QUADRATURE of the CONIC SECTIONS, &c. = 27 
5. To proceed now in the inveftigation of the different fe- 
ries, for the re¢tification of an arch of a circle, let A denote 
any arch, the radius being fuppofed unity. Then, from the 
arithmetic of fines, we have 
I I I I 
S| ——————-_ tan — A. 
tanA 2tanzA 2 2 
In this formula let each term of the feries of arches 
I I I I I 
(which is a geometrical progreflion, having the number of its 
terms 7, and its common ratio 4,) be fucceffively fubftituted for 
A, and let the refults be multiplied by the terms of the corre- 
fponding feries of fractions 
Bay Ale a I re 
SMe Dg ONG | Gaceah gaat 
then we fhall obtain the following feries of equations : 
I I 
: = — —- tanya, 
tana 2tan+4 2 
I 
Gey ie eae t tanta, 
2tant+a 4tanta 4 
x = z T tanta 
4tanta ~ 8tanja Sige oe 
= : tan +z @ 
Stan¢a — i6tan;;a Re og ie 
e ° e e ° ° ° ° ° - 
I a, I ma a 
Qn—2 tan 2"—I tan a Qi 2Qn—V? 
Qi—a Qn. 
> 
I mn I a 
ar—! tan 2" tan — - 
gn—t1 2% 
Let 
