A history of Greek mathematics Vol.II from Aristarchus to Diophantus by Heath, Thomas Little, Sir, 1921

DISCOVERY OF THE CONIC SECTIONS 111
observation that an ellipse can be obtained from a cylinder
as well as a cone is actually made by Euclid in his Phaeno-
'if, says Euclid, ' a cone or a cylinder be cut by
mena :
a plane not parallel to the base, the resulting section is a
section of an acute-angled cone which is
similar to a Ovpeos
(shield).' After this would doubtless follow the question
what sort of curves they are which are produced if we
cut a cone by a plane which does not cut through the
cone completely, but is either parallel or not parallel to
a generator of the cone, whether these curves have the
same property with the ellipse and with one another, and,
if not, what exactly are their fundamental properties respectively.
As it is, however, we are only told how the first writers on
conies obtained them in actual practice. We learn on the
authority of Geminus l that the ancients defined a cone as the
surface described by the revolution of a right-angled triangle
about one of the sides containing the right angle, and that
they knew no cones other than right cones. Of these they
distinguished three kinds ; according as the vertical angle of
the cone was less than, equal to, or greater than a right angle,
they called the cone acute-angled, right-angled, or obtuseangled,
and from each of
these kinds of cone they produced
one and only one of the three sections, the section being
always made perpendicular to one of the generating lines of
the cone ;
the curves were, on this basis, called section of an
'
acute-angled cone' (= an ellipse), ' section of a right-angled
cone' (= a parabola), and 'section of an obtuse-angled cone '
(= a hyperbola) respectively. These names were still used
by Euclid and Archimedes.
Menaechmuss probable procedure.
Menaechmus's constructions for his curves would presumably
be the simplest and the uyost direct that would show the
desired properties, and for the parabola nothing could be
simpler than a section of a right-angled cone by a plane at right
angles to one of its generators. Let OBG (Fig. 1) represent
1
Eutocius, Comm. on Conies of Apollonius.
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
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
F
= 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