Manchester Memoirs, Vol. lix. (1915), No. 13- H 



produce EM to meet the circle HVK at P". Draw BF 

 and BF' cutting the circle ERG at R and S respectively. 

 Draw the chord RS. With F' as centre and radius equal 



Fig. 25. Modification of Durer's Method. 



to RS describe an arc cutting the arc FF' at a point P. 

 Connect B to P, then BP is one of the approximate tri- 

 sectors. 



It will now be proved that the angle ABP only 



exceeds one-third of the angle ABC by JL (*-) 



648 \3/ 



radians, where the angle ABC is /3 radians and the terms 

 neglected involve (^ J to at least the 13th power. Draw 

 BV perpendicular to HK and P" W parallel to HK. 

 Let the circular measure of angles ABC, ABP, PBP" , 

 P'BP" and ABP' be /3, a, y, <S and respectively. 

 Then, since angle P"BW=QJ2, 



3 sin 0/2 = sin/3/2 ; 

 hence by expanding, 



But 



• -£-!(£)' neglecting.^) 7 . 



3 3^3' ^3' 



