180 



Then a"reeably to the Corollary to the II. Lemma, the difference 

 between the sums of those chords, alternately taken, is equal to the 

 radius of the circle ; that is . . 



(AI + AG+AE+AC)— (AH+AF + AD+AB) = AO 

 or . (AG+AE+ AC-AD) — (AH+ AB + AF - AI) = AO . Theor. I. 



But a remarkable property of these chords, 

 (AG + AE + AC — AD) and (AH+AB + AF — AI) will be 

 found to result from the application of the 3d and 4th Lemmas, viz. 



The sum of all the rectangles under each of the first four, and 

 each of the four following, is equal to four times the square of the 

 radius of the circle. — (See Note 1.) 



That is, . (AG + AE + AC — AD) x (AH + AB + AF — AI) 

 = 4 AO' . . . Theor. II. 



Again, by the application of the same Lemmas, it will appear 

 that . . (AH + AB)x(AI — AF) = AO^ . . Theor. III. See Note 2. 

 and . . (AG4-AE)x(AD — AC)= AO^ . . Theor. IV. See Note 3. 



^"4lh^^'Ta^t}^^^(^^-^^)= ^" X AB . . . Theor. V. 

 Since therefore in Theorems I. and II. are given 



(AG+AE+ AC — AD) — (AH+ AB+AF— AI) = AO 1 ^ 

 and (AG+AE+AC — ADJ) x (AH+AB+AF— Al)= 4 AO^ i 



by the construction of a well known simple problem of finding two 



lines of which their difference and rectangle are given, 



AG + AE + AC— AD? 



, i are given, 

 and AH + AB + AF — AIJ 



but AH + AB + AF — Al 



is the same as . . (AH + AB) — (AI — AF) 



therefore. . . . . (AH+ AB) — (AI — AF) is given| ^ 



and Theor. III. . (AH + AB) x (AI — AF) is given^ 



therefore, as before AH + AB) 



^ V are given, 

 and AI — AF^ " 



