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 Retarded Publication of the method of Fluxions. 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 the project with regard to fluxions. In 1680 Collins sought the assistance of the Royal Society for the publication of the tract, and this was granted in 1682. Yet it remained unpublished.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 introduce 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 diferentialis, 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 on the occasion of his first visit to London, early in Leibnitz’s course of discovery. 1673, 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’ Géométrie. He also read Pascal’s Lettres de Dettonville, Gregory of St Vincent’s Opus geometricum, Cavalieri’s Indivisibles and the Synopsis geometrica of Honoré 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 demonstration 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
1⁄4π = 1 − 1⁄3 + 1⁄5 − 1⁄7 + ...
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 omnia y for that which we write “∫ ydx,” but within a day or two he wrote “∫ y”. He regarded the symbol “∫” 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 “∫” represents “sum,” “d” represents “difference.” His notation appears to have been practically settled before the end of 1675, for in November he wrote ∫ ydy = ½ y2, 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 statement of the binomial theorem and many results Correspondence of Newton and Leibnitz. relating to series. He stated that by means of such series he could find areas and lengths of curves, centres of gravity and volumes and surfaces of solids, but, as 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 quantitates moratur, et singulare pro illis calculi genus. Leibnitz’s Differential Calculus. In this memoir the differential dx of a variable x, considered as the abscissa of a point of a curve, is said 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(xm) = mxm−1 dx.
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 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, in which Leibnitz described the method of using his new differential calculus for the problem of quadratures. Development of the Calculus. This was the first publication of the notation ∫ ydx. 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 “∫.” In 1696 the marquis de l’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 (ax) of becoming infinitely great in comparison with any power (xn) when x is 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.