(related to Proposition: Prop. 8.24: If Ratio of Square to Number is as between Two Squares then Number is Square)

- For let two numbers, $A$ and $B$, have to one another the ratio which the square number $C$ (has) to the square number $D$.
- And let $A$ be square.
- I say that $B$ is also square.

- For since $C$ and $D$ are square, $C$ and $D$ are thus similar plane (numbers) .
- Thus, one number falls (between) $C$ and $D$ in mean proportion [Prop. 8.18].
- And as $C$ is to $D$, (so) $A$ (is) to $B$.
- Thus, one number also falls (between) $A$ and $B$ in mean proportion [Prop. 8.8].
- And $A$ is square.
- Thus, $B$ is also square [Prop. 8.22].
- (Which is) the very thing it was required to show.∎

**Fitzpatrick, Richard**: Euclid's "Elements of Geometry"