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 : OF 2 , 
and in Prop. 17, where the property is put in the Apollonian 
form involving the latus rectum, QV 2 = PV. 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 P Fas breadth and falling 
short by a rectangle similar to the rectangle contained by the 
diameter PP' and the latus rectum PL (which is determined 
by the condition PL . PP' — DO' 2 and is drawn at right angles 
to PV). Prop. 18 proves the corresponding property with 
reference to the conjugate diameter 1)1)' and the correspond 
ing latus rectum, 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).