him a method which was nearly the same as his own, and in his reply to Leibnitz's letter of 9 April 1716 (Raphson, History of Fluxions, p. 122) we find Newton saying, ‘And as for the Scholium … which is so much wrested against me, it was written, not to give away that lemma to Mr. Leibnitz, but, on the contrary, to assert it to myself.’ And again (p. 115), writing of the same scholium, he says: ‘I there represent that I sent notice of my method to Mr. Leibnitz before he sent notice of his method to me, and left him to make it appear that he had found his method before the date of my letter,’ while in an unpublished manuscript, entitled ‘A Supplement to the Remarks,’ part of which is quoted by Brewster (Life of Newton, vol. ii. chap. xiv.), Newton explains that Leibnitz's silence in 1684 as to who was the author of the ‘methodus similis’ mentioned by him in his first paper on the calculus put on Newton himself ‘a necessity of writing the scholium … lest it should be thought that I borrowed that lemma from Mr. Leibnitz.’ In the Portsmouth papers there are various suggested forms for the new scholium (ib. vol. ii. chap. xiv.). In the end all reference to Leibnitz was omitted, and the scholium only contains a paragraph from the letter to Collins of 10 Dec. 1672, explaining that the method of tangents was a particular case or corollary of a general method of solving geometrical and mechanical problems.
The main facts of this controversy establish without any doubt that Newton's invention of fluxions was entirely his own. It is not so easy to decide how much Leibnitz owed to Newton.
Oldenburg clearly sent to Leibnitz on 26 July 1676, along with Newton's letter of the preceding 13 June giving a brief account of his method, a collection made by Collins from the writings of James Gregory, and a copy of part of a letter from Newton to Collins, dated 10 Dec. 1672, ‘in quâ Newtonus se Methodum generalem habere dicit ducendi Tangentes, quadrandi curvilineas et similia peragendi.’ The ‘Commercium Epistolicum’ and Newton himself assumed that the complete letter of 1672 was forwarded. It is, however, practically certain that the whole was not sent. The example of the method given by Newton was omitted. In Leibnitz's ‘Mathematical Works,’ published at Berlin in 1849, there are printed from manuscripts left by him the papers said to have been received by him from Oldenburg in 1676. In these, as in a draft by Collins known as the ‘Abridgement,’ preserved at the Royal Society (MSS. vol. lxxxi.), we find a list of problems from Newton's letter of 10 Dec. 1672, but not the example of the method of drawing a tangent which formed the second part of the letter. In the second edition of the ‘Commercium’ (p. 128), it is stated that a much larger ‘Collectio’ made by Collins, and also preserved at the Royal Society (MSS. vol. lxxxi.), was sent to Leibnitz, but there is no evidence of this, and it is almost certainly an error (Edleston, Cotes Corr. n. 35).
The papers in their possession bearing on the subject were in 1880 examined for the Royal Society by Mr. Rix, clerk of the society. They tend to prove that Leibnitz did not get that full information about Newton's method which Newton believed him to have derived from the letter of 1672.
But if Leibnitz had not seen the whole of that letter, there can be little doubt, especially after Gerhardt's discovery of Leibnitz's autograph copy of part of it at Hanover among his autograph letters, that Collins had shown him in 1676 the no less important manuscript ‘De Analysi per Æquationes.’ Dealing with the matter in the preface to the Portsmouth collection, Dr. Luard, Sir G. Stokes, Professor Adams, and Professor Liveing express the view ‘that Newton was right in thinking that Leibnitz had been shown his manuscript’ (the ‘Tract de Analysi’). Mr. Ball (Short Hist. of Math. p. 366) comes to the same conclusion. Dr. Brewster, who wrote before Gerhardt's discovery, thought that Newton and Leibnitz borrowed nothing from each other. But it is almost certain that Leibnitz owed much to Newton, though the form in which he presented the calculus is, to quote Mr. Ball (Short Hist. of Math. p. 367), ‘better fitted to most of the purposes to which the infinitesimal calculus is applied than that of fluxions.’
In the same year (1705) in which the two struggles with Flamsteed and Leibnitz respectively began, Newton was knighted by Queen Anne on the occasion of her visit to Cambridge (15 April), and a month later, 17 May, he was defeated in the university election. The tory candidates were successful with the cry of ‘The church in danger;’ it is said they were carried by the votes of the non-residents against the wishes of the residents (Brewster, Life of Newton, ii. 162). In 1709 the correspondence relative to the second edition of the ‘Principia’ commenced. Dr. Bentley had succeeded in the summer of 1708 in obtaining a promise to republish the work, and it was arranged that Roger Cotes, then a fellow of Trinity College, and the first Plumian professor, should edit the book.