# If Ratio of Square to Number is as between Two Squares then Number is Square

Jump to navigation
Jump to search

## Theorem

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

- $\dfrac a b = \dfrac {c^2} {d^2}$

Let $a$ be a square number.

Then $b$ is also a square number.

In the words of Euclid:

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

(*The Elements*: Book $\text{VIII}$: Proposition $24$)

## Proof

- $\tuple {c^2, c d, d^2}$

is a geometric sequence.

- $\tuple {a, m, b}$

is a geometric sequence for some $m$.

We have that $a$ is a square number.

- $b$ is a square number.

$\blacksquare$

## Historical Note

This proof is Proposition $24$ 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