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

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## Contents

## 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

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

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*.

## Sources

- 1926: Sir Thomas L. Heath:
*Euclid: The Thirteen Books of The Elements: Volume 2*(2nd ed.) ... (previous) ... (next): Book $\text{VIII}$. Propositions