# 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.

## Proof

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

is a geometric sequence.

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

is a geometric sequence for some $m$.

We have that $a$ is a cube number.

$b$ is a cube number.

$\blacksquare$

## Historical Note

This theorem is Proposition $25$ of Book $\text{VIII}$ of Euclid's The Elements.