# Between two Squares exists one Mean Proportional

## Theorem

In the words of Euclid:

Between two square numbers there is one mean proportional number, and the square has to the square the ratio duplicate of that which the side has to the side.

## Proof

Let $a^2$ and $b^2$ be square numbers.

Consider the number $a b$.

We have:

$\dfrac {a^2} {a b} = \dfrac a b = \dfrac {a b} {b^2}$

By definition, it follows that $a b$ is the mean proportional between $a^2$ and $b^2$.

Then:

$\paren {\dfrac a b}^2 = \dfrac {a^2} {b^2}$

By definition, it follows that $a^2$ has to $b^2$ the duplicate ratio that $a$ has to $b$.

$\blacksquare$

## Historical Note

This proof is Proposition $11$ of Book $\text{VIII}$ of Euclid's The Elements.