Proposition: 5.08: Relative Sizes of Ratios on Unequal Magnitudes
(Proposition 8 from Book 5 of Euclid's “Elements”)
For unequal magnitudes, the greater (magnitude) has a greater ratio than the lesser to the same (magnitude). And the latter (magnitude) has a greater ratio to the lesser (magnitude) than to the greater.
 Let $AB$ and $C$ be unequal magnitudes, and let $AB$ be the greater (of the two), and $D$ another random magnitude.
 I say that $AB$ has a greater ratio to $D$ than $C$ (has) to $D$, and (that) $D$ has a greater ratio to $C$ than (it has) to $AB$.
Modern Formulation
In modern notation, this proposition reads that if \(\alpha > \beta\) then \[\frac\alpha\gamma > \frac\beta\gamma\] and
\[\frac\gamma\beta > \frac\gamma\alpha\]
for all positive real numbers \(\alpha,\beta,\gamma\).
Generalized Modern Formulation
see rules of calculation with inequalities (Rules 6 and 11)
Table of Contents
Proofs: 1
Mentioned in:
Proofs: 1 2 3 4 5
Sections: 6
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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