Proposition: 5.21: Relative Sizes of Elements in Perturbed Proportion

Euclid's Formulation

If there are three magnitudes, and others of equal number to them, (being) also in the same ratio taken two by two, and (if) their proportion (is) perturbed, and (if), via equality, the first is greater than the third then the fourth will also be greater than the sixth. And if (the first is) equal (to the third then the fourth will also be) equal (to the sixth). And if (the first is) less (than the third then the fourth will also be) less (than the sixth).


Modern Formulation

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

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

Proofs: 1

Proofs: 1
Sections: 2

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