Page:Encyclopædia Britannica, Ninth Edition, v. 17.djvu/476

Rh 446 to high posts in church or state, he still remained without any mark of national gratitude. But this blot upon the English name was at last removed by Montague in 1694, when he was appointed chancellor of the exchequer. He had previously consulted Newton upon the subject of the recoinage, and on the opportunity occurring he appointed Newton to the post of warden of the mint. In a letter to Newton announcing the news, Montague writes : &quot; I am very glad that at last I can give you a good proof of my friendship, and the esteem the king has of your merits. Mr Over- ton, the warden of the mint, is made one of the Commissioners of Customs, and the king has promised me to make Mr Newton warden of the mint. The office is the most proper for you. Tis the chief office in the mint: tis worth five or six hundred pounds per annum, and has not too much business to require more attend ance than you can spare.&quot; This letter must have convinced Newton of the sincerity of Montague s good intentions towards him ; we find them living as friends on the most intimate terms until Halifax s death in 1715. The chemical and mathematical knowledge of Newton proved of great use in carrying out the recoinage. This was completed in about two years, and such was the zeal and devotion with which Newton discharged the laborious duties of his office that he was in 1697 appointed to the mastership of the mint, a post worth between 1200 and 1500 per annum. While he held the latter office, Newton drew up a very extensive table of assays of foreign coins, and composed an official report on the coinage. Up to the time of the publication of the Principia in 1687 the method of fluxions which had been invented by Newton, and had been of great assistance to him in his mathematical investigations, was still, except to Newton and his friends, a secret. One of the most important rules of the method forms the second lemma of the second book of the Principia. Though this new and powerful method was of great help to Newton in his work, he did not exhibit it in the results. He was aware that the well- known geometrical methods of the ancients would clothe his new creations in a garb which would appear less strange and uncouth to those not familiar with the new method. The Principia gives no information on the subject of the notation adopted in the new calculus, and it was not until 1693 that it was communicated to the scientific world in the second volume of Dr Wallis s works. Newton s admirers in Holland had informed Dr Wallis that Newton s method of fluxions passed there under the name of Leibnitz s Calculus Differentialis. It was there fore thought necessary that an early opportunity should be taken of asserting Newton s claim to be the inventor of the method of fluxions, and this was the reason for this method first appearing in Wallis s works. A further account of the method was given in the first edition of Newton s Optics, which appeared in 1704. To this work were added two treatises, entitled Tractatus duo de speciebus et magnitudine figurarum curvilinearnm, the one bearing the title Tractatus de Quadratura Curvarum, and the other Enumeratio linearum tertii ordinis. The first contains an explanation of the doctrine of fluxions, and of its application to the quadrature of curves ; the second, a classification of seventy-two curves of the third order, with an account of their properties. The reason for publishing these two tracts in his Optics, from the subsequent editions of which they were omitted, is thus stated in the advertisement : &quot;In a letter written to M. Leibnitz in the year 1679, and published by Dr Wallis, I mentioned a method by which I had found some general theorems about squaring curvilinear figures on comparing them with the conic sections, or other the simplest figures with which they might be compared. And some years ago I lent out a manuscript containing such theorems ; and having since met with some things copied out of it, I have on this occasion made it public, prefixing to it an introduction, and joining a Scholium concerning that method. And I have joined with it another small tract concerning the curvilineal figures of the second kind, which was also written many years ago, and made known to some friends, who have solicited the making it public.&quot; In the year 1707 Whiston published the algebraical lectures which Newton had delivered at Cambridge, under the title of Arithmetica Universalis, sive de Compositione et Resolutione Arithmetica Liber. We are not accurately informed how Whiston obtained possession of this work ; but it is stated by one of the editors of the English edition &quot; that Mr Whiston, thinking it a pity that so noble and useful a work should be doomed to a college confinement, obtained leave to make it public.&quot; It was soon afterwards translated into English by Mr Raphson ; and a second edition of it, with improvements by the author, was published at London in 1712, by Dr Machin, secretary to the Royal Society. W T ith the view of stimu lating mathematicians to write annotations on this ad mirable work, the celebrated S Gravesande published a tract, entitled Specimen Commentarii in Arithmeticam Universalem ; and Maclaurin s Algebra seems to have been drawn up in consequence of this appeal. In mentioning the mathematical works of our author, we must not omit his solution of the celebrated problems proposed by John Bernoulli and Leibnitz. In June 1696 Bernoulli addressed a letter to the mathematicians of Europe challenging them to solve two problems (1) to determine the brachistochrone between two given points not in the same vertical line, (2) to determine a curve such that, if a straight line drawn through a fixed point A meet it in two points P 1; P 2, then AP^ + AP 2 m will be constant. This challenge was first made in the Acta Lipsiensia for June 1696. Six months were allowed by Bernoulli for the solution of the problem, and in the event of none being sent to him he promised to publish his own. The six months elapsed without any solution being produced ; but he received a letter from Leibnitz, stating that he had &quot; cut the knot of the most beautiful of these problems,&quot; and requesting that the period for their solution should be extended to Christmas next, that the French and Italian mathematicians might have no reason to complain of the shortness of the period. Bernoulli adopted the sugges tion, and publicly announced the prorogation for the infor mation of those who might not see the Acta Lipsiensia. On the 29th January 1696-97 Newton received from France two copies of the printed paper containing the problems, and on the following day he transmitted a solu tion of them to Montague, then president of the Royal Society. He announced that the curve required in the first problem must be a cycloid, and he gave a method of determining it. He solved also the second problem, and he showed that by the same method other curves might be found which shall cut off three or more segments having the like properties. Solutions were also obtained from Leibnitz and the Marquis de L Hopital ; and, although that of Newton was anonymous, yet Bernoulli recognized the author in his disguise ; &quot; tanquam,&quot; says he, &quot; ex ungue leonem.&quot; In the year 1699 Newton s position as a mathematician and natural philosopher of the first order were recog nized in a very honourable manner by the French Academy of Sciences. In that year the Academy was remodelled, and eight foreign associates were created. Leibnitz, Guglielmini, Hartsoeker, and Tschirnhausen were appointed on February 4, James Bernoulli and John Bernoulli on February 14, and Newton and Roemer on February 21. While Newton held the office of warden of the mint, he retained his chair of mathematics at Cambridge, and dis charged the duties of the post, but shortly after he was