It is but rarely that we can lay hold with certainty of the thread by which genius has been guided in its first discoveries. Here we are proceeding on the authority of the author himself, for in a letter to Oldenburgh,' Secretary of the Royal Society of London, he has entered into considerable detail' on this subject, adding (so ready are the steps of invention to be forgotten), that the facts would have entirely escaped his memory, if he had not been reminded of them by some notes which he had made at the time, and which he had accidentally fallen on. The whole of the letter just referred to is one of the most valuable documents to be found in the history of invention.
In all this, however, nothing occurs from which it can be inferred that the method of flexions had yet occurred to the inventor. His discovery consisted in the method of reducing the value of y, the ordinate of a curve, into an infinite series of the integer powers of x the abscissa, by division, or the extraction of roots, that is, by the Bi nomial Theorem ; after which, the part of the area belonging to each term could be assigned by the arithmetic of infinites, or other methods already known. He has assured us himself, however, that the great principle of the new geometry was known to him, and applied to investigation as early as 1665 or 1666.' Independently of that authority, we also know, on the testimony of Barrow, that Boon after the period just mentioned, there was put into his hands by Newton a manuscript treatise,' the same which was afterwards published under the title of Analysis per dEquationes Numero Terminorum Infinitas, in which, though the instrument of investigation is nothing else than infinite series, the principle of fluxions, if not fully explained, is at least distinctly pointed out. Barrow strongly exhorted his young friend to publish this treasure to the world ; but the modesty of the author, of which the excess, if not culpable, was certainly in the present instance very unfortunate, prevented his com pliance. All this was previous to the year 1669 ; the treatise itself was not published till 1711, more than forty years after it was written.
For a long time, therefore, the discoveries of Newton were only known to his friends, and the first work in which he communicated any thing to the world on the subject of fluxions was in the first edition of the Principia, in 1687, in the second Lemma of the second book, to which, in the disputes that have since arisen about the invention of the new analysis, reference has been so often made. The principle of the fluxionary calculus was there pointed out, but nothing appeared that indicated the peculiar al gorithm, or the new notation, which is so essential to that calculus. About this Newton had yet given no information ; and it was only from the second voluthe of Wallis's Works, in 1693, that it became known to the world.' It was no less than
ten years after this, in 1704, that Newton himself first published a work on the new calculus, his Quadrature of Curves, more than twenty-eight years after it wgs written.
These discoveries, however, even before the press was employed as their vehicle, could not remain altogether unknown in a country where the mathematical sciences were cultivated with zeal and diligence. Barrow, to whom they were first made known by the author himself, communicated them to Oldenburgh, the secretary of the Royal Society, who had a very extensive correspondence all over Europe. By him the series for the quadrature of the circle were made known to James Gregory,' in Scotland, who had occupied himself very much with the same subject. They were also com municated to Leibnitz in Germany, who had become acquainted with Oldenburgh in a visit which he made to England in 1674." At the time of that visit, Leibnitz was but little conversant with the mathematics ; but having afterwards devoted his great talents to the study of that science, he was soon in a condition to make new discoveries. He invented a method of squaring the circle, by transforming it into another curve of an equal area, but having the ordinate expressed by a rational fraction of the abscise, so that its area could be found by the methods already known. In this way he discovered the series, so remarkable for its siinpli city, which gives the value of a circular arch in terms of the tangent. This series he communicated to Oldenburgh in 1674, and received from him in return an account of the progress made by Newton and Gregory in the invention of series. In 1676, Newton descsibed his method of quadratures at the request of Oldenburgh, in order that it might be transmitted to Leibnitz in the two letters already mentioned, as of such value by recording the views which guided that great geometer in his earliest, and SOW of his most important discoveries. The method of fluxion is not communicated in these letters; nor are the principles of it in any way suggested ; though there are, in the last letter, two sentences in transposed characters, which ascertain that New ton was then in possession of that method, and employed in speaking of it the same Ian.. page in which it was afterwards made known. In the following year, Leibnitz, in a letter to Oldenburgh, introduces differentials, and the methods of his calculus for the first time. This letter,' which is very important, clearly proves that the author was then in full possession of the principles of his calculus ; and had even invented the algorithm and notation.