Manchester Memoirs, Vol. lix, (191 5), No. 1*5. 5 



and is obtained by using the radius AB as the constant 

 length. 



Let ABC (or /3) be the angle to be trisected. Pro- 

 duce CB to meet the trisectrix at D. Join DA, cutting 

 the circle at E. Then, since DE equals AB, the angle 

 ADB, denoted by a, is one-third of the angle ABC. 



On comparing Fig. 3 with Fig. 1 it will be apparent 

 that this solves the vevans of Archimedes. 



III. Use of the Conchoid of Nicomedes? 



This curve is such that the straight line joining any 

 point on the curve with a given point is cut by a given 

 straight line so that the segment between the curve and 

 the straight line is constant. 



The conchoid has been used in several ways to solve 

 the trisection problem. 



Proclus states that Nicomedes, the inventor of the 

 curve, applied it to trisect an angle, but Pappus claims 

 this application. It is possible, however, that Archimedes 

 himself used a curve of the nature of the conchoid, in 

 order to solve the vevcris problem already mentioned. 



The simplest application of the conchoid is as follows : 



/'>; r . 4. Use of Conchoid, 

 Circa 1 So B.C. 



