Proposition: 5.18: Magnitudes Proportional Separated are Proportional Compounded
(Proposition 18 from Book 5 of Euclid's “Elements”)
If separated magnitudes are proportional then they will also be proportional (when) composed.
 Let $AE$, $EB$, $CF$, and $FD$ be separated magnitudes (which are) proportional, (so that) as $AE$ (is) to $EB$, so $CF$ (is) to $FD$.
 I say that they will also be proportional (when) composed, (so that) as $AB$ (is) to $BE$, so $CD$ (is) to $FD$.
Modern Formulation
In modern notation, this proposition reads that if \[\frac\alpha\beta=\frac\gamma\delta,\] then \[\frac{\alpha+\beta}\beta=\frac{\gamma+\delta}\delta,\]
for all positive real numbers \(\alpha,\beta,\gamma,\delta\).
Table of Contents
Proofs: 1
Mentioned in:
Proofs: 1 2 3 4 5 6 7
Sections: 8
Thank you to the contributors under CC BYSA 4.0!
 Github:

 nonGithub:
 @Fitzpatrick
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