[ 437 .1 



the fine of the arc GI to the fine of the arc IK, 

 that is, as the fine of the angle GAI to the fine 

 of the angle I A K. Therefore, the fine of the 

 angle I A K (— 2IAC 4- GAI) is equal to three 

 times the fine of the angle GAI; but GAI is the 

 complement of AGB v and IAC the complement of 

 ABGL 



C o r o r. 31 



If (Fig. 3.) any line BN be drawn to divide the 

 angle ABG, and AN be joined, alfo AO be drawn 

 perpendicular to BN, and continued to the circle 

 in P, the fine of the angle compofed of NAP 

 and 2PAC will be lefs than three times the fine of 

 the angle NAP. Draw NQ\R perpendicular to 

 AB, cutting AP in Sj join AR, and draw QT 

 perpendicular to B N, and parallel to A 0,j then 

 BQ?_= NBT. But BQ^is greater than the red-, 

 angle E B C, that is, greater than the rectangle 

 NBV» under the two fegments of the line BN 

 drawn from B* to cut the- circle in N- and V : 

 therefore, TB is greater than VB, and NO greater 

 than O T. Confequently N-S is greater than S Q* 

 Hence RS is lefs than three times N-S; and there- 

 fore, the fine of the angle PAR (-NAP-H2PAC) 

 is lefs- than three times the fine. of NAP. 



Pro b l s m 



