QUADRATURE of the CONIC SECTIONS, &c. 313 
Now, as we have found (Art. 36.) that 2" tan = exprefles 
twice the area of the polygon AFF’F’F” (Plate IX.), the 
numerical value of which we have there denoted by Q, it 
Ay ° 
follows, that 2 tan’ —= = Q:. Moreover, the geometrical fe- 
Hae Ty 
{ANS apy 
Mi ad aaopabpagheds a 
* is equivalent to ~ (z —<); 
2.4 a daod i 
therefore, by fubftitution and tranfpofition, we get 
I I 
i (= tan’ = s+ tame ae + tan’ss....+-—tan?=), 
4 8 4” 2” 
4X | 
44. LET us now conceive 2 to be indefinitely great, then, as 
upon this hypothefis, Q becomes s, and : (e+ becomes 
fimply 2 and the feries whofe terms were 2 in number, now 
3 
goes on ad infinitum, we have at laft, after multiplying the 
oes expreffion by 4, 
fig es 
tan’ s 3 
x, b> 
Q (tan = s nin tan? 2 ae 1 tan?2s5 ae tan? 
— - - = A + tan seit &e. ). 
And this is one am of the feries to be eke 
45. Tur 
