112 
CONIC SECTIONS 
a section through the axis OL of a right-angled cone, and 
conceive a section through AG (perpendicular to OA) and at 
right angles to the plane of the paper. 
o 
Fig. 1. 
If P is any point on the curve, and PN perpendicular to 
A G, let BG he drawn through N perpendicular to the axis of 
the cone. Then P is on the circular section of the cone about 
BG as diameter. 
Draw AD parallel to BG, and DF, GG parallel to OL meet 
ing AL produced in P, G. Then AD, AF are both bisected 
by OL. 
If now PN = y, AN — x, 
y 2 = PN 2 = BN. NG. 
But B, A, G, G are concyclic, so that 
BN.NG= AN.NG 
= AN.AF 
= AN.2AL. 
Therefore ' y 1 = AN.2AL 
= 2 AL. x, 
and 2 AL is the f parameter 5 of the principal ordinates y. 
In the case of the hyperbola Menaechmus had to obtain the