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 $E$.

fig25e

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

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