2 Gee and Adamson, Trisecting an Angle. 



geometers is expressed by the word veixris, which presents 

 difficulties of exact translation. It has been rendered by 

 inclinalio in Latin, and by einschiebung in German. This 

 latter word signifies "putting in," "intercalation," or 

 "interpolation," meanings only partly expressing what is 

 implied by a vevcri^ which requires the interpolation of a 

 line in a diagram under special conditions, as is evident 

 in the following construction that Archimedes used to 

 solve the trisection problem. 



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

 B as centre and any radius BA, describe a circle. Produce 

 CB to F. then from A draw a line AED so that the 



D F 



Fig. I. Method of Archir^edes. 



intercept ED is equal to the radius of the circle. Here 

 the i>eu(ris requires the interpolation of a line passing 

 through a given point A and drawn so as to cut the 

 circle in a point E that must satisfy the condition that 

 ED is of the specified length. This can only be done 

 by certain indirect methods as stated below. If we 

 suppose the operation be accomplished, then — 



The angle EBF= - the angle ABC, 



or a = - \d 

 3 



for y = 2« and (3 = a + y. 

 Hence BG drawn parallel to AD will be a trisector of 

 the angle ABC. 



