280 NEW SERIES for the 
Let the fum of thefe equations be taken, as in the inveftigation 
of the firft formula, and obferving that the feries 
I I I I 
Favte Gare” Stan ta" + 
22"—2 tan? —-— 
n 
—TI 
is found on both fides of the refulting equation, let it be reject- 
ed from both; then we obtain 
past 
hee I T 
tan 4—= + tan*= 2 + —tan® SRE tan Pee Ee ent 
4 
A ng 2 iy AeA cog UE es Blan 16 
. 2 I oa) te I I 
tan — — GC Bagi ag as e * +). 
Now it appears, that one part of this expreflion, viz. 
ee pha oie ela 
x I 
2 © 248) 2.4 2.43 2.42 
is a geometrical feries, the firft term of which is =; the laft 
4 : ‘ 
term — et and common ratio = 3 therefore its fum_ is 
I 
a (x = et 
3 4 
Alfo, fince 2” tan = is the expreffion for the perimeter of a 
polygon, formed by dividing the arch @ into 2"—1 equal parts, 
by drawing tangents at the points of divifion, and producing 
the 
