# Numbers whose Product is Square are Similar Plane Numbers

## Theorem

In the words of Euclid:

If two numbers by multiplying one another make a square, they are similar plane numbers.

## Proof

Let $a$ and $b$ be natural numbers such that $a b$ is square.

$a : b = a^2 : a b$

We have that $a b$ and $a^2$ are both square.

there exists one mean proportional between $a^2$ and $a b$.
there exists one mean proportional between $a$ and $b$.

$\blacksquare$

## Historical Note

This proof is Proposition $2$ of Book $\text{IX}$ of Euclid's The Elements.
It is the converse of Proposition $1$: Product of Similar Plane Numbers is Square.