---
book: 12
number: 13
id: "XII.13"
kind: "theorem"
uses: ["[[book-12/proposition-11]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:12.prop.13"
license: "CC-BY-SA-4.0"
---

# XII.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.*

## Proof

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. [[book-12/proposition-11|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*. [[book-5/definitions#Definition 5|V. Def. 5]] Q. E. D.