Proposition: 5.10: Relative Sizes of Magnitudes on Unequal Ratios
Euclid's Formulation
For (magnitudes) having a ratio to the same (magnitude), that > For (magnitudes) having a ratio to the same (magnitude), that (magnitude which) has the greater ratio is (the) greater. And that (magnitude) to which the latter (magnitude) has a greater ratio is (the) lesser.
 For let $A$ have a greater ratio to $C$ than $B$ (has) to $C$.
 I say that $A$ is greater than $B$.
Modern Formulation
In modern notation, this proposition reads that if \[\frac\alpha\gamma > \frac\beta\gamma,\] then
\[\alpha > \beta,\]
for all positive real numbers \(\alpha,\beta,\gamma\).
Generalized Modern Formulation
see rules of calculation with inequalities (Rules 6)
Table of Contents
Proofs: 1
Mentioned in:
Proofs: 1 2 3
Sections: 4
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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