VOL. XXIX.] PHILOSOPHICAL TRANSACTIONS. 123 



gory sent in the said letter, were these two: let the radius of a circle be r, the 

 arc a, and the tangent t, the equations for finding the arc whose tangent is 

 given, and the tangent whose arc is given, will be these : 



/-«-l--^4-^4-i^4- -5^ + &c 



' ~~ " ^ 3r^ ^ 15H ^ 315rf' ^ 2835r8 T «-^. 



In this year, 1671, Mr. Leibnitz published two Tracts at London, the one 

 dedicated to the Royal Society, the other dedicated to the Academy of Sciences 

 at Paris ; and in the dedication of the first he mentioned his correspondence 

 with Mr. Oldenburg. 



In February 1 672-8, Mr. Leibnitz meeting Dr. Pell at Mr. Boyle's, he pre- 

 tended to the differential method of Mouton. And though he was shown by 

 Dr. Pell, that it was Mouton's method, he persisted in maintaining it to be 

 his own invention, because he had found it out himself, without knowing what 

 Mouton had done before, and had much improved it. 



When one of Mr. Newton's Series was sent to Mr. Gregory, he tried to 

 deduce it from his own Series combined together, as he mentions in his letter 

 dated Dec. 19, 1670. And by some such method, Mr. Leibnitz, before he 

 left London, seems to have found the sum of a series of fractions decreasing 

 in infinitum, whose numerator is a given number, and denominators are tri- 

 angular, or pyramidal, or triangulo-triangular numbers, &c. Behold the 

 mystery ! from the Series 4--f-^-j-^-j-4.-f--«.-|- &c. subduct all the terms 

 except the first (viz. i -f- ^ -|- ^ + i &c.) and there will remain 1 = 1 — 4- -|- 



o-Ta ^^* ^^^*"'^'— 1X2^2X3^3X4.^4X5^*^^* 



And from this Series take all the terms except the first, and there will remain 

 4. = -— - — - 4- - — - — - -|- - — - — - -}- - — - — ^ -|- &c. And from the first 



* 1x2x3 ' 2x3x4 ' 3x4x5 ' 4x5x6 ' 



Series take all the terms except the first two, and there will remain 



J. — __? I - I - I L &c. 



^ ~~ Ix3'^2x4'3x5^4x6^ 



About the end of February or beginning of March, 1 67 2-3, Mr. Leibnitz 

 went from London to Paris, and continuing his correspondence with Mr. 

 Oldenburg and Mr. Collins, wrote in July 1674, that he had a wonderful 

 Theorem, which gave the area of a circle, or any sector of it exactly, in a 

 Series of rational numbers; and in October following, that he had found the 

 circumference of a circle in a Series of very simple numbers; and that by the 

 same method (so he calls the said Theorem) any arc whose sine was given, 

 might be found in a like Series, though the proportion to the whole circum- 

 ference be not known. His Theorem therefore was for finding any sector or 



R 2 



