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

FOCUS-DIRECTRIX PROPERTY 121
Similarly it can be shown that
2
(1 *e):e = XK':A'N.
By multiplication, XK . XK' :AN. A'N = (1 - e ) e2 : ;
and it follows from above, ex aequali, that
PN 2 :AN.A'N=(l~e 2 ):l,
which is the property of a central conic.
When e < 1, A and A' lie on the same side of 1, while
N lies on A A', and the conic is an ellipse ; when e > 1, A and
A / lie on opposite sides of X, while N lies on A'A produced,
and the conic is a hyperbola.
The case where e = 1 and the curve is a parabola is easy
and need not be reproduced here.
The treatise would doubtless contain other loci of types
similar to that which, as Pappus says, was used for the
trisection of an angle : I refer to the proposition already
quoted (vol. i, p. 243) that, if A, B are the base angles of
a triangle with vertex P, and AB = 2 A A, the locus of P
is a hyperbola with eccentricity 2.
Propositions included in Euclid's Conies.
That Euclid's
Conies covered much of the same ground as
the first three Books of Apollonius is clear from the language
of Apollonius himself. Confirmation is forthcoming in the
quotations by Archimedes of propositions (1) 'proved in
the elements of conies ', or (2) assumed without remark as
already known. The former class include the fundamental
ordinate properties of the conies in the following forms
(1) for the ellipse,
PN 2 : AN. A'N = P'N' 2 : AN'. A'N' = BC 2 :AG 2 ;
(2) for the hyperbola,
PN 2 : AN. A'N = P'N' 2 : AN' .A'N';
(3) for the parabola, PN 2 = pa
. AN;
the principal tangent properties of the parabola
the property that, if there are two tangents drawn from one
point to any conic section whatever, and two intersecting
122 CONIC SECTIONS
chords drawn parallel to the tangents respectively, the rectangles
contained by the segments of the chords respectively
are to one another as the squares of the parallel tangents
the by no means easy proposition that, if in a parabola the
diameter through P bisects the chord QQ' in V, and QD is
drawn perpendicular to PV, then
where pa is the parameter of the principal ordinates and p is
the parameter of the ordinates to the diameter PV.
Conic sections in Archimedes.
But we must equally regard Euclid's Conies as the source
from which Archimedes took most of the other ordinary
properties of conies which he assumes without proof. Before
summarizing these it will be convenient to refer to Archimedes's
terminology. We have seen that the axes of an
ellipse
are not called axes but diameters, greater and lesser
the axis of a parabola is likewise its diameter and the other
diameters are ' lines parallel to the diameter ', although in
a segment of a parabola the diameter bisecting the base is
the ' diameter ' of the segment. The two ' diameters ' (axes)
of an ellipse are conjugate. In the case of the hyperbola the
'
diameter ' (axis) is the portion of it within the (single- branch)
hyperbola ; the centre is not called the ' centre ', but the point
in which the nearest '
lines to the section of an obtuse-angled
cone' (the asymptotes) meet; the half of the axis (CA) is
'
the line adjacent to the axis ' (of the hyperboloid of revolution
obtained by making the hyperbola revolve about its 'diameter'),
and A'A is double of this line. Similarly GP is the line
'
adjacent to the axis ' of a segment of the hyperboloid, and
P'P double of this line. It is clear that Archimedes did not
yet treat the two branches of a hyperbola as forming one
curve ; this was reserved for Apollonius.
The main properties of conies assumed by Archimedes in
addition to those above mentioned may be summarized thus.
Central
Conies.
1. The property of the ordinates to any diameter PP\
QV 2 :PV.P / V = Q'V' 2 :PV'.P'V.