Between two Squares exists one Mean Proportional

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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.

(The Elements: Book $\text{VIII}$: Proposition $11$)


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


Sources