Proof: By Euclid
(related to Proposition: Prop. 12.14: Volumes of Cones or Cylinders on Equal Bases are in Same Ratio as Heights)
 For let $EB$ and $FD$ be cylinders on equal bases, (namely) the circles $AB$ and $CD$ (respectively).
 I say that as cylinder $EB$ is to cylinder $FD$, so $GH$ (is) to $KL$.
 For let the $KL$ have been produced to point $N$.
 And let $LN$ be made equal to $GH$.
 And let the cylinder $CM$ have been conceived about $LN$.
 Therefore, since cylinders $EB$ and $CM$ have the same height they are to one another as their bases [Prop. 12.11].
 And the bases are equal to one another.
 Thus, cylinders $EB$ and $CM$ are also equal to one another.
 And since cylinder $FM$ has been cut by the plane $CD$, which is parallel to its opposite planes, thus as cylinder $CM$ is to cylinder $FD$, so $LN$ (is) to $KL$ [Prop. 12.13].
 And cylinder $CM$ is equal to cylinder $EB$, and $LN$ to $GH$.
 Thus, as cylinder $EB$ is to cylinder $FD$, so $GH$ (is) to $KL$.
 And as cylinder $EB$ (is) to cylinder $FD$, so cone $ABG$ (is) to cone $CDK$ [Prop. 12.10].
 Thus, also, as $GH$ (is) to $KL$, so cone $ABG$ (is) to cone $CDK$, and cylinder $EB$ to cylinder $FD$.
 (Which is) the very thing it was required to show.
∎
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References
Adapted from (subject to copyright, with kind permission)
 Fitzpatrick, Richard: Euclid's "Elements of Geometry"