Similar Solid Numbers have Same Ratio as between Two Cubes

Theorem

In the words of Euclid:

Similar solid numbers have to one another the ratio which a cube number has to a cube number.

(The Elements: Book $\text{VIII}$: Proposition $27$)

Proof

Let $a$ and $b$ be similar solid numbers.

From Between two Similar Solid Numbers exist two Mean Proportionals, there exists two mean proportionals $m_1$ and $m_2$ between them.

By definition of mean proportional:

$\left({a, m_1, m_2, b}\right)$

is a geometric sequence.

From Form of Geometric Sequence of Integers:

$\exists k, p, q \in Z: a = k p^3, b = k q^3, m_1 = k p^2 q, m_2 = k p q^2$

Thus:

$\dfrac a b = \dfrac {k p^3} {k q^3} = \dfrac {p^3} {q^3}$

Hence the result.

$\blacksquare$

Historical Note

This proof is Proposition $27$ 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