250 
THE DUPLICATION OF THE CUBE 
The projection on the plane ABC oi the intersection between 
the cone and the tore is seen, by means of their equations 
(1) and (3) above, to be 
X 2 = — V (x 2 + y' 2 ), 
a 
or, in polar coordinates referred to A as origin and AC as axis, 
_ fe 2 
^ a cos 2 6 
It is easy to find any number of points on the curve. Take 
the circle ABC, and let AG the diameter and AB a chord 
be the two given straight lines between which two mean 
proportionals have to be found. 
With the above notation 
AG = a, AB — h ; 
and, if BF be drawn perpendicular to AG, 
AB 2 = AF. AC, 
or AF = h 2 / a. 
Take any point G on BF and join AG. 
Then, if Z GAF =6, AG = AFsec (9. 
With A as centre and AC as radius draw a circle meeting 
AC in H, and draw HL at right angles to AG, meeting AG 
produced in L.