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.


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)

Proofs: 1

Proofs: 1 2 3
Sections: 4

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