Part 17

a result which Newton obtained by expanding (x = [.x]o)^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 _ / x^(m+1) | x^m 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.

Publication of the Fluxional Notation.

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], [.y], [.z], ... are the fluxions, so [.x], [.y], [.z], ... can be treated as fluents of which the fluxions may be denoted by [:x], [:y], [:z],... In like manner the fluxions of these may be denoted by [:x], [:y], [:z], ... and so on. Again x, y, z, ... may be regarded as fluxions of which the fluents may be denoted by ['x], ['y], ['z], ... and these again as fluxions of other quantities denoted by ["x], ["y], ["z], ... and so on. No use was made of the notation ['x], ["x], ... 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 0, [.x], [:x], ... occur, but not the symbols ['x], ["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.

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 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."

Leibnitz's course of discovery.

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 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 1 1 1 --- [pi] = 1 - --- + --- - --- + ... 4 3 5 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 "[int] ydx," but within a day or two he wrote "[int] y". He regarded the symbol "[int]" 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 "[int]" represents "sum," "d" represents "difference." His notation appears to have been practically settled before the end of 1675, for in November he wrote [int] y dy = ½y², just as we do now.

Correspondence of Newton and Leibnitz.

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 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.

Leibnitz's Differential Calculus.

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_. 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(x^m) = mx^(m-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.

Development of the Calculus.

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. This was the first publication of the notation [int] 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 "[int]." 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 (a^x) of becoming infinitely great in comparison with any power (x^n) 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.

Dispute concerning Priority.

28. The famous dispute as to the priority of Newton and Leibnitz in the invention of the calculus began in 1699 through the publication by Nicolas Fatio de Duillier of a tract in which he stated that Newton was not only the first, but by many years the first inventor, and insinuated 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 Honoré 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 Keill, 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 _fluxions_, 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 _Acta 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."

British and Continental Schools of Mathematics.

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 notation of Leibnitz was universally adopted. This severance did not at first prevent a great advance in mathematics in Great Britain. So long as attention 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 fluxions. 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 Colin 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.

Oppositions to the calculus.

The "Analyst" controversy.

Cauchy's method of limits.