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 squared^{1} 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].
Modern Formulation
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Table of Contents
Proofs: 1
Mentioned in:
Proofs: 1 2 3 4 5
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References
Adapted from (subject to copyright, with kind permission)
 Fitzpatrick, Richard: Euclid's "Elements of Geometry"
Adapted from CC BYSA 3.0 Sources:
 Prime.mover and others: "Pr∞fWiki", https://proofwiki.org/wiki/Main_Page, 2016
Footnotes