THE PROGRESS OF 



howd that the cubic parabola might be rectified 

 by one of Wallis's methods; and Van Heuraet 

 applied the same method to the rectification of 

 several other parabolas. 



We must pass over the discoveries of Barrow, 

 though they are highly entitled to attention, and 

 hasten to Sir Isaac Newton, who was destined to 

 chancre the. face of physical science and who 

 hitherto stands alone among mankind as the in- 

 uT.tor of a theory to explain the motions of the 

 iuMvenly bodies, which has stood the test of the 

 most rigid examination, and which every subse- 

 quent improvement and discovery has served 

 only to confirm and establish. 



Newton w;:s horn in the year 1642, on christ- 

 nias, old style, at Woolsthorpe, in the county of 

 Lincoln. The family came originally from 

 NiM\ton, in Lancashire, and it appears from a let- 

 ter published by Dr Heid, that, by the mother's 

 side, he was of Scottish extraction. When young, 

 he discovered a surprising turn for making 

 models of mechanical instruments. His mathe- 

 matical knowledge came to him with incredible 

 facility. Euclid he merely turned over, consid- 

 ering it to contain nothing but common things. 

 The first mathematical book that he read was 

 Descartes' Geometry, and the second Wallis's 

 Arithmetic of Infinites. On these books he wrote 

 commentaries as he read them, and reaped a rich 

 harvest of discoveries ; or, more properly speak- 

 ing, he made almost all his mathematical disco- 

 veries as he proceeded in their perusal In 

 1667, he was elected fellow of Trinity College, 

 Cambridge, and in 1 669, Dr Barrow resigned his 

 mathematical professorship to him. He resigned 

 this chair in 1703, two years after Mr Whiston 

 had taught for him. He died in the year 1727. 

 He was a member of the convention parliament, 

 in 1789, and was, for a good many years, master 

 of the mint 



His first mathematical discovery of import- 

 ance wai the binomial theorem. But his great 

 mathematical discovery was the method of fac- 

 tions, or the integral and differential calculus, as 

 it is called on the continent. This, he assures us, 

 he was in possession of as early as 1665 or 1666. 

 Barrow informs us, that soon after that period, 

 there was put into his hands, by Newton, a manu- 

 script treatise, the same which was afterwards 

 published under the title of Analysis per JEqua- 

 tiones Numero terminorum infinitas, in which the 

 principle of fluxions, though not fully explained, 

 is yet distinctly pointed out Barrow strongly 

 exhorted Newton to publish this treasure to the 

 world ; but an excess of modesty, almost amount- 

 Ing to a disease, prevented his compliance. 

 For a long time, the mathematical discoveries 

 ewton were known only to his friends. The 

 tirst work in'which he communicated any thing to 



the world on the subject, was the first edition ot 

 the Principia, in 1687, in the second lemma of 

 the second book. 'Hie principle of the calculus is 

 there pointed out ; but nothing is said of the al- 

 gorithm, which is so essential to that calculus. 

 This only became known to the world in 1693, 

 by the publication of the second volume of Wal- 

 lis's works (p. 390, &c.) There is no evidence 

 that this notation existed earlier than 1692, 

 though it is highly probable that it did. It was 

 no less than ten years after this, or in 1704, that 

 Newton himself published a work on 1 the new 

 calculus, his Quadrature of Curves, more than 

 twenty-eight years after it had been written. 



These discoveries, however, even before 

 they were committed to the press, could not re- 

 main altogether unknown in a country where 

 mathematics were cultivated with such zeal 

 and success. Barrow communicated them to 

 Oldenburgh, secretary of the Royal Society. By 

 him they were partly communicated to Mr James 

 Gregory, and likewise to Leibnitz, who had be- 

 come acquainted with Oldenburgh during a visit 

 which he had made to England in 1673. At that 

 time Leibnitz knew very little of the mathema- 

 tics ; but having afterwards turned his attention 

 to that science, he was soon in a condition to 

 make discoveries. One of the first of these was 

 a very remarkable series, which gives the value 

 of a circular arch in terms of the tangent. This 

 series he communicated to Oldenburgh, in 1674, 

 and received, in return, an account of the pro- 

 gress made by Newton and Gregory, in the in- 

 vention of series. In 1676, Newton described 

 his method of quadratures, at the request of Ol- 

 denburgh, in order that it might be transmitted 

 to Leibnitz. 



The method of Auctions is not communicated 

 in these letters ; nor are the principles in any 

 way suggested ; though there are, in the last let- 

 ter, two sentences, in transposed characters, 

 which ascertain that Newton was then in posses- 

 sion of that method, and employed, in speaking 

 of it, the same language in which it was after- 

 wards made known. In the following year, 

 Leibnitz, in a letter to Oldenburgh, introduced 

 differentials, and the method of his calculus, for 

 the first time. This letter clearly proves, that, 

 in 1677, Leibnitz was in full possession of the 

 principles of his calculus, and had even invented 

 the algorithm and notation. 



From these facts, it cannot, we think, be 

 doubted, that Newton was the first inventor of the 

 fluxionary calculus ; but that nothing was com- 

 municated to Leibnitz regarding the principles 

 of that analysis. Leibnitz, therefore, when he 

 invented the differential calculus, was not assist- 

 ed by any communications that could give him 

 any idea of what had been done by Newton. 



