In equal cones and cylinders the bases are reciprocally proportional to the heights; and those cones and cylinders in which the bases are reciprocally proportional to the heights are equal.

Let there be equal cones and cylinders with the circular bases *ABCD* and *EFGH.* Let *AC* and *EG* be the diameters of the bases, and *KL* and *MN* the axes, which are also the heights of the cones or cylinders

Complete the cylinders *AO* and *EP.*

I say that in the cylinders *AO* and *EP* the bases are reciprocally proportional to the heights, that is, the base *ABCD* is to the base *EFGH* as the height *MN* is to the height *KL.*

For the height *LK* is either equal to the height *MN* or unequal.

First, let it be equal.

Now the cylinder *AO* also equals the cylinder *EP.* But cones and cylinders of the same height are to one another as their bases, therefore the base *ABCD* equals the base *EFGH.*

Hence, reciprocally, the base *ABCD* is to the base *EFGH* as the height *MN* is to the height *KL.*

Next, let the height *LK* be unequal to *MN,* and let *MN* be greater.

Cut *QN* off the height *MN* equal to *KL.* Through the point *Q* let the cylinder *EP* be cut by the plane *TUS* parallel to the planes of the circles *EFGH* and *RP.* Erect the cylinder *ES* from the circle *EFGH* as base and with height *NQ.*

Now, since the cylinder *AO* equals the cylinder *EP,* therefore the cylinder *AO* is to the cylinder *ES* as the cylinder *EP* is to the cylinder *ES.*

But the cylinder *AO* is to the cylinder *ES* as the base *ABCD* is to the base *EFGH,* for the cylinders *AO* and *ES* are of the same height. And the cylinder *EP* is to the cylinder *ES* as the height *MN* is to the height *QN,* for the cylinder *EP* is cut by a plane parallel to its opposite planes. Therefore the base *ABCD* is to the base *EFGH* as the height *MN* is to the height *QN.*

But the height *QN* equals the height *KL,* therefore the base *ABCD* is to the base *EFGH* as the height *MN* is to the height *KL.*

Therefore in the cylinders *AO* and *EP* the bases are reciprocally proportional to the heights.

Next, in the cylinders *AO* and *EP* let the bases be reciprocally proportional to the heights, that is, as the base *ABCD* is to the base *EFGH,* so let the height *MN* be to the height *KL.*

I say that the cylinder *AO* equals the cylinder *EP.*

With the same construction, since the base *ABCD* is to the base *EFGH* as the height *MN* is to the height *KL,* and the height *KL* equals the height *QN,* therefore the base *ABCD* is to the base *EFGH* as the height *MN* is to the height *QN.*

But the base *ABCD* is to the base *EFGH* as the cylinder *AO* is to the cylinder *ES,* for they have the same height. And the height *MN* is to *QN* as the cylinder *EP* is to the cylinder *ES,* therefore the cylinder *AO* is to the cylinder *ES* as the cylinder *EP* is to the cylinder *ES.*

Therefore the cylinder *AO* equals the cylinder *EP.*

And the same is true for the cones also.

Therefore, *in equal cones and cylinders the bases are reciprocally proportional to the heights; and those cones and cylinders in which the bases are reciprocally proportional to the heights are equal.*

Q.E.D.

This proposition completes the theory of the volumes of cones and cylinders. The remaining three propositions in this book concern the volume of spheres.