A history of Greek mathematics - Wilbourhall.org

THE C0NIC8, BOOK I 135
the centre of the circular base. After proving that all
sections parallel to the base are also circles, and that there
is another set of circular sections subcontrary to these, he
proceeds to consider sections of the cone drawn in any
manner. Taking any triangle through the axis (the base of
the triangle being consequently a diameter of the circle which
is the base of the cone), he is careful to make his section cut
the base in a straight line perpendicular to the particular
diameter which is the base of the axial triangle. (There is
no loss of generality in this, for, if any section is taken,
without reference to any axial triangle, we have only to
select the particular axial triangle the base of which is that
diameter of
the circular base which is
at right angles to the straight line in
which the section of the cone cuts the
base.) Let ABC be any axial triangle,
and let any section whatever cut the
base in a straight line DE at right
angles to EC; if then PM be the intersection
of the cutting plane and the
/
axial triangle, and if QQ be any chord
in the section parallel to DE, Apollonius
proves that QQ' is bisected by PM. In
other words, PM is a diameter of the section. Apollonius is
careful to explain that,
'
if the cone is a right cone, the straight line in the base (DE)
will be at right angles to the common section (PM) of the
cutting plane and the triangle through the axis, but, if the
cone is scalene, it will not in general be at right angles to PM,
but will be at right angles to it only when the plane through
the axis (i.e. the axial triangle) is at right angles to the base
of the cone ' (I. 7).
That is to say, Apollonius works out the properties of the
conies in the most general way with reference to a diameter
which is not one of the principal diameters or axes, but in
general has its ordinates obliquely inclined to it. The axes do
not appear in his exposition till much later, after it has been
shown that each conic has the same property with reference
to any diameter as it has with reference to the original
diameter arising out of the construction ;
the axes then appear