071 the Duplication of the Cube. 123 



For from V, erect the perpendiculars VG, VH ; then 

 the angles HCV, VCG, being- each equal to 60° ; HC, 

 or CG, will be equal to 2 CV=4 a, and HV^ = 16 a^ — 

 4a^ = 12 a^ but by conie s, CV='(4a^): HV^(12a^):: 

 2 CV + Yd X Vd (4 a + X X x) : dP^ ; putting tlierefore 

 dP, the ordinate of the curve, as usual, equal to y, we 

 shall have y^==12ax + 3x'' for the equation showing 

 the relation of the ordinates and abscissas of this curve, 

 ^vhich is the same as that above determined for the tri- 

 secting curve; consequently, this is an hyperbola, whose 

 assymptotes are inclined in an angle of 120 degrees, and 

 whose axes are 4 a, 4av/3, respectively. 



October 2^ 1809. 



