126 PHILOSOPHICAL TRANSACTIONS. [aNNO 1714. 



nicating to me so many curious things." And towards the end of the letter, 

 after he had done with the contents of Mr. Newton's letter, he proceeds thus: 

 " I come now to other things contained in your letter, which the learned 

 Mr. Collins was pleased to communicate ; I wish he had added, the demonstra- 

 tion of Mr. Gregory's Linear Approximation ; for he certainly had a genius 

 for promoting such speculations." And the answer of Mr. Tschurnhause, 

 dated Sept, I, ]676, after he had done with Mr. Newton's letter about Series, 

 concludes thus: " And what that excellent geometrician Mr. Gregory has done 

 in this matter are certainly extraordinary. And indeed those who shall cause 

 his MS. to be published, will do the greatest service to his reputation." In 

 the first part of this letter, where Mr. Tschurnhause speaks of Mr. Newton's 

 Series, he says, that he looked over them cursorily, to see if he could find 

 the Series of Mr. Leibnitz for squaring the circle or hyperbola. If he had 

 searched for it in the Extracts of Gregory's Letters, he might have found it 

 in the letter of Feb. 15, 1671, abovementioned. For the MS. of those 

 Extracts, with that letter in it, is still extant in the hand-writing of Mr. 

 Collins. 



And though Mr. Leibnitz had now received this Series twice from Mr. 

 Oldenburg, yet in his letter of August 27, 1676, he sent it back to him by 

 way of recompence for Mr. Newton's Method, pretending that he had com- 

 municated it to his friends at Paris 3 years before, or more ; that is, 2 years 

 before he received it in Mr. Oldenburg's letter of April 15, 1^75; at which 

 time he did not know it to be his own, as appears by his answer of May 20, 

 1675, abovementioned. He might receive this Series at London, and commu- 

 nicate it to his friends at Paris, above 3 years before he sent it back to Mr. 

 Oldenburg: but it does not appear that he had the demonstration of it so early. 

 When he found the demonstration, then he composed it in his Opusculum, 

 and communicated that also to his friends; and he himself has told us that 

 this was in the year 1675. However, it lies upon him to prove that he had 

 this Series before he received it from Mr. Oldenburg. For in his answer to 

 Mr. Oldenburg he did not know any of the Series then sent him to be his 

 own ; and concealed from the gentlemen at Paris his having received it from 

 Mr. Oldenburg with several other Series, and his having seen a copy of the 

 letter in which Mr. Gregory had sent it to Mr. Collins, in the beginning of 

 the year 167 1. 



In the same letter, of August 27, 1676, after Mr. Leibnitz had described 

 his quadrature of the circle and equilateral hyperbola, he adds : 



" Again from the Series of regressions I found the following for the hyper- 

 bola, viz. if any number be less than unity, as 1 — w, and its hyperbolic loga- 



