If Ratio of Cube to Number is as between Two Cubes then Number is Cube

Theorem

Let $a, b, c, d \in \Z$ be integers such that:

$\dfrac a b = \dfrac {c^3} {d^3}$

Let $a$ be a cube number.

Then $b$ is also a cube number.

In the words of Euclid:

If two numbers have to one another the ratio which a cube number has to a cube number, and the first be cube, the second will also be cube.

(The Elements: Book $\text{VIII}$: Proposition $25$)

Proof

From Proposition $19$ of Book $\text{VIII} $: Between two Similar Solid Numbers exist two Mean Proportionals:

$\left({c^3, c^2 d, c d^2, d^3}\right)$

is a geometric sequence.

From Proposition $8$ of Book $\text{VIII} $: Geometric Sequences in Proportion have Same Number of Elements:

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

is a geometric sequence for some $m$.

We have that $a$ is a cube number.

From Proposition $23$ of Book $\text{VIII} $: If First of Four Numbers in Geometric Sequence is Cube then Fourth is Cube:

$b$ is a cube number.

$\blacksquare$

Historical Note

This proof is Proposition $25$ 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