MENAECHMUS’S PROCEDURE 
113 
dcd cone, and 
bo OA) and at 
rpendicular to 
to the axis of 
the cone about 
particular hyperbola which we call rectangular or equilateral, 
and also to obtain its property with reference to its asymp 
totes, a considerable advance on what was necessary in the 
case of the parabola. Two methods of obtaining the particular 
hyperbola were possible, namely (1) to obtain the hyperbola 
arising from the section of any obtuse-angled cone by a plane 
at right angles to a generator, and then to show how a 
rectangular hyperbola ' can be obtained as a particular case 
by finding the vertical angle which the cone must have to 
give a rectangular hyperbola when cut in the particular way, 
or (2) to obtain the rectangular hyperbola direct by cutting 
another kind of cone by a section not necessarily perpen 
dicular to a generator. 
(1) Taking the first method, we draw (Fig. 2) a cone with its 
vertical angle BOC obtuse. Imagine a section perpendicular 
to the plane of the paper and passing through AG which is 
perpendicular to OB. Let G A produced meet CO produced in 
A', and complete the same construction as in the case of 
the parabola. 
el to OL meet- 
bo th bisected 
G 
Fig. 2. 
■dinates y. 
d to obtain the 
In this case we have 
* 
PN 2 = BN. NC = AN.NG. 
1523.2 J