Proposition: 5.13: Relative Sizes of Proportional Magnitudes

(Proposition 13 from Book 5 of Euclid's “Elements”)

If a first (magnitude) has the same ratio to a second that a third (has) to a fourth, and the third (magnitude) has a greater ratio to the fourth than a fifth (has) to a sixth, then the first (magnitude) will also have a greater ratio to the second than the fifth (has) to the sixth.


Modern Formulation

In modern notation, this proposition reads that if \[\frac\alpha\beta=\frac\gamma\delta\text{ and }\frac\gamma\delta > \frac\epsilon\zeta\] then \[\frac\alpha\beta > \frac\epsilon\zeta\]

for all positive real numbers \(\alpha,\beta,\gamma,\delta,\epsilon,\zeta\).

Generalized Modern Formulation

see rules of calculation with inequalities (Rule 11)

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