i8 



Gee and A damson, Trisecting an Angle. 



(a) Let X = i ; the conic is then the parabola 



2x 2 + $crx + by = 0. 



(b) Let \= — I ; the equation then becomes : 



x~ + ax - yr - by = 0. 

 The conic is thus a rectangular hyperbola, and the 

 solution of the trisection problem is the same as that of 

 Chasles (Fig. J). 



(c) Let A =3/2 ; then 



gx' 2 + 1 4ax +y- + 6by = 0. 

 This is the equation of an ellipse of a form convenient for 

 the purpose of construction. 



V. Use of the Quadratrix of Hippias, the Sine Curve and 

 the Spiral of A rdiimedes. 



In order to construct these curves, the circumference 

 of a circle must be divided into a large number of equal 

 parts. Consequently, these curves may be used to divide 

 any angle at the centre of the circle into any number of 

 equal parts. 



Hippias of Elias (about 420 B C.) used the Quadratrix 

 in connection with the quadrature of a circle. The curve 

 may be defined as the locus of a point P (Fig. 12) which 



ON MB 



Fig. 12. Quadratrix, Sine Curve ar.d Spiral of Archimedes. 



