Volumes of Similar Cones and Cylinders are in Triplicate Ratio of Diameters of Bases

Proof

 * Euclid-XII-12.png

Let there be cones and cylinders which are solid.

Let the circles $c \left({ABCD}\right)$ and $c \left({EFGH}\right)$ be their bases.

Let $BD$ and $FH$ be the diameters of the bases $c \left({ABCD}\right)$ and $c \left({EFGH}\right)$.

Let $KL$ and $MN$ be the axes of the cones and cylinders.

Let $L$ and $N$ be the apices of the cones.

It is to be demonstrated that:
 * $c \left({ABCDL}\right) : c \left({EFGHN}\right) = BD^3 : FD^3$

where $c \left({ABCDL}\right)$ and $c \left({EFGHN}\right)$ are the cones on $c \left({ABCD}\right)$ and $c \left({EFGH}\right)$ respectively.