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

PL is
THE CONICS, BOOK I 139
called the latus rectum (opQia) or the parameter of
the ordinates (nap' t)v Bvvavrai at Karayonevcu reTay/Jiei'cos) in
each case. In the case of the central conies, the diameter PP'
is the transverse (fj irXayLa) or transverse diameter', while,
even more commonly, Apollonius speaks of the diameter and
the corresponding parameter together, calling the latter the
latus rectum or erect side (6p6la TrXevpd) and the former
the transverse side of the figure (e?#o?) on, or applied to, the
diameter.
Fundamental properties equivalent to Cartesian equations.
If p is the parameter, and d the corresponding diameter,
the properties of the curves are the equivalent of the Cartesian
equations, referred to the diameter and the tangent at its
extremity as axes (in general oblique),
y 2
= px (the parabola),
y 2 =.px ±--jX 2 (the hyperbola and ellipse respectively).
Thus Apollonius expresses the fundamental property of the
central conies, like that of the parabola, as an equation
between areas, whereas in Archimedes it appears as a
proportion
y 2 : (a 2 + x 2 )
= b 2 : a 2 ,
which, however, is equivalent to the Cartesian equation
referred to axes with the centre as origin. The latter property
with reference to the original diameter is separately
proved in I. 21, to the effect that QV 2 varies as PV.P'V, as
is really evident from the fact that QV 2 :PV .P'V = PL: PP',
seeing that PL : PP' is constant for any fixed diameter PP'.
Apollonius has a separate proposition (I. 14) to prove that
the opposite branches of a hyperbola have the same diameter
and equal latera recta corresponding thereto. As he was the
first to treat the double-branch hyperbola fully, he generally
discusses the hyperbola (i.e. the single branch) along with
the ellipse, and the opposites, as he calls the double-branch
hyperbola, separately. The properties of the single-branch
hyperbola are, where possible, included in one enunciation
with those of the ellipse and circle, the enunciation beginning,
'
140 APOLLONIUS OF PERGA
If in a hyperbola, an ellipse, or the circumference of a circle '
sometimes, however, the double-branch hyperbola and the
ellipse come in one proposition, e.g. in I. 30: 'If in an ellipse
or the opposites (i. e. the double hyperbola) a straight line be
drawn through the centre meeting the curve on both sides of
the centre, it will be bisected at the centre.'
The property of
conjugate diameters in an ellipse is proved in relation to
the original diameter of reference and its conjugate in I. 15,
where it is shown that, if DD' is the diameter conjugate to
PP' (i.e. the diameter drawn ordinate- wise to PP'), just as
PP' bisects all chords parallel to DD', so DD' bisects all chords
parallel to PP'
;
also, if DL' be drawn at right angles to DD'
and such that DL' . DD' = PP' 2 (or DL' is a third proportional
to DD', PP'), then the ellipse
has the same property in relation
to DD' as diameter and DL' as parameter that it
has in
relation to PP' as diameter and PL as the corresponding parameter.
Incidentally it appears that PL . PP' = DD' 2 , or PL is
a third proportional to PP', DD', as indeed is obvious from the
property of the curve QV 2 : PV. PV'= PL : PP' = DD' 2 : PP' 2 .
The next proposition, I. 16, introduces the secondary diameter
of the double-branch hyperbola (i.e. the diameter conjugate to
the transverse diameter of reference), which does not meet the
curve; this diameter is defined as that straight line drawn
through the centre parallel to the ordinates of the transverse
diameter which is bisected at the centre and is of length equal
to the mean proportional between the ' sides of the figure ',
i.e. the transverse diameter PP' and the corresponding parameter
PL. The centre is defined as the middle point of the
diameter of reference, and it is proved that all other diameters
are bisected at it (I. 30).
Props. 17-19, 22-9, 31-40 are propositions leading up to
and containing the tangent properties. On lines exactly like
those of Eucl. III. 1 6 for the circle, Apollonius proves that, if
a straight line is drawn through the vertex (i. e. the extremity
of the diameter of reference) parallel to the ordinates to the
diameter, it will fall
outside the conic, and no other straight
line can fall between the said straight line and the conic
therefore the said straight line touches the conic (1.17, 32).
Props. I. 33, 35 contain the property of the tangent at any
point on the parabola, and Props. I. 34, 36 the property of