282 



mall have 



NEW SERIES for the 



tan a 



+=? 



a* 



( - tan - a + — a tan 2 - « -f- — tan* - a -f -i tan 2 4= « 

 v 4 2 4 4 43 8 44 16 



+, &c), 



and this may be confidered as a fecond formula for the rectifi- 

 cation of any arch of a circle ; for the procefs by which an 

 arch is found from the fquare of its reciprocal is fo rlmple, 

 that the latter being known, the former may alfo be regarded 

 as known. 



16. Instead of expreffing the fquare of the reciprocal of 

 the arch in this manner, by the fquares of the tangents of its 

 fub-multiples, we may exprefs it otherwife by the fquares of 



their fecants. For fince tan 2 - a — fee* - a — 1, and tan 2 i a 



22 4 



— fee 2 - a — 1, and fo on, therefore the feries 

 4 



1 „ 2 1 ii 2I ii ii 1 1 



- tan 2 _ a -f- _ tan 2 - a -f - tan 2 - a -f — tan 4 ~ a -f , &c. 

 424 4 43 8 44 l6 "r, «^. 



1 



■ tan* - a -+- 



424* 



is equivalent to 



2 fee* Li +J fee* l^a + 1 fee' la+l fee- ^ a +, &e. 



But the latter part of this expreflion is evidently an infinite 



geometrical 



