Oldenburg. Next year he claimed to have arrived at ‘methodos quasdam analyticas generales et late fusas, quas majoris facio quam Theoremata particularia et exquisita.’ On his return to Paris he maintained through Oldenburg a correspondence with various English mathematicians, and heard of Newton and his great power of analysis. Thus he wrote, 30 March 1675 (Comm. Epist. p. 39): ‘Scribis clarissimum Newtonium vestrum habere methodum exhibendi quadraturas omnes;’ and a year later, May 1676, referring to a series due to Newton, ‘ideo rem gratam mihi feceris, vir clarissime, si demonstrationem transmiseris.’ Collins urged Newton to comply with Leibnitz's wishes, and Newton wrote, 13 June 1676, a letter giving a brief account of his method. This was read before the Royal Society on 15 June, and was sent to Leibnitz 26 July (ib. p. 49), together with a manuscript of Collins, containing extracts from the writings of James Gregory, and a copy of a letter, with a highly important omission, from Newton to Collins, dated 10 Dec. 1672, about his methods of drawing tangents and finding areas. Newton's example of drawing a tangent was omitted, as has been subsequently proved. Leibnitz replied to Oldenburg on 27 Aug. 1676, asking Newton to explain some points more fully, and giving some account of his own work. Newton replied through Collins on 24 Oct., expressing his pleasure at having received Leibnitz's letter, and his admiration of the elegant method used by him (ib. p. 67). He gives a brief description of his own procedure, mentioning his method of fluxions, which, he says, was communicated by Barrow to Collins about the time at which Mercator's ‘Logarithmotechnia’ appeared (i.e. in 1669). He does not describe the method, but added an anagram containing an explanation. This is not intelligible without the key, but Newton gives some illustrations of its use (see Ball, Short Hist. of Math., 2nd ed. p. 328).
Leibnitz was in London for a week in October 1676, and saw Collins, who had not then received Newton's letter of 24 Oct., and there was some delay in forwarding it to Leibnitz. But on 5 March 1677 Collins wrote to Newton that it would be sent within a week, and on 21 June 1677 Leibnitz, writing to Oldenburg, acknowledged its receipt: ‘Accepi literas tuas diu expectatas cum inclusis Newtonianis sane pulcherrimis.’ He then proceeded to explain his own method of drawing tangents, ‘per differentias ordinatarum,’ and to develop from this the fundamental principles of the differential calculus with the notation still employed by mathematicians. A second letter followed from Hanover, dated 12 July 1677, and dealt with other points. The death of Oldenburg in September 1677 put a stop to the correspondence.
Collins had in his possession a copy of Newton's manuscript ‘De Analysi per Æquationes,’ containing a full account of his method of fluxions, which was published in 1711. Leibnitz, in a letter to the Abbé Conti, written in 1715, and published in Raphson's ‘History of Fluxions,’ p. 97, admits that ‘Collins me fit voir une partie de son commerce.’ He states that during his first visit he had nothing to do with mathematics, and in a second letter, 9 April 1716, he writes (Raphson, History of Fluxions, p. 106): ‘Je n'ay jamais nié qu'à mon second voyage en Angleterre j'ai vu quelques lettres de M. N. chez Monsieur Collins, mais je n'en ay jamais vu où M. N. explique sa methode de Fluxions.’
Leibnitz's recent editor, Gerhardt, found, however, among the Leibnitz papers at Hanover, a copy of a part of the tract ‘De Analysi’ in Leibnitz's own handwriting. The copy contains notes by Leibnitz expressing some of Newton's results in the symbols of the differential calculus (Ball, Short Hist. of Math. p. 364; Portsmouth Catalogue, p. xvi). The date at which these extracts were made is important. They must, of course, have been taken from Newton's published edition of 1704, or else, as the Portsmouth MSS. prove that Newton suspected, Leibnitz must have copied the tract when in London in 1676. The last hypothesis seems the more probable.
Leibnitz published his differential method in the ‘Acta Lipsica’ in 1684.
Many of the results in Newton's ‘Principia,’ 1687, had been obtained by the method of fluxions, though exhibited in geometrical form, and the second lemma of book ii. concludes with the following scholium: ‘In literis quæ mihi cum geometra peritissimo G. G. Leibnitio annis abhinc decem intercedebant, cum significarem me compotem esse methodi determinandi Maximas et Minimas ducendi Tangentes et similia peragendi quæ in terminis Surdis æque ac in rationalibus procederet, et literis transpositis hanc sententiam involventibus [Data Æquatione quotcunque Fluentes quantitates involvente, Fluxiones invenire et vice versâ] eandem celarem; rescripsit Vir Clarissimus se quoque in ejusmodi methodum incidisse, et methodum suam communicavit a mea vix abludentem præterquam in verborum et notarum formulis. Utriusque fundamentum continetur in hoc Lemmate.’