CONIC SECTIONS IN ARCHIMEDES 
123 
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In the case of the hyperbola Archimedes does not give 
any expression for the constant ratios PN 2 : AN. A'N and 
QV 2 :PV. P'V respectively, whence we conclude that he had 
no conception of diameters or radii of a hyperbola not meeting 
the curve. 
2. The straight line drawn from the centre of an ellipse, or 
the point of intersection of the asymptotes of a hyperbola, 
through the point of contact of any tangent, bisects all chords 
parallel to the tangent. 
3. In the ellipse the tangents at the extremities of either of two 
conjugate diameters are both parallel to the other diameter. 
4. If in a hyperbola the tangent at P meets the transverse 
axis in T, and PN is the principal ordinate, AN > AT. (It 
is not easy to see how this could be proved except by means 
of the general property that, if PP' be any diameter of 
a hyperbola, Q V the ordinate to it from Q, and QT the tangent 
at Q meeting P'P in T, then TP: TP' = PV: P'V.) 
5. If a cone, right or oblique, be cut by a plane meeting all 
the generators, the section is either a circle or an ellipse. 
6. If a line between the asymptotes meets a hyperbola and 
is bisected at the point of concourse, it will touch the 
hyperbola. 
7. If x, y are straight lines drawn, in fixed directions respec 
tively, from a point on a hyperbola to meet the asymptotes, 
the rectangle xy is constant. 
8. If PN be the principal ordinate of P, a point on an ellipse, 
and if NP be produced to meet the auxiliary circle in p, the 
ratio pN: PN is constant. 
9. The criteria of similarity of conics and segments of 
conics are assumed in practically the same form as Apollonius 
gives them. 
The Parabola. 
1 • the fundamental properties appear in the alternative forms 
PN 2 : P'N' 2 = AN: AN', or PN 2 = p a . AN, 
QV 2 :Q'V' 2 = PV:PV', or QV 2 = p.PV. 
Archimedes applies the term parameter (a 7rap’ av Svvavrai 
al dnb ray royds) to the parameter of the principal ordinates