VOL. XXIX.] PHILOSOPHICAL TRANSACTIONS. 131 



equations, he reduces the equations into converging series by the binomial 

 theorem, and by the extraction of fluents out of equations involving or not in- 

 volving their fluxions. And when finite equations are wanting, he deduces 

 converging series from the conditions of the problem, by assuming the terms 

 of the series gradually, and determining them by those conditions. And when 

 fluents are to be derived from fluxions, and the law of the fluxions is wanting, 

 he finds that law very nearly, by drawing a parabolic line through any num- 

 ber of given points. And by these improvements Mr. Newton had, in those 

 days, made his method of fluxions much more universal, than the diff^erential 

 method of Mr. Leibnitz is at present. 



This letter of Mr. Newton's, dated Oct. 24, 1676, came to the hands of 

 Mr. Leibnitz in the end of the winter or beginning of the spring following ; 

 and Mr. Leibnitz soon after, viz. in a letter dated June 21, 1677^ wrote back: 

 *^ I agree with Mr. Newton, that Slusius's method of tangents is still imperfect; 

 and I have long since treated the subject more generally, viz. by the differences 

 of the ordinates. — Hence in future, calling dy the difference of the two nearest 

 2/, &c." Here Mr. Leibnitz began first to propose his differential method, and 

 there is not the least evidence that he knew it before the receipt of Mr. 

 Newton's last letter. He says indeed, '' That he had long since treated the 

 subject of tangents more generally, viz. by the differences of the ordinates :" 

 and so he affirmed in other letters, that he had invented several converging 

 series direct and inverse, before he had the method of inventing them ; and 

 had forgot an inverse method of series before he knew what use to make of it. 

 But no man is a witness in his own cause. A judge would be very unjust, and 

 act contrary to the laws of all nations, who should admit any man to be a wit- 

 ness in his own cause. And therefore it is incumbent on Mr. Leibnitz to prove 

 that he found out this method long before the receipt of Mr. Newton's letters. 

 And if he cannot prove this, the question, who was the first inventor of the 

 method, is decided. 



The Marquis de I'Hospital, in the preface to his Analyse des infiniments petits, 

 published A. C. 1696, tells us, " that a little after the publication of the me- 

 thod of tangents of Descartes, Mr. Fermat found also a method, which 

 Descartes himself at length allowed to be, for the most part, more simple than 

 his own. But that it was not yet so simple as Mr. Barrow afterwards made it, 

 by considering more nearly the nature of polygons, which offers naturally to 

 the mind a little triangle, composed of a particle of the curve lying between 

 two ordinates infinitely near each other, and of the difference of these two 

 ordinates, andot that of the two correspondent abscissas. And this triangle is 



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