Between two Cubes exist two Mean Proportionals

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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.

(The Elements: Book $\text{VIII}$: Proposition $12$)


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

From the corollary to Form of Geometric Sequence of Integers:

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

is a geometric sequence.

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


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

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


Historical Note

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