Proposition: Prop. 8.18: Between two Similar Plane Numbers exists one Mean Proportional

(Proposition 18 from Book 8 of Euclid's “Elements”)

There exists one number in mean proportion to two similar plane numbers. And (one) plane (number) has to the (other) plane (number) a squared¹ ratio with respect to (that) a corresponding side (of the former has) to a corresponding side (of the latter).

Let $A$ and $B$ be two similar plane numbers.
And let the numbers $C$, $D$ be the sides of $A$, and $E$, $F$ (the sides) of $B$.
And since similar numbers are those having proportional sides [Def. 7.21] , thus as $C$ is to $D$, so $E$ (is) to $F$.
Therefore, I say that there exists one number in mean proportion to $A$ and $B$, and that $A$ has to $B$ a squared ratio with respect to that $C$ (has) to $E$, or $D$ to $F$ - that is to say, with respect to (that) a corresponding side (has) to a corresponding [side].