Numbers between which exist two Mean Proportionals are Similar Solid

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