PROPOSITION 13.
If a cylinder be cut by a plane which is parallel to its opposite planes, then, as the cylinder is to the cylinder, so will the axis be to the axis.
For let the cylinder
AD be cut by the plane
GH which is parallel to the opposite planes
AB,
CD, and let the plane
GH meet the axis at the point
K; I say that, as the cylinder
BG is to the cylinder
GD, so is the axis
EK to the axis
KF.
For let the axis
EF be produced in both directions to the points
L,
M, and let there be set out any number whatever of axes
EN,
NL equal to the axis
EK, and any number whatever
FO,
OM equal to
FK; and let the cylinder
PW on the axis
LM be conceived of which the circles
PQ,
VW are the bases.
Let planes be carried through the points
N,
O parallel to
AB,
CD and to the bases of the cylinder
PW, and let them produce the circles
RS,
TU about the centres
N,
O.
Then, since the axes
LN,
NE,
EK are equal to one another, therefore the cylinders
QR,
RB,
BG are to one another as their bases. [
XII. 11]
But the bases are equal; therefore the cylinders
QR,
RB,
BG are also equal to one another.
Since then the axes
LN,
NE,
EK are equal to one another, and the cylinders
QR,
RB,
BG are also equal to one another, and the multitude of the former is equal to the multitude of the latter, therefore, whatever multiple the axis
KL is of the axis
EK, the same multiple also will the cylinder
QG be of the cylinder
GB.
For the same reason, whatever multiple the axis
MK is of the axis
KF, the same multiple also is the cylinder
WG of the cylinder
GD.
And, if the axis
KL is equal to the axis
KM, the cylinder
QG will also be equal to the cylinder
GW, if the axis is greater than the axis, the cylinder will also be greater than the cylinder, and if less, less.
Thus, there being four magnitudes, the axes
EK,
KF and the cylinders
BG,
GD, there have been taken equimultiples of the axis
EK and of the cylinder
BG, namely the axis
LK and the cylinder
QG, and equimultiples of the axis
KF and of the cylinder
GD, namely the axis
KM and the cylinder
GW; and it has been proved that, if the axis
KL is in excess of the axis
KM, the cylinder
QG is also in excess of the cylinder
GW, if equal, equal, and if less, less.
Therefore, as the axis
EK is to the axis
KF, so is the cylinder
BG to the cylinder
GD. [
V. Def. 5] Q. E. D.