184 



Put AG + AE = cTN Then (x — ij) — (z — t>J= r \ 



AD — AC = i/ 1 C^ — y)'^(z — 'i'J = 4 r^. ^whence z and y 



AH + AB=;s> zv = ,-2 ^ are determined ; 



AI— AF=ul xy = r) 



and . . . . AO = r J 



then AH + AB rr ^| Whence AH and AB are determined, 

 and AH x AB = j/j 



I shall conclude this paper with the following numerical calcula- 

 tion of the length of the side of the Polygon : 



Let the Radius AO = 1 ; and suppose the chords AR, AP, KP, 

 and AN, to be drawn ; then followiag the several constructions, 



OD -i, DC =yoC + OD'=v/l + !6 =iVir; 



OF = DC + DO = iv/l7 +i=i(\/l7+l;) 



FC =v/oF+OC = i V34+2 ^/T7; 



OH=FC — OF =i(v/:34 +2v/l7 — v^n — 0; 

 Again EO =DC — DO = i (77?— 1) ; 



EC = VEO + OC - i V 34 - 2^17, 

 and OG = EC + EO = i (\/34 - 2 v'ly + v/Tr - 1) 

 and hence 01 + IG . (OG) =\ (^34 - 2 ^Ty + v/TZ - I ) 



Whence, by the well known division of a right line into two parts 

 containing a given rectangle, we shall have 



01 = i 0G + V loG" - OH X AO 

 and IG = i OG ~\/^0(P — OH x AO 



