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.


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)

Proofs: 1

Proofs: 1 2 3 4 5
Sections: 6

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