# Numbers between which exist two Mean Proportionals are Similar Solid

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## Theorem

Let $a, b \in \Z$ be the extremes of a geometric sequence of integers whose length is $4$:

- $\tuple {a, m_1, m_2, b}$

That is, such that $a$ and $b$ have $2$ mean proportionals.

Then $a$ and $b$ are similar solid numbers.

In the words of Euclid:

*If two mean proportional numbers fall between two numbers, the numbers will be similar solid numbers.*

(*The Elements*: Book $\text{VIII}$: Proposition $21$)

## Proof

From Form of Geometric Sequence of Integers:

- $\exists k, p, q \in \Z: a = k p^3, b = k q^3$

So $a$ and $b$ are solid numbers whose sides are:

- $k p$, $p$ and $p$

and

- $k q$, $q$ and $q$

respectively.

Then:

- $\dfrac {k p} {k q} = \dfrac p q$

demonstrating that $a$ and $b$ are similar solid numbers by definition.

$\blacksquare$

## Historical Note

This proof is Proposition $21$ 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