A history of Greek mathematics - Wilbourhall.org

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.
a/
L
F
Fig. 1.
If P is any point on the curve, and PN perpendicular to
A G, let BG be 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 GL meeting
AL produced in F, G. Then AD, AF are both bisected
by OL.
If now PN = y, AN = x,
1/= PN 2 = BN.NG.
But B, A, G, G are concyclic, so that
Therefore
BN.NG=AN.NG
= AN.AF
= AN.2AL.
f = AN.2AL
= 2AL.x,
and 2AL is the ' parameter ' of the principal ordinates y.
In the case of the hyperbola Menaechmus had to obtain the