VOL. XXIX.] PHILOSOPHICAL TRANSACTIONS. 125 



and especially the latter Series, which has a certain peculiar elegance ; there- 

 fore. Sir, you will oblige me very much, if you send me the demonstration ; 

 in return 1 shall send you my thoughts on this matter, which are very different 

 from these, and of which I think I have written you now some years ago, 

 without adding the demonstration, which I am now about polishing ; I intreat 

 you would remember me very kindly to Mr. Collins, who can enable you to 

 satisfy my desira" 



Here, by the word Inquam (I say), one would think that he had never seen 

 these two Series before, and that his diversa circa banc rem meditata were some- 

 thing else than one of the Series which he had received from Mr. Oldenburg the 

 year before, and a demonstration of it, which he was now polishing, to make 

 the present an acceptable recompence for Mr. Newton's method. 



On the receipt of this letter, Mr. Oldenburg and Mr. Collins wrote press- 

 ingly to Mr. Newton, desiring that he himself would describe his own method, 

 to be communicated to Mr. Leibnitz. On which Mr. Newton wrote his letter, 

 dated June 13, 1676, describing the method of Series, as he had done before 

 in the Compendium abovementioned, but with this difference, that here he 

 described at large the reduction of the power of a binomial into a Series, and 

 only touched on the reduction by division and extraction of affected roots ; 

 because in the Compendium he had already described these latter : there he 

 described at large the reduction of fractions and radicals into Series by division 

 and extraction of roots, and only set down the first two terms of the Series 

 into which the power of a binomial might be reduced. And among the 

 examples in this letter, there were Series for finding the number whose loga- 

 rithm is given, and for finding the versed sine whose arc is given. This letter 

 was sent to Paris June 26, 1676, with a MS. drawn up by Mr. Collins, con- 

 taining Extracts of Mr. James Gregory's Letters. For Mr. Gregory died near 

 the end of the year l675; and Mr. Collins, at the request of Mr. Leibnitz, 

 and some others of the Academy of Sciences, drew up Extracts of his Letters, 

 and the collection is still extant in the hand writing of Mr. Collins, with this 

 title ; " Extracts of Mr. Gregory's Letters, to be lent to Mr. Leibnitz to 

 peruse, who is desired to return the same to you." And that they were sent, 

 is affirmed by Mr. Collins, in his letter to Mr. David Gregory, the brother of 

 the deceased, dated August 11, 1676; and appears further by the answers of 

 Mr. Leibnitz and Mr. Tschurnhause, concerning them. 



The answer of Mr. Leibnitz, directed to Mr. Oldenburg, and dated August 27, 

 1676, begins thus ; " Your letter, dated July 26, contains more, and those more 

 remarkable things in analysis, than many voluminous books published on this 

 subject. Therefore I thank you, Mr. Newton, and Mr. Collins, for commu-^ 



