# Numbers whose Product is Square are Similar Plane Numbers

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## Theorem

In the words of Euclid:

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

(*The Elements*: Book $\text{IX}$: Proposition $2$)

## Proof

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

From Proposition $17$ of Book $\text{VII} $: Multiples of Ratios of Numbers:

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

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

By Square Numbers are Similar Plane Numbers they are similar plane numbers.

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

- there exists one mean proportional between $a$ and $b$.

The result follows from Proposition $20$ of Book $\text{VIII} $: Numbers between which exists one Mean Proportional are Similar Plane.

$\blacksquare$

## 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.

## Sources

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