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

CONIC SECTIONS IN ARCHIMEDES 123
In the case of the hyperbola Archimedes does not give
A'N and
any expression for the constant ratios PN 2 : AN.
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 f 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 respectively,
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 conies and segments of
conies 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 pa
. AN,
QV 2 :Q'V' 2 = PV:PV, or QV 2 = p.PV.
Archimedes applies the term parameter (a irap
av Bvvclvtcu
at oltto t&s rofxds) to the parameter of the principal ordinates
124 CONIC SECTIONS
only :
p is simply the line to which the rectangle equal to QV 2
and of width equal to PFis applied.
2. Parallel chords are bisected by one straight line parallel to
the axis, which passes through the point of contact of the
tangent parallel to the chords.
3. If the tangent at Q meet the diameter PV in T, and QV be
the ordinate to the diameter, PV = PT.
By the aid of this proposition a tangent to the parabola can
be drawn (a) at a point on it, (b) parallel to a given chord.
4. Another proposition assumed is equivalent to the property
of the subnormal, NG = \
rpa .
5. If QQ' be a chord of a parabola perpendicular to the axis
and meeting the axis in M, while QVq another chord parallel
to the tangent at P meets the diameter through P in V, and
RHK is the principal ordinate of any point R on the curve
meeting PV in H and the axis in K, then PV :PH > or
= MK : KA '
;
for this is proved ' (On Floating Bodies, II. 6).
Where it was proved we do not know ; the proof is not
altogether easy. 1
6. All parabolas are similar.
As we have seen, Archimedes had to specialize in the
parabola for the purpose of his treatises on the Quadrature
of the Parabola, Conoids and Spheroids, Floating Bodies,
Book II, and Plane Equilibriums, Book II ; consequently he
had to prove for himself a number of special propositions, which
have already been given in their proper places. A few others
are assumed without proof, doubtless as being. easy deductions
from the propositions which he does prove. They refer mainly
to similar parabolic segments so placed that their bases are in
one straight line and have one common extremity.
1. If any three similar and similarly situated parabolic
segments BQ
X 3
lying along the same straight line
, BQ
2
,
BQ
as bases (BQ 1
< BQ 2
< BQ 3 ), and if E be any point on the
tangent at B to one of the segments, and EO a straight line
through E parallel to the axis of one of the segments and
meeting the segments in R%, R 2
,
R 1
respectively and BQ 3
in 0, then
R,R :
2
R 2
R, = (Q2 BQ (BQ, Q3 : 3 ) . Q, Q : 2 ).
1
See Apollonius of Perga, ed. Heath, p. liv.