Proposition: 5.25: Sum of Antecedent and Consequent of Proportion

(Proposition 25 from Book 5 of Euclid's “Elements”)

If four magnitudes are proportional then the (sum of the) largest and the smallest [of them] is greater than the (sum of the) remaining two (magnitudes). * Let $AB$, $CD$, $E$, and $F$ be four proportional magnitudes, (such that) as $AB$ (is) to $CD$, so $E$ (is) to $F$. * And let $AB$ be the greatest of them, and $F$ the least. * I say that $AB$ and $F$ is greater than $CD$ and $E$.


Modern Formulation

In modern notation, this proposition reads that if $\alpha > \beta,\gamma >\delta > 0$ and if \[\frac{\alpha}{\beta}=\frac{\gamma}{\delta},\] then \[\alpha+\delta > \beta+\gamma.\]

Proofs: 1

Sections: 1

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