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Rh recognition of the fact that differentiation, known to be a

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

19. Newton was Barrow’s pupil, and he knew to start with in 1664 all that Barrow knew, and that was practically all that was known about the subject at that time. His original thinking on the subject dates from the year of the great plague (1665–1666), and it issued in the

invention of the “Calculus of Fluxions,” the principles and methods of which were developed by him in three tracts entitled De analysi per aequationes numero terminorum infinitas, Methodus fluxionum et serierum infinitarum, and De quadratura curvarum. None of these was published until long after they were written. The Analysis per aequationes was composed in 1666, but not printed until 1711, when it was published by William Jones. The Methodus fluxionum was composed in 1671 but not printed till 1736, nine years after Newton’s death, when an English translation was published by John Colson. In Horsley’s edition of Newton’s works it bears the title Geometria analytica. The Quadratura appears to have been composed in 1676, but was first printed in 1704 as an appendix to Newton’s Opticks.

20. The tract De Analysi per aequationes was sent by Newton to Barrow, who sent it to John Collins with a request that it might be made known. One way of making it known would have been to print it in the Philosophical Transactions 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 in the form  In the proof he transformed this formula to the form zn = cn xp, where n and p are positive integers, substituted x + o for x and z + ov for z, and expanded by the binomial theorem for a positive integral exponent, thus obtaining the relation

$z^n+nz^{n-1}ov+\ldots=c^n(x^p+px^{p-1}o+\ldots) \,$,

from which he deduced the relation

$nz^{n-1}v=c^npx^{p-1} \,$

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

21. In the Methodus fluxionum (1671) Newton introduced his characteristic notation. He regarded variable quantities as generated by the motion of a point, or line, or plane, and called the generated quantity a “fluent” and its rate of generation a “fluxion.” The fluxion of a fluent x is represented

by x, and its moment, or “infinitely” small increment accruing in an “infinitely” short time, is represented by ẋo. The problems of the calculus are stated to be (i.) to find the velocity at any time when the distance traversed is given; (ii.) to find the distance traversed when the velocity is given. The first of these leads to differentiation. In any rational equation containing x and y the expressions x + ẋo and y +ẏo are to be substituted for x and y, the resulting equation is to be divided by o, and afterwards o is to be omitted. In the case of irrational functions, or rational functions which are not integral, new variables are introduced in such a way as to make the equations contain rational integral terms only. Thus Newton’s rules of differentiation would be in our notation the rules (i.), (ii.), (v.) of § 11, together with the particular result which we write

$\frac{dx^m}{dx}=mx^{m-1}$, (m integral).

a result which Newton obtained by expanding (x + ẋo)m by the binomial theorem. The second problem is the problem of integration, and Newton’s method for solving it was the method of series founded upon the particular result which we write

$\int x^m \,dx=\frac{x^{m+1}}{m+1}.$|undefined

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 ẋ, ẏ, ż, are the fluxions, so ẋ, ẏ, ż,  can be treated as fluents of which the fluxions may be denoted by ẍ, ÿ, z̈,... In like manner the fluxions of these may be denoted by ẍ, ÿ, z̈, and so on. Again x, y, z, may be regarded as fluxions of which the fluents may be denoted by x́, ý, ź, and these again as fluxions of other quantities denoted by x&#779;, y&#779;, z&#779;, and so on. No use was made of the notation x́, x&#779;, 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, ẋ, ẍ, occur, but not the symbols x́, x&#779;, Wallis’s treatise also contains Newton’s formulation of the problems of the calculus in the words Data aequatione fluentes quotcumque quantitates involvente fluxiones invenire et vice versa (“an equation containing any number of fluent quantities being given, to find their fluxions and vice versa”). In the Philosophiae naturalis principia mathematica (1687), commonly called the “Principia,” the words “fluxion” and “moment” occur in a lemma in the second book; but the notation which is characteristic of the calculus of fluxions is nowhere used.

23. It is difficult to account for the fragmentary manner of publication of the Fluxional Calculus and for the long delays which took place. At the time (1671) when Newton composed the Methodus fluxionum he contemplated 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