Proposition: 7.14: Proportion of Numbers is Transitive
(Proposition 14 from Book 7 of Euclid's “Elements”)
If there are any multitude of numbers whatsoever, and (some) other (numbers) of equal multitude to them, (which are) also in the same ratio taken two by two, then they will also be in the same ratio via equality.
 Let there be any multitude of numbers whatsoever, $A$, $B$, $C$, and (some) other (numbers), $D$, $E$, $F$, of equal multitude to them, (which are) in the same ratio taken two by two, (such that) as $A$ (is) to $B$, so $D$ (is) to $E$, and as $B$ (is) to $C$, so $E$ (is) to $F$.
 I say that also, via equality, as $A$ is to $C$, so $D$ (is) to $F$.
Modern Formulation
In modern notation, this proposition states that if \[\frac AB=\frac DE\wedge \frac BC=\frac EF,\] then \[\frac AC=\frac DF.\]
Table of Contents
Proofs: 1
Mentioned in:
Proofs: 1 2 3 4 5 6 7 8
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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