14 GEE AND Adamson, Trisecting an Angle. 



Method of Newton. — Let ABC {Fig. 9) be the angle 

 to be trisected. On BC take any point E, and through 

 E draw EM perpendicular to BC Construct a hyper- 

 bola of eccentricity =2, with EM as directrix and B as 

 corresponding focus. Let F and H be the vertices, so 

 that BF= 2FE and BH= 2HE. Draw BO perpendicular 

 to AB, meeting ME at 0. With as centre and OB or 

 OH as radius, describe a circle cutting at P the branch of 



Fig. 9. Method of Sir I. Newton. 



the hyperbola which passes through F. Join PB. Then 

 the angle ABP will be one-third of the angle ABC To 

 prove this, draw PQ parallel to BH, intersecting ME at 

 N. join QH, PH. Then PQ=2PN=PB= QH. There- 

 fore the angle ABP, which is equal to the angle BHP, is 

 equal to half the angle PBH and to one-third of the 

 angle ABC. 



Method of Pappus. — Let ABC {Fig. 10) be the angle 

 to be trisected. As before {Fig. 9) construct the hyper- 

 bola and determine the point 0. With as centre and 

 OB or OH as radius describe a segment of a circle cutting 

 ME produced at 5. Join SB, SH. With 5 as centre 

 and SB or SH as radius describe a circle cutting at P 



