# Between two Cubes exist two Mean Proportionals

## Theorem

In the words of Euclid:

Between two cube numbers there are two mean proportional numbers, and the cube has to the cube the ratio triplicate of that which the side has to the side.

## Proof

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

$\left({a^3, a^2 b, a b^2, b^3}\right)$

is a geometric sequence.

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 $b$.

$\blacksquare$

## Historical Note

This proof is Proposition $12$ of Book $\text{VIII}$ of Euclid's The Elements.