A history of Greek mathematics - Wilbourhall.org

134 APOLLONIUS OF PERGA
straight line through the centre of
the circle perpendicular to
its plane, a straight line passing through the point and produced
indefinitely in both directions is made to move, while
always passing through the fixed point, so as to pass successively
through all the points of the circle ; the straight line
thus describes a double cone which is in general oblique or, as
Apollonius calls it, scalene. Then, before proceeding to the
geometry of a cone, Apollonius gives a number of definitions
which, though of course only required for conies, are stated as
applicable to any curve.
1
In any curve,' says Apollonius, ' I give the name diameter to
any straight line which, drawn from the curve, bisects all the
straight lines drawn in the curve (chords) parallel to any
straight line, and I call the extremity of the straight line
(i.e. the diameter) which is at the curve a vertex of the curve
and each of the parallel straight lines (chords) an ordinate
(lit. drawn ordinate- wise, reray/zej/o)? KaTrj-^Oai) to the
diameter/
He then extends these terms to a pair of curves (the primary
reference being to the double-branch hyperbola), giving the
name transverse diameter to any straight line bisecting all the
chords in both curves which are parallel
to a given straight
line (this gives two vertices where the diameter meets the
curves respectively), and the name erect diameter (6p6ia) to
any straight line which bisects all straight lines drawn
between one curve and the other which are parallel to any
straight line ; the ordinates to any diameter are again the
parallel straight lines bisected by it. Conjugate diameters in
any curve or pair of curves are straight lines each of which
bisects chords parallel to the other. Axes are the particular
diameters which cut at right angles the parallel chords which
they bisect ; and conjugate axes are related in the same way
as conjugate diameters. Here we have practically our modern
definitions, and there is a great advance on Archimedes's
terminology.
The conies obtained in the
oblique cone.
most general way from an
Having described a cone (in general oblique), Apollonius
defines the axis as the straight line drawn from the vertex to