Proposition: Prop. 10.009: Commensurability of Squares

(Proposition 9 from Book 10 of Euclid's “Elements”)

Squares on straight lines (which are) commensurable in length have to one another the ratio which (some) square number (has) to (some) square number. And squares having to one another the ratio which (some) square number (has) to (some) square number will also have sides (which are) commensurable in length. But squares on straight lines (which are) incommensurable in length do not have to one another the ratio which (some) square number (has) to (some) square number. And squares not having to one another the ratio which (some) square number (has) to (some) square number will not have sides (which are) commensurable in length either.

For let $A$ and $B$ be (straight lines which are) commensurable in length.
I say that the square on $A$ has to the square on $B$ the ratio which (some) square number (has) to (some) square number.
And so let the square on $A$ be to the (square) on $B$ as the square (number) on $C$ (is) to the [square] (number) on $D$.
I say that $A$ is commensurable in length with $B$.
And so let $A$ be incommensurable in length with $B$.
I say that the square on $A$ does not have to the [square] on $B$ the ratio which (some) square number (has) to (some) square number.
So, again, let the square on $A$ not have to the [square] on $B$ the ratio which (some) square number (has) to (some) square number.
I say that $A$ is incommensurable in length with $B$.