A history of Greek mathematics - Wilbourhall.org

520 COMMENTATORS AND BYZANTINES
proof, that the said oblique sections cutting all the generators
are equally ellipses whether they are sections of a cylinder or
of a cone. He begins with ' a more general definition ' of a
cylinder to include any oblique circular cylinder. ' If in two
equal and parallel circles which remain fixed the diameters,
while remaining parallel to one another throughout, are moved
round in the planes of the circles about the centres, which
remain fixed, and if they carry round with them the straight line
joining their extremities on the same side until they bring it
back again to the same place, let the surface described by the
straight line so carried round be called a cylindrical surface!
The cylinder is the figure contained by the parallel circles and
the cylindrical surface intercepted by them ; the parallel
circles are the bases, the axis is the straight line drawn
through their centres; the generating straight line in any
position is a side. Thirty-three propositions follow. Of these
Prop. 6 proves the existence in an oblique cylinder of the
parallel circular sections subcontrary to the series of which
the bases are two, Prop. 9 that the section by any plane not
parallel to that of the bases or of one of the subcontrary
sections but cutting all the generators is not a circle ;
the
next propositions lead up to the main results, namely those in
Props. 14 and 16, where the said section is proved to have the
property of the ellipse which we write in the form
QV 2 :PV.P'V = CD 2 :CP 2 ,
and in Prop. 17, where the property is put in the Apollonian
form involving the latus rectum, QV = PV 2 . VR (see figure
on p. 137 above), which is expressed by saying that the square
on the semi-ordinate is equal to the rectangle applied to the
latus rectum PL, having the abscissa PV as breadth and falling
short by a rectangle similar to the rectangle contained by the
diameter PP f and the latus rectum PL (which is determined
by the condition PL .
PP'= DD' 2 and is drawn at right angles
to PV). Prop. 18 proves the corresponding property with
reference to the conjugate diameter DD' and the corresponding
latus rectum t
and Prop. 19 gives the main property in the
form QV 2 :PV.P'V = Q'V' 2 :PV. P'V. Then comes the
proposition that '
it is possible to exhibit a cone and a cylinder
which are alike cut in one and the same ellipse ' (Prop. 20).