20 Gee AND Adamson, Trisecting an Angle. 



Therefore at any point P 



r sin 6 — b 8 = a 6 x b\a. 

 Therefore 



OJV=slyc DR x OE/OD, 

 or 



0E:0D=0N:mcDR. 



Also area of sector DOR 



= L0Dx arc DR 



2 



= J -0Dx ONx ^- 

 2 OE 



= area of triangle ODPx OD/OE. 



Trial Methods. 



Several of the methods of trisecting an angle involve 

 a problem which may be solved, not only by using a 

 locus, as in the examples already given, but also by 

 making a series of trials until the diagram is correctly 

 drawn. In the following examples the trials are such as 

 can easily be made. 



I. The construction of Archimedes (Fig. i) can readily 

 be accomplished by trial. Straight lines may be drawn 

 through A until it is found by measurement that the 

 intercept ED is equal in length to the radius of the circle. 

 The actual drawing of the lines may be avoided by 

 marking on the edge of a ruler the length of the radius 

 and applying the ruler to the diagram so that the edge 

 passes through the point A and the two marks lie, one on 

 the circle at E, the other on CB produced at D. 



Another appliance which can be used to determine 

 the points E and D was invented by Hermes 9 and is 

 shown in Fig. 13. The instrument is a pair of dividers 

 one leg of which is provided with two points P and Q 

 separated by a fixed distance. The legs are hinged so 

 that the three points are always co-linear when touching 



8 W. Dyck. " Katalog Mathematischer unci Mathematisch-physika- 

 lischer Modelle, Apparate und Instrumente." Munich, 1892. 



