Between two Cubes exist two Mean Proportionals

Proof
Let $a^3$ and $b^3$ be cube numbers.

From the corollary to Form of Geometric Progression of Integers:
 * $\left({a^3, a^2 b, a b^2, b^3}\right)$

is a geometric progression.

It follows by definition that $a^2 b$ and $a b^2$ are mean proportionals of $a^3$ and $b^3$.

Then:
 * $\left({\dfrac a b}\right)^3 = \dfrac {a^3} {b^3}$

By definition, it follows that $a^3$ has to $b^3$ the triplicate ratio that $a$ has to $b$.