Series.

20. The tract _De Analysi per aequationes ..._ was sent by Newton to Barrow, who sent it to John Collins with a request that it might be made known. One way of making it known would have been to print it in the _Philosophical Transactions_ of the Royal Society, but this course was not adopted. Collins made a copy of the tract and sent it to Lord Brouncker, but neither of them brought it before the Royal Society. The tract contains a general proof of Barrow's inversion-theorem which is the same in principle as that in § 6 above. In this proof and elsewhere in the tract a notation is introduced for the momentary increment (_momentum_) of the abscissa or area of a curve; this "moment" is evidently meant to represent a moment of time, the abscissa representing time, and it is effectively the same as our differential element--the thing that Fermat had denoted by E, and Barrow by e, in the case of the abscissa. Newton denoted the moment of the abscissa by o, that of the area z by ov. He used the letter v for the ordinate y, thus suggesting that his curve is a velocity-time graph such as Galileo had used. Newton gave the formula for the area of a curve v = x^m (m ± -1) in the form z = x^(m+1)/(m + 1). In the proof he transformed this formula to the form z^n = c^n x^p, where n and p are positive integers, substituted x + o for x and z + ov for z, and expanded by the binomial theorem for a positive integral exponent, thus obtaining the relation

z^n + nz^(n-1) ov + ... = c^n (x_p + px^(p-1)o + ...),

from which he deduced the relation

nz_(n-1)v = c^n px^(p-1)

by omitting the equal terms z^n and c^n·x^p and dividing the remaining terms by o, tacitly putting o = 0 after division. This relation is the same as v = x^m. Newton pointed out that, conversely, from the relation v = x^m the relation z = x^(m+1) / (m + 1) follows. He applied his formula to the quadrature of curves whose ordinates can be expressed as the sum of a finite number of terms of the form ax^m; and gave examples of its application to curves in which the ordinate is expressed by an infinite series, using for this purpose the binomial theorem for negative and fractional exponents, that is to say, the expansion of (1 + x)^n in an infinite series of powers of x. This theorem he had discovered; but he did not in this tract state it in a general form or give any proof of it. He pointed out, however, how it may be used for the solution of equations by means of infinite series. He observed also that all questions concerning lengths of curves, volumes enclosed by surfaces, and centres of gravity, can be formulated as problems of quadratures, and can thus be solved either in finite terms or by means of infinite series. In the _Quadratura_ (1676) the method of integration which is founded upon the inversion-theorem was carried out systematically. Among other results there given is the quadrature of curves expressed by equations of the form y = x^n·(a + bx^m)^p; this has passed into text-books under the title "integration of binomial differentials" (see § 49). Newton announced the result in letters to Collins and Oldenburg of 1676.

Newton's method of Fluxions.

21. In the _Methodus fluxionum_ (1671) Newton introduced his characteristic notation. He regarded variable quantities as generated by the motion of a point, or line, or plane, and called the generated quantity a "fluent" and its rate of generation a "fluxion." The fluxion of a fluent x is represented by x, and its moment, or "infinitely" small increment accruing in an "infinitely" short time, is represented by [.x]o. The problems of the calculus are stated to be (i.) to find the velocity at any time when the distance traversed is given; (ii.) to find the distance traversed when the velocity is given. The first of these leads to differentiation. In any rational equation containing x and y the expressions x + [.x]o and y +[.y]o are to be substituted for x and y, the resulting equation is to be divided by o, and afterwards o is to be omitted. In the case of irrational functions, or rational functions which are not integral, new variables are introduced in such a way as to make the equations contain rational integral terms only. Thus Newton's rules of differentiation would be in our notation the rules (i.), (ii.), (v.) of § 11, together with the

## particular result which we write

dx^m ---- = mx^(m-1), (m integral). dx