182 



And hence it is evident that in Analysis A, where half the differ- 

 ence of the chords, as specified, is = 20D, and their lectangle 4AO^, 

 or 40C-, 



that . AG + AE+AC — AD is = 2 (DC+DO) = 20F| by construc- 

 and . AH+ AB-^AF — AI . . 2(DC -DO) = 20Ei tion ; 

 and therefore now Analysis B 



(AH + AB) - (AI — AF) is given = 20E 



and (AH + AB) x (AI— AF) = AO' = OC^ 



and Analysis C(AG+ AE) — (AD— AC) = 20F 



and (AG + AE) x (AD - AC) = AO' = OC 



Whence under the construction referred to 

 there is given in the former AH + AB = EC+ EO = OG, i by construc- 



and in the latter AD - AC = FC — FO = OH j tion. 



and hence Theor. V. . . AH x AB =(AD — AC) x AO-=OH x AO 



and therefore finally .... AH + AB =OG 



and AH X AB =0H x AO 



But since OG has been divided into two parts 01, IG, the rec- 

 tangle of which is equal to OH x AO, consequently 01 is = AH, 

 and IG = AB, but AH is the supplemental choi'd of twice the 

 arch, and AB the chord of half the arch, subtended by the side of 

 the polygon, by hypothesis ; and hence, 01 and IG are equal to 

 those chords, respectively, and since 01 has been applied on the 

 circumference from A to R, and the arch KR bisected in P ; and IG 

 taken twice on the circumference from A to N, it is evident that the 

 arches KP and AN are each the l7th part of the whole circum- 

 ference. 



CoROLLAUY. — IG is the side of a regular polygon of 3t sides 

 inscribed in the circle. 



Scholium. — Such is the construction of this interesting problem 

 to which we are led by the preceding Analysis: a construction in 

 itself exceedingly simple, falling little short of the well known 



