A history of Greek mathematics - Wilbourhall.org

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,