Manchester Memoirs, Vol. Ex. (191 5), No. 13. 31 



trisected. With O as centre and any radius OA describe a 

 circle cutting B O produced at E. Make EC=OE. Join 

 AC. Then angle A CO is nearly one-third of angle A OB. 

 If angle AOB = fi and angle ACO = a , then 



sin (ft - a) = 2 sin a, 

 from which it follows that 



neglecting ' 



3 3^3', v 3' 



Let AC intersect the circle at D. Join 01). Then if 



the angle DOC=a, ft + a = 2(/3 -a), therefore 



2//3 v; 

 a =/:> - 2a— - + 



and 



or 



• ft 



sin - 



+ a' 



. h — « 

 2 s 1 n 









2 



2 





sin 



{ft 



- «') ■ 



_ sin /3 j_ sin 



r/ 



2 2 



Thus a is the same approximation to yS/3 as that obtained 

 by Method A. 



Method E. — An obvious method of trisecting an angle 

 or a circular arc is to first find, by estimation or con- 

 struction, an approximation to one-third, and then to add 

 to or subtract from this, as nearly as possible one-third of 

 the difference between the whole angle or arc and three 

 times the first approximation. 



The following methodical way of performing these 

 operations is given by Cantor and attributed to Albrecht 

 Diirer 19 (1471-1528), 



Let ABC (Fig. 24) be the angle to be trisected. With 

 B as centre and any radius describe a circle HVK cutting 

 AB at H and CB at IC Join UK. Trisect the chord 

 HK at M and N. With H as centre and HM as radius 



1 '■' M. Cantor, '.' Vorlesungen liber Geschichte der Mathematik," Vol. 

 II., p. 425, 1892. 



