186 



NOTE I. 



This property is thus derived. 



From the third and fourth lemmas it will appear 

 that ... AG X AHrr AO x XI+AE" 

 AG X AB=AO X AE^^D 

 AG X AF=AO X AI+ AC 



AG X— AI =AO X— AH— AF 



AE X AH=AO X AG+ AC 

 AE X AB=AO X AG— AF 

 AE X AF=AOx AI— AB 

 AE X— Al =AO X— AF— AD 



AC X AH=AO X AE— AB 

 AC X AB = AO X AI— AH 

 AC X AF = AO X AG— AD 

 AC X— AI = AO X— AD— AB 



— AD X AH = AO X — AF— AB 



— ADx AB=AOx— AH+AG 



— ADx AF=AO^— AH + AC 



— AD X _AI =AO X AE+AC, 



Now in the above rectangles under the radius and the sum, or 

 difference of the chords as specified, it will be observed that each 

 chord is four times taken ; whence by addition it is evident, that these 

 rectangles will become ... 



AO X (4AG+ 4AEi-4AC-4AD)-(4AH + 4AB +4AF— 4AI) ; 



that is by Lemma II. AO x 4AO or 4AO\ whence the truth of 

 this property is manifest. 



