Manchester Memoirs, Vol. /2V. ( 1 9 1 5 ), No. 13. 

 Thus after 2n bisections : — 



35 



+ P»n-*OP» 



/3 + ^ 

 4 16 



64 





£ 1- 

 4V 



_ 1 \ 



4 n J 







1 - 



i/4 



3 



„2» 



3V 1 2*0 



In conclusion, the problem of trisecting an angle is 

 essentially, as Pappus noted, <z solid problem or one of the 

 third degree, requiring the solution of a cubic equation. 

 Thus an exact trisection must be impossible in practice 

 with the aid only of a ruler and a pair of compasses. 

 Owing to this not having been realised, much time and 

 labour has been wasted in striving to obtain an exact 

 solution of the problem.- 



20 De Morgan in his "Budget of Paradoxes," pp. 255-258 (1872), cites 

 some examples. 



