124 PHILOSOPHICAL TRANSACTIONS. [anNO 1714. 



arc whose sine was given. If the proportion of the arc to the whole circum- 

 ference was not known, the Theorem or method gave him only the arc ; if it 

 was known, it gave him also the whole circumference: and therefore it was the 

 first of Mr. Newton's two Theorems abovementioned. But the demonstration 

 of this Theorem Mr. Leibnitz wanted. For in his letter of May 12, 1676, he 

 desired Mr. Oldenburg to procure the demonstration from Mr. Collins, 

 meaning the method by which Mr, Newton had invented it. 



In a letter written by Mr. Collins, and dated April 15, 1675, Mr. Olden- 

 burg sent to Mr. Leibnitz eight of Mr. Newton's and Mr. Gregory's Series; 

 among which were Mr. Newton's two Series abovementioned, for finding the 

 arc whose sine is given, and the sine whose arc is given ; and Mr. Gregory's 

 two Series abovementioned for finding the arc whose tangent is given, and the 

 tangent whose arc is given. And Mr. Leibnitz, in his answer, dated May 20, 

 1675, acknowledged the receipt of this letter in these words. 



" I received your letter, containing a great deal of algebraical knowledge; 

 for which I thank you and the learned Mr. Collins. But being now very nmch 

 occupied; besides my ordinary business, especially with mechanical affairs, I 

 could not examine the Series you sent me, and compare them with my own; 

 as soon as I shall have done it, I shall give you my opinion ; for, it is some 

 years ago since I invented my Series in a certain very peculiar way." 



But yet Mr. Leibnitz never took any further notice of his having received 

 these Series, nor how his own differed from them, nor ever produced any other 

 Series than those which he received from Mr. Oldenburg, or numeral Series 

 deduced from them in particular cases. And what he did with Mr. Gregory's 

 Series, for finding the arc whose tangent is given, he has told us in the Acta 

 Eruditorum for the month of April 1691, p. 178. Now in the year 1675, 

 says he, I had composed a small tract of Arithmetical Quadrature, which from 

 that time was read by my friends, &c. By a Theorem for transmuting figures, 

 like those of Dr. Barrow and Mr. Gregory, he had now found a demonstration 

 of this Series ; and this was the subject of his Opusculum. But he still wanted 

 a demonstration of the rest : and meeting with a pretence to ask for what he 

 wanted, he wrote to Mr. Oldenburg the following letter, dated at Paris, 

 May 12, 1676: " Since M. George Mohr, a Dane, brought me the method 

 of expressing the ratio between the arch and the sine by the following infinite 

 Series, which had been communicated to him by your learned Mr. Collins; 

 viz. putting X for the sine, z for the arch, and 1 for the radius, 

 2 = jr + ^j^ + ^a7^H--r4^z7 + -j-f ^ J7» + &c. 

 :c = z-i2'+-r^oZ*- ToW z' -f- ,,,\,, z"" - &c. 

 since, I say, he brought me these, which appear to me to be very ingenious. 



