finding the Value of an infinite Series , See. i r - 
thofe firft twelve terms to be = rx .764,600,691,483. 
1 3 f t 7 t \ 
Therefore the whole feries t ~— +.— «+ 
3rr 5^ 7 r 
1 ir 
8cc. infinitum is in this cafe = r x .764,600,691,483, 
+ r x .020,797,473,793, = r x -785,398,165,276, 
which is true to eight places of figures, the more exadt 
value of that feries being r x -785,398,163,397, 8cc.; 
fo that the value here found for this feries, by the help 
of only eight terms of the differential feries, differs from 
its true value by lefs than an unit in the eighth place of 
decimal figures, that is, by lefs than an hundred-mil- 
lionth part of the radius r, which is a degree of exadt-. 
nefs that could not have been attained by the mere com- 
/ 3 t 7 t 9 
putation of the feries /-•— +— ; 1+ — - — — 8cc. it- 
1 3 rr S r l r 9 r i Jr 
felf without computing fifty millions of its terms. There 
cannot be a ftronger inftance of the utility of that dif- 
ferential feries. 
Computation of the feries which eUpreJfes the time of the 
dejcent of a pendulum through the arch of a circle . 
Art. 16. As another example of the utility of the 
foregoing differential feries in finding the value of a fe- 
ries that converges very flowly, I will now apply it to 
the feries which expreffes the time of defeeat of a 
heavy body through a circular arch of 90°, which de- 
creafes 
