QUADRATURE' of the CONIC SECTIONS, &c. 271 



5. To proceed now in the inveftigation of the different fe- 

 ries, for the rectification of an arch of a circle, let A denote 

 any arch, the radius being fuppofed unity. Then, from the 

 arithmetic of lines, we have 



— I tan - A. 



tan A ~ 2 tan 4- A 2 2 

 In this formula let each term of the feries of arches 



111 1 1 



'248 2 »— 2 ' 2 «—i ' 



(which is a geometrical progreffion, having the number of its 

 terms n> and its common ratio ^,) be fucceffively fubftituted for 

 A, and let the refults be multiplied by the terms of the corre- 

 fponding feries of fractions 



111 1 1 



*> Zi *!» 



2 4 8 2"— 2 2"— l 



then we mall obtain the following feries of equations : 



1 1 i*i 



— - tan 7 a, 



tan a 2 tan ±a 2 



tan J #, 



2 



1 



2 tan 7 # 4 tan •£ a 4 



1 1 



4 tan ±a 8 tan | a 8 T 



I _ I £ 



8tan£* 16 tan T V a l6 tan t<t *,. 



1 . o 



tan 



^ ^« t ^. # 2" — i 2 n — l ' 



2 «— 2 tan 2«— x tan * *■ 



2i — a 2 n ~~ l 



1 1 r , <z 



— — tan—. 



ci a in on 



2«— > tan 2» tan — * z 



2«— 1 2 



Let 



