QUADRATURE of the CONIC SECTIONS, &c. 299 
mi Ty) {* tanta "8 ‘tanta 
tana srtan 33—tan’>a ~ 3° 3—tan'ta 
+ eee ef e 8 era 
—tan’ 54 a a 
3 3 3 — tan’ — 
And this is true, 2 being any number whatever. Now, if we | 
confider that 3” tan = exprefles the fum of the fides of a figure 
formed by dividing the arch into 3” equal parts, and drawing 
tangents at the points of divifion, whofe orders, reckoned from 
one end of the arch, are indicated by even numbers, (that end 
itfelf being reckoned one of them), and producing each to meet 
thofe adjoining to it, and the laft to meet a radius of the circle 
produced through the other end of the arch, it will be obvious, 
that being fuppofed to increafe indefinitely, the expreffion 
3r tan =will have for its limit the arch a, and in this cafe the 
feries will go on ad infinitum. Thus we fhall have 
Toy E 8 tania ano £ tan< a 8 tan 3. a4 afiga 
tana 4 3.3—tan’y a7) 3} 3—tan*2a 3 3—tan’ 5a’ 
and by tranfpofition, 
1 ii Ny 8 tanta — tan! a tan 
= +5 chef er | nite: 
a tana,’ 3 tan AON: j—tan™ 34 =693° 3—tan’ oa 
and this is the feries which I propofed to inveftigate. 
Vou. VI.—P. II. Pp 31. THE 
