$VADRA?URE of the CONIC SECTIONS, &c. 273 



arch a into 2"-"- 1 equal parts, and drawing tangents at the 

 points of divifion, and the extremities of the arch. Therefore, 

 denoting the perimeter of the figure thus conftrufted by P, we 

 have 



L — — L_ "* - tan - a -f- - tan - a -f- - tan - a 

 P tanfl 2 2 44 8 8 



-f- JL tan — > tf • . . -| tan - ; 



' l6 10 2" 2? 



and this is true,- whatever be the number of terms in the feries 

 - tan - a -f- - tan - a .... -j tan — *. 



2 2 4 4 2" 2" 



7. Now fuppofing n the number of terms in the feries, to in- 

 creafe, then a"— 1 , the number of equal parts into which the 

 arch is conceived to be divided, will alfo increafe, and may be- 

 come greater than any affignable number. But it is a principle 

 admitted 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 n inde- 

 finitely great, fo that the feries may go on ad infinitun, then, in- 

 ftead of P in the formula of the laft article, we may fubftitute 

 its limit, namely, the arch a, and thus we mail have 



I III I , I „. I , I .. I . 



_ — — - — -f- - tan - a -f- _ tan - a + - tan - a -f- 

 a tan a 2 2 4 4 8 8 



— - tan — - a -\-, &c. 

 16 16 



Thus 



* We may here obferve, that this formula may be confidered as the analy- 

 tic expreffion of a general theorem (which is not meleganlj relating to regular 

 figures defcribed about any arch of a circle ; and others analogous to it will 

 occur in the following inveftigations. 



