QUADRATURE of the CONIC SECTIONS, &c. 299 



1 1 f 8 tan 1 a 



j_ 8 tan ~ a 

 3 Z 3 — tan 2 -^ 



tan - / 



,-tan^\ 



tan ^~ s«tan" L33 — tan z f a 



6 3 a 



, 8 tan T V ^ ,8 



~T" 1' ■ ' ' " • • • • ~ r~ 



3' 3 — tan i T a y 



3" 

 And this is true, n being any number whatever. Now, if we 



confider that 3" tan — exprefTes the fum of the fides of a figure 



formed by dividing the arch into y 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 n being fuppofed to increafe indefinitely, the expreffion 



2» tan —will have for its limit the arch a, and in this cafe the 

 3 n 



feries will go on ad infinitum. Thus we fhall have 



1 = 1 _ f 8 tan j- a ,8 tan ~ a ,8 tan -^ 7 a f 



tan* a "I33— tan 2 ^ 3*3— tan^a 3 3 _ tan 2 T y* ' \ 



and by tranfpofition, 



1 ___ 1 _^_ 8 tan-i-^ , 8 tan ± a 8^ tan T y a 



a " tan** 3 3— tan a f* 3* 3— tan 2 -^ J" 3 3 3— tanV 7 « "*"' &C * 



and this is the feries which I propofed to invefligate. 



Vol. VI.— P. II. P p 3I . t he 



