Neil and Van-Heuract gave the first example of a curve that may be rectified, (one of the cubic parabolas.) Van Heuraet's method reduced the problem of rectifications to that of quadratures. Brouncker and Mercator, proceeding in the path of Wallis's discoveries, found the first series known for the rectification of the circle, and hyperbola. Brouncker also first noticed continued fractions; and he spewed that the fundamental principle employed by Neil in the rectification of curves, and that by which Mercator squared the hyperbola, were to be found in the works of Wallis.
Mercator published his Logarithmotechnia in September 1668, which contained his quadrature of the hyperbola ; and soon after the book came out, Mr Collins, secretary to the Royal Society, sent a copy to Barrow, at Cambridge, who put it into the hands of Sir Isaac, then Mr Newton, and a fellow of Trinity College. Presently afterwards, viz. in July 1669, Barrow wrote to Collins, that a friend of his (Newton,) who had an excellent genius to these things, had brought him some papers, wherein he had set down methods of calculating the dimensions of magnitudes, like that of Mr Mercator for the hyperbola, but very general ; as also of resolving equations: Barrow afterwards sent these papers to Collins, saying, that he presumed he would be much pleased with them, and requesting him to shew them to Lord Brouncker. Their title was De analysi per equationes numero terminorum infinitas. In this manu script, the method of fluxions was first indicated, and rules deduced from it given for the quadrature of curves, to which it was observed, their rectification, and the deter mination of the quantity and the superficies of solids, and of the centre of gravity, may be all reduced : moreover, the author there asserted, that he knew no problems rela ting to the quadrature or rectification of curves, to which his method would not apply ; and that by means of it, he could draw tangents to mechanical curves ; so there can be no doubt, but that then Newton possessed the method or fluxions, and therefore he must be reckoned the first inven tor. Indeed it appears that although his discovery was pro• mulgated then for the first time, he had been in possession of it from about the year 1666, which was two years before Mercator published his quadraturc of the hyperbola. And although the MS. memoir De analysi per (equationes, Etc. professes to explain the method briefly, rather than to de monstrate it accurately, yet there was enough to show, that the author was aware of its great importance as an instru of investigation, and that he had reduced it in some measure to the form of an analytical theory.
Barrow, Collins, and Oldenburg, (another Secretary to the Royal Society,) disseminated the analytical discoveries of Newton by their correspondence, and communicated them to several geometers on the continent, such as Slus sius, and Borclli.
In the year 1672, the celebrated Leibnitz, who afterwards also claimed the honour of the discovery of the method of fluxions, appeared for the first time upon the scene. Hap pening to be in London, he communicated to sonic mem bers of the Royal Society, certain researches relating to the differences of numbers ; but he was given to understand. that this subject had been already treated by Mouton, an' astronomer of Lyons : upon this, he turned his attention to the doctrine of infinite series, which, at that time, engaged all the mathematicians ; and, in 1674, he announced to Ol denburg, that he possessed important theorems relative to the quadrature of the circle by series ; and that he had very general analytic methods. Oldenburg, in answer, in timated to him, that Gregory and Newton had also found methods, which gave the quadrature of curves, whether they were geometrical or mechanical, and which extend ed to the circle.
The first direct communication which Newton had with Leibnitz, was in 1676. On the 13th June, in that year. Newton sent a letter to Oldenburg, which was to be shewn to Leibnitz : This contained his celebrated binomial theo rem, which he appears to have known in 1669 ; and a vari ety of other matters relating to infinite series, and quadra tures, but nothing directly relating to the theory of fluxions: and it is worthy of remark, that in this letter, Newton speaks of Leibnitz with great respect ; so that the suspicion which afterwards arose in his mind, that Leibnitz was not dealing fairly with him in respect of his discoveries, does net ap pear to have then existed. In a second letter from New ton to Oldenburg, to be also communicated to Leibnitz, he still speaks of his rival with respect; and he here, in compli ance with a wish expressed by Leibnitz, explains the man ner in which he found the binomial theorem. He also de scribes the properties of his method of flux ions, as well for the determination of tangents, as the quadrature of curves; but he conceals it under an anagram of transposed letters. Here we have positive evidence that Newton was now in possession of his calculus.