4 16 Mr. vince’s new Method , &c. 
(cries continually approach to - , the correction, to be added , 
4 
will be 4 ^ therefore — - — H — --f — b &c. +-J- is = 
to the fum of the given feries ; but by prop. i. part I. the fum 
of — 1 8 — 4 — f &c. is equal to— S + 4 " (S being 
1- 3- 5 5*7-9 9-n* I 3 1 4 8 ^ 6 
a circular arc of 45 0 , whofe radius is unity) hence the fum of 
the given feries is ~ S -f - . 
This method is not only applicable to thofe cafes, where the 
given feries refolves itfelf into another, whofe fum is either 
accurately known or can be exprefled by circular arcs and loga- 
rithms, but alfo to thofe cafes where we want to approximate 
to the value of the given feries, as it muft, in general, be 
neceffary firft to render the terms of the feries converging, by 
collecting two into one, before the operation of approximation 
begins, and confequently a correction of this latter is neceffary 
■in order to exhibit the value of the given feries. 
