---
book: 12
number: 14
id: "XII.14"
kind: "theorem"
uses: ["[[book-12/proposition-11]]", "[[book-12/proposition-13]]", "[[book-12/proposition-10]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:12.prop.14"
license: "CC-BY-SA-4.0"
---

# XII.14

*Cones and cylinders which are on equal bases are to one another as their heights.*

## Proof

For let *EB*, *FD* be cylinders on equal bases, the circles *AB*, *CD*; I say that, as the cylinder *EB* is to the cylinder *FD*, so is the axis *GH* to the axis *KL*.

For let the axis *KL* be produced to the point *N*, let *LN* be made equal to the axis *GH*, and let the cylinder *CM* be conceived about *LN* as axis.

Since then the cylinders *EB*, *CM* are of the same height, they are to one another as their bases. [[book-12/proposition-11|XII. 11]]

But the bases are equal to one another; therefore the cylinders *EB*, *CM* are also equal.

And, since the cylinder *FM* has been cut by the plane *CD* which is parallel to its opposite planes, therefore, as the cylinder *CM* is to the cylinder *FD*, so is the axis *LN* to the axis *KL*. [[book-12/proposition-13|XII. 13]]

But the cylinder *CM* is equal to the cylinder *EB*, and the axis *LN* to the axis *GH*; therefore, as the cylinder *EB* is to the cylinder *FD*, so is the axis *GH* to the axis *KL*.

But, as the cylinder *EB* is to the cylinder *FD*, so is the cone *ABG* to the cone *CDK*. [[book-12/proposition-10|XII. 10]]

Therefore also, as the axis *GH* is to the axis *KL*, so is the cone *ABG* to the cone *CDK* and the cylinder *EB* to the cylinder *FD*. Q. E. D.