QUADRATURE of the GONIC SECTIONS, &c. 273 
arch a into’ 2*+! equal parts, and drawing tangents at the 
points of divifion, and the extremities of the arch. Therefore, 
denoting the a ya of the figure thus so by P, we 
have 
2. 35 ook Fe tes aac titap hg. 1 2 tan ie 
P tang 2 4 8 8 
® 
I I a 
tan — a... — tan — 
ee 3 ype oma 
and this is true; whatever be the number of terms in the feries 
ae me 3 PB, I a 
T tan -a+ -tan-4a....-+ —tan—*, 
2 2 4 4 20 Qn 
7. Now fuppofing » the numberof terms in the feries, to, in- 
creafe, then 2"—', the number of equal parts into which the 
arch is conceived to be divided, will alfo increafe, and may be- 
come greater than any aflignable number. But it is a principle 
admitted i in the elements of geometry, that/an ‘arch being divi- 
ded, and a polygon’ defcribed about’ it in the manner fpecified 
in. ‘article 6., the perimeter of the polygon will continually ap- 
proach to the circular arch, and will at laft differ from it by 
lefs than, any given quantity. Therefore, if we fuppofe » inde- 
finitely great, fo that the feries may go on ad infinitum, then,!in- 
ftead of P in the formula of the laft aiticle, we may fubftitute 
its limit, namely, the arch a, and thus we fhiall have’ 
mol S07 orpish Hi yt I I I 
=o -+ ctan-a-+ -tan = a+ >-tan = a 
a oe el 2 aa 4 13 8 34 
~ 
poy pe ) Thus 
> * WE may here obferve, that this formula may be confidered as the analy- 
tic expreffion of a general theorem (which is not inelegant) relating to regular. 
figures defcribed about any arch of a circle; and others analogous to it will 
occur in the following inveftigations. 
