278 NEW SERIES for the 



which formula, by reduction, will be found to be the very fame 

 as the other. 



12. The remit, then, of the whole inveftigation, may be 

 briefly ftated as follows : Let a denote any arch of a circle of 

 which the radius is unity, then fhall 



h - tan - a -f- - tan - a -\- - tan- a A — y tan -><*..,. 



I =z i tSLna 2 2 4 4 8 8 n 16 16 



a ] 



+ T( W ) -f- T (w + I ) -f S; 



where T( m ) and T( w +ij) denote any two fucceeding terms 



of the feries - tan - a -f - tan - a -f> &c, their places in it 

 2244 r 



being expreffed by the numbers m and flz+i ■> and where S is 

 put for the fum of all the remaining terms ; and the limits of S 

 are the two quantities 



I To+i) and - T(«+o — ( T ("Q — 4Tq+o) T( m + I ) 

 3 3 3 (T W -T c , + ) '* 



is, S is lefs than the former, but greater than the latter. 



The expreflions tan - a, tan - a, tan - a, &c. are eafily de- 



240 



duced from tan a, and from one another, by a well-known for- 

 mula in the arithmetic of lines, which may be exprefTed thus, 



at 



tan 



2 tan 1 A tan A 



13. I now proceed to the inveftigation of a fecond formula 

 for the rectification of the circle ; and for this purpofe refume 



the 



