129 



FLUXIONS. 



FLUXIONS. 



130 



fluxions in the year 1665 ; and that in a tract written in 1666 he had 

 begun partially to use the notation of fluxions. In 1669 Barrow 

 communicated to Collins the tract of Newton, afterwards published 

 under the title of ' De Analysi per Equationes numero termiuorum 

 infinitas ;' of which he afterwards says, " I am glad my friend's paper 

 gives you so much satisfaction : bis name is Mr. Newton, a fellow of 

 our college, and very young (being but the second year Master of Arts), 

 but of an extraordinary genius and proficiency in these things." This 

 tract contains a method of series, and many problems solved by 

 application of limits to differences obtained by expansion ; but no 

 direct method of fluxions. It was first published by William Jones, 

 who had become the possessor of Collins's papers, in 1711, in a tract 

 which is little known, having been superseded in the following year by 

 the publication of the Commercium Epistolicum. Various letters of 

 Newton, Collins, and others, up to the beginning of 1676, state that 

 the first-named had invented a method by which tangents could be 

 drawn, &c., without the necessity of freeing their equations from 

 irrational terms. Among them is a letter from Newton to Collins, 

 dated December 10, 1672, in which he states the fact of his discovery, 

 with one example. This letter the committee [COMMERCIUM EPIS- 

 TOLICUM] assert to have been sent to Leibnitz, but without proof : 

 and it has been since ascertained that nothing but an imperfect extract 

 was sent to Leibnitz. Leibnitz desired to have this method communi- 

 cated to him ; and Newton, at the request of Oldenburg and Collins, 

 wrote to the former the celebrated letters of June 13 and August 24, 

 1676. In the first he states the binomial theorem, and various 

 consequences of it in combination with his method, but without 

 giving any information as to that method. Leibnitz in a reply, also 

 addressed to Oldenburg, speaks in the highest terms of what Newton 

 had sent, and requests further explanation. Newton, hi the second 

 letter just mentioned, then explained how he arrived at the binomial 

 theorem [BINOMIAL THEOREM], and gives various results of his 

 method. He also communicated his method of fluxions and fluents 

 in ripher (as was often practised at the time), if cipher it could be 

 called, which had no method by which it could be deciphered. It 

 consisted in placing in alphabetical order all the letters in the sentence 

 communicated. Thus Newton gravely tells Oldenburg that hia method 

 of drawing tangents was 



6 ace dec ISeff 7 31 9n 4o 4jrr 4 9< 12 vx; 



or, that if any one could arrange six a t, two c g, one d, ftc., into a 

 certain sentence, he would see the method. That sentence was, Data 

 jEquatione quotcunque fluentes quantitates involvente fluxiones 

 invenire, et vice versa. If Leibnitz could have taken a hint either 

 from the preceding letters in alphabetical order, or (had he known it) 

 iu their significant arrangement, he would have deserved as much 

 credit for his sagacity, as if he had made the invention independently. 

 We cannot find anything in the rest of the letter which could give any 

 such hint; and certainly Newton, who showed himself desirous to 

 conceal the method, and knew that his letter was to come under the 

 acute eye of Leibnitz, did not imagine that he had in any part of it 

 betrayed his secret. This letter, of October 24, 1676, had not been 

 gent to Leibnitz, March 5, 1677, as Collins informs Newton by letter 

 of that date. So early as June 21, of the same year, however, Leibnitz 

 had received that letter and written an answer to Collins, in which, 

 without any desire of concealment, he explains the principle, notation, 

 and use of his differential calculus : this letter was published in the 

 ' Commercium Epistolicum.' It is of this correspondence that Newton 

 wrote the celebrated scholium ; of which, as we shall see, he was after- 

 wards weak enough, first to deny the plain and obvious meaning, and 

 secondly, to omit it entirely from the third edition of the ' Principia.' 

 This scholium, very literally translated, is as follows (book ii. prop. 

 7, scholium). 



A.D. 1687. " In letters which went between me and that most 

 excellent geometer, G. O. Leibnitz, ten years ago, when I signified 

 that I was in the knowledge of a method of determining maxima and 

 minima, of drawing tangents, and the like, and when I concealed it in 

 transposed letters involving this sentence {Data icquatione, &c., above 

 cited), that most distinguished man wrote back that he had also 

 fallen upon a method of the same kind, and communicated his 

 method, which hardly differed from mine/ except in his forms of words 

 and symbols." 



It will be convenient here to give Newton's subsequent explanation, 

 given in the year 1716, taken from his remarks on Leibnitz's letter to 

 Conti of April 9, 1716, published in 1716 in the appendix to 

 Raphson's ' History of Fluxions.' 



" He pretends that in my book of principles I allowed him the 

 invention of the calculus differentialis, independently of my own : 

 and that to attribute this invention to myself is contrary to my 

 knowledge there avowed. But in the paragraph there referred unto I 

 <U> not find one word to this purpose. On the contrary, I there 

 represent that I Kent notice of my method to Mr. Leibnitz before he 

 sent notice of IUH method to me : and left him to make it appear that 

 he had found his method before the date of my letter ; that is, eight 

 months * at the least before the date of his own. And, by referring 



He mnt h known by Collim'i letter tht itwa 

 AKTS AMD SCI. DIV. VOL. IV. 



not three. 



to the letters which passed between Mr. Leibnitz and me ten years 

 before, I left the reader to consult those letters * and interpret the 

 paragraph thereby. For by those letters he would see that I wrote 

 a tract on that method and the method of series together, five years 

 before the writing of these letters; that is, in the year 1671. And 

 these hints were as much as was proper in that short paragraph, it 

 being beside the design of that book to enter into disputes about these 

 matters." 



Nothing material passed till 1684, in which Leibnitz gave his first 

 paper on the Differential Calculus in the Leipzig Acts. In 1687 the 

 ' Principia ' was published by Newton ; and Leibnitz continued to give 

 papers on the subject of his new calculus. The Bernoullis began to 

 cultivate the subject about the year 1691, and as they were on terms 

 of correspondence with Leibnitz, he was the source from whence they 

 drew, and to which they returned, additional ideas on the subject. 

 The Marquis de 1'Hopital was employed in writing his elementary 

 treatise (the first written), which was published in 1696. All these 

 considered Leibnitz as their chief ; and the consequence was that Dr. 

 Wallis informs Newton, by letter of April 10, 1695, that "he had 

 heard that his notions of fluxions passed in Holland with great applause 

 by the name of Leibnitz's Calculus Differentialis." Accordingly, Wallis, 

 who had just completed printing the first volume of his works (the 

 third, which contains Newton's letters to Oldenburg, having been pre- 

 viously printed), inserted in the preface, as a reason for not mentioning 

 the differential calculus, that it was Newton's method of fluxions 

 which had been communicated to Leibnitz in the Oldenburg Letters. 

 A review of Wallis's works, in the 'Acta Eruditorum, or Leipzig 

 Acts," for 1696, reminds the reader of Newton's own admission 

 above cited. On this Newton (Raphson, supplement above cited) 

 remarks, " Whether Mr. Leibnitz invented it after me, or had it 

 from me, is a question of no consequence, for second inventors have 

 no right." 



In 1699 Fatio de Duillier, a Genevese, settled in England, stated iu 

 a mathematical work his conviction that Newton was the first inventor, 

 adding that he left it to those who had seen the manuscripts and 

 letters to say whether Leibnitz borrowed from Newton. This was the 

 first distinct suspicion of plagiarism ; and Leibnitz, who had never 

 contested the priority of Newton's discovery, and who appeared to be 

 quite satisfied by Newton's admission, now appears for the first time in 

 the controversy. In a reply to Duillier (Leipzig Acts, 1700), after 

 calling attention to Newton's scholium, he declares that when he pub- 

 lished his method, in 1684, he knew nothing more of any method of 

 Newton, except that the latter had written to him that he could dis- 

 pense with the removal of irrational terms ; and that, though ou the 

 publication of the ' Principia ' he became aware how much further its 

 author had pushed his discoveries, he did not know that Newton 

 possessed a calculus (or organised method) like the differential, till the 

 publication of Wallis's preface. 



The ' Quadrature of Curves ' was published by Newton in 1704, at 

 the end of his ' Optics.' It contains a formal exposition (the first 

 published) of the method and notation of fluxions. Some propositions 

 had been already published by Wallis. But in all that Newton had 

 previously allowed to be published, as well as in his early papers which 

 have been published in our own time, he uses the language and ideas 

 of infinitely small quantities. These he now rejects. 



Since so gVeat a stress was laid by the parties to the quarrel on the 

 introduction of specific notation, we may remark that Newton himself 

 did not very soon adopt such a course. He says that in 1666 he 

 " sometimes used a letter with one prick for quantities involving first 

 fluxions ; and the same quantity with two pricks for quantities 

 involving second fluxions." Even so late as 1687 he does not (in the 

 ' Principia ') give any notation for the momenta to which he had given 

 a name, and (though not laying any stress on it) we doubt whether 

 Newton would ever have systematised his notation if he had not seen 

 the letter of Leibnitz referred to in the scholium. 



A review of the above work appeared in the ' Leipzig Acts," January, 

 1705, in which, after stating that the differential calculus had been 

 explained in that work by Leibnitz, its iuveutor, and further by the 

 Bernoullis, and De 1'Hdpital, the author proceeds as follows : "Instead 

 of the Leibnitian differences Newton applies and always has applied 

 (adhibet semperque adhibuit) fluxions, which are quam proximi as the 

 increments of flowing quantities generated in infinitely small times, 

 and has used them with elegance both in his ' Principia ' and in subse- 

 quent writings, just as (quemadmodum et) Fabri in his synopsis has 

 substituted (substituit) motion for the method of Cavalieri." This was 

 considered by Newton's friends as an imputation of plagiarism on their 

 chief; but such a construction was always strenuously resisted by 

 Leibnitz. On the one hand, it was declared that Newton was repre- 

 sented in the same light with regard to Leibnitz as Fabri to Cavalieri, 

 by the force of " quemadmodum et ; " on the other, it was replied 

 that the distinction between separate invention and borrowing was 

 preserved in adhibuit and substituit. We are inclined to suspect that 

 the meaning of the writer was not very fair, though the words semper- 

 nue adhibuit are rather in his favour. Be this as it may, the preceding 

 sentence called forth the assertion of Keill (' Phil. Trans.,' 1708), that 



* They hd not then been published, nor was it known that they were to bo 

 published. 



K 



