# Between two Cubes exist two Mean Proportionals

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## 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.*

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

## Proof

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 $b^3$.

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*.

## Sources

- 1926: Sir Thomas L. Heath:
*Euclid: The Thirteen Books of The Elements: Volume 2*(2nd ed.) ... (previous) ... (next): Book $\text{VIII}$. Propositions