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

Let *EB* and *FD* be cylinders on equal bases, the circles *AB* and *CD.*

I say that the cylinder *EB* is to the cylinder *FD* as the axis *GH* is to the axis *KL.*

Produce the axis *KL* to the point *N,* make *LN* equal to the axis *GH,* and construct the cylinder *CM* about *LN* as the axis.

Then, since the cylinders *EB* and *CM* are of the same height, therefore they are to one another as their bases.

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

And, since the cylinder *FM* has been cut by the plane *CD* parallel to its opposite planes, therefore the cylinder *CM* is to the cylinder *FD* as the axis *LN* is to the axis *KL.*

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

But the cylinder *EB* is to the cylinder *FD* as the cone *ABG* is to the cone *CDK.* Therefore the axis *GH* is to the axis *KL* as the cone *ABG* is to the cone *CDK* and as the cylinder *EB* is to the cylinder *FD.*

Therefore, *cones and cylinders on equal bases are to one another as their heights.*

Q.E.D.