VOL. XXIX.] PHILOSOPHICAL TRANSACTIONS. 129 



And the same is manifest also by what Mr. Leibnitz wrote in the Acta 

 Eruditorum, anno 169I, concerning this matter. 



" Now in 1675, says he, I had composed a small treatise of arithmetical 

 quadrature, which from that time had been perused by my friends ; but the 

 matter enlarging under my hands, I had not leisure to prepare it for the press, 

 and other avocations interfered afterwards; especially now as it does not seem 

 worth while to explain prolixly in the common way, what my analysis performs 

 briefly." This quadrature, composed in the common manner, he began to 

 communicate at Paris in the year 1675. The next year he was polishing the 

 demonstration of it, to send it to Mr. Oldenburg in recompence for Mr. 

 Newton's method, as he wrote to him May 12, 1676; and accordingly in his 

 letter of August 27, 1676, he sent it, composed and polished in the common 

 manner. The winter following he returned into Germany, by England and 

 Holland, to enter on public business, and had no longer any leisure to fit it for 

 the press, nor thought it afterwards worth his while to explain those things 

 prolixly in the vulgar manner, which his new analysis exhibited in short. He 

 found out this new analysis therefore after his return into Germany, and con- 

 sequently not before the year l677- 



The same is further manifest by the following consideration. Dr. Barrow 

 published his method of tangents in the year 1670. Mr. Newton in his letter 

 dated December 10, 1672, communicated his method of tangents to Mr. 

 Collins, and added : " This is one particular, or rather corollary, of the gene- 

 ral method, which extends without any troublesome calculation, not only to 

 the drawing of tangents to any kind of curves, either geometrical, or mechani- 

 cal, or any how regarding right lines, or other curves ; but also to the solving 

 of other more abstruse problems, of curvatures, areas, lengths, centres of 

 gravity of curves, &c. Nor is it confined, (like Hudden's method de maximis 

 et minimis) only to such equations as have no surd quantities. I have added 

 this method to that other, in which I give an exegesis of equations, by re- 

 ducing them into infinite series. M. Slusius sent his method of tangents to 

 Mr. Oldenburgh. Jan. 17, 167^, and the same was soon after published in the 

 Transactions. It proved to be the same with that of Mr. Newton. It was 

 founded on three lemmas; the first of which was this, " The difference of two 

 powers of the same degree, divided by the difference of the sides or roots, 

 gives the members of the next lower degree of the binomial of the sides ; as 

 =. yy -{• yx ■\- xx^ that is, in the notation of Mr. Leibnitz —- = Syz/." 



A copy of Mr. Newton's letter, of Dec. 10, 1672, was sent to Mr. Leibnitz 

 by Mr. Oldenburg, among the papers of Mr. James Gregory, at the same time 



VOL. VI. S 



