Segar. — On the Trisection of an Angle. 



219 



Let AOB be the given angle. With as centre, describe a circle 

 meeting OA, OB, in C, D, and these lines produced through in E, F, 

 respectively. Let G, J, be the middle points of the arcs CD, EF ; and 



Fig. 1. 



H, I, those of the arcs CG, GD, respectively. Now draw JH, EG, meeting 

 in K; and JI, FG, meeting in L. Then OK, OL, are the trisectors as 

 given by this construction. 



We shall now proceed to investigate the degree of approximation. 

 We shall use the methods of co-ordinate geometry. 



Take OK as the axis of x, and a line througli O perpendicular to OK 

 as the axis of y. Let the angle GOH be a. Then we easily obtain the 

 following system of co-ordinates : — 



where is the original angle AOB. 

 The equation of JH is 



cos a + cos I a — ^ I 

 X — a cos a V 4/ 



y — a sin a , ■ I g\ 



•^ sin a + .«in I a — "1 



X - a cos a / n\ 



■■ = cot (a-^) (1) 



y — a siti a \ 8/ ^ 



which reduces to 



