Volumes of Cones or Cylinders on Equal Bases are in Same Ratio as Heights

Proof

 * Euclid-XII-14.png

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

It is to be shown that:
 * $EB : FD = GH : KL$

where $GH$ and $KL$ are the axes of $GH$ and $KL$.

Let the axis $KL$ be produced to the point $N$.

Let $LN = GH$.

Let the cylinder $CM$ be described about the axis $LN$.

We have that:
 * $EB$ and $CM$ have the same height

and
 * $EB$ and $CM$ are on equal bases.

From :
 * $EB$ and $CM$ are equal.

We have that the cylinder $FM$ has been cut by the plane $CD$ which is parallel to its bases.

Therefore by :
 * $CM : FD = LN : KL$

But:
 * $CM = EB$

and:
 * $LN = GH$

Therefore:
 * $EB : FD = GH : KL$

But from :
 * $EB : FD = ABG : CDK$

where $ABG$ and $CDK$ are cones whose bases are $AB$ and $CD$ and whose apices are $G$ and $K$.

Therefore:
 * $GH : KL = ABG : CDK$

and:
 * $GH : KL = EB : FD$