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 \(\alpha,\beta,\gamma,\delta\).

Proofs: 1

Proofs: 1 2 3 4 5 6 7
Sections: 8

Thank you to the contributors under CC BY-SA 4.0!



Adapted from (subject to copyright, with kind permission)

  1. Fitzpatrick, Richard: Euclid's "Elements of Geometry"

Adapted from CC BY-SA 3.0 Sources:

  1. Prime.mover and others: "Pr∞fWiki",, 2016