86 PHILOSOPHICAL TRANSACTIONS, [anNO 1776. 



The slower of which converges ahnost thrice as fast as Dr. Halley's, raised 

 from the tangent of 30°. The latter of these two series converges still a great 

 deal quicker ; but then the large incomposite number 239, by the reciproca Is o 

 the powers of which the series converges, occasions such long, tedious divisions, 

 as to counterbalance its quickness of convergency ; so that the former series is 

 summed with rather more ease than the latter, to the same number of places of 

 figures. Mr. Jones, in his Synopsis, mentions other series' besides this, which 

 he had received from Mr. Machin for the same purpose, and drawn from the 

 same principle. But we may conclude this to be the best of them all, as he did 

 not publish any other besides it. 



M. Euler too, in his Introductio in Analysin Infinitorum, by a contrivance 

 something like Mr. Machin's, discovers, that 1 and -l are the tangents of two 

 arcs, the sum of which is just 45°; and that therefore the diameter is to the 



circumference, as 1 to quadruple the sum of the two series', - X (1 — f- 



5T^ -71'+ si^ ^^-^"^3- X -rp + .7 -7^ +5i-^'=- Both Which 

 series' converge much faster than Dr. Halley's, and are yet at the same time 

 made to converge by the powers of numbers producing only short divisions ; 

 that is, divisions performed in one line, or without writing down any thing be- 

 sides the quotients. ,1 . .. 



I come now to explain my own method, which indeed bears some little resem- 

 blance to the methods of Machin and Euler : but then it is more general, and 

 discovers, as particular cases of it, both the series' of those gentlemen, and 

 many others, some of which are fitter for this purpose than theirs are. This 

 method then consists in finding out such small arcs, as have for tangents some 

 small and simple vulgar fractions, the radius being denoted by 1, and such also 

 that some multiple of those arcs shall differ from an arc of 45°, the tangent of 

 which is equal to the radius, by other small arcs, which also shall have tangents 

 denoted by other such small and simple vulgar fractions. For it is evident, that 

 if such a small arc can be found, some multiple of which has such a proposed 

 difference from an arc of 45°, then the lengths of these two small arcs will be 

 easily computed from the general series, because of tlie smallness and simplicity 

 of their tangents; after which, if the proper multiple of the first arc be in- 

 creased or diminished by the other arc, the result will be the length of an arc 

 of 45°, or ^th of the circumference. And the manner in vviiich I discover such 

 arcs is thus : 



Let T, /, denote any two tangents, o\' which x is the greater, and t the less ; 

 then it is known, that the tangent of the difference of the corresjjonding arcs is 

 equal to ^ ,^.^ . Hence, if t, the tangent of the smaller arc, be successively 

 denoted by each of the simple fractions ; , i, 4, |, &c. the general expression 



