Proposition: 5.23: Equality of Ratios in Perturbed Proportion

(Proposition 23 from Book 5 of Euclid's “Elements”)

If there are three magnitudes, and others of equal number to them, (being) in the same ratio taken two by two, and (if) their proportion is perturbed, then they will also be in the same ratio via equality.


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 \[\frac\alpha\gamma=\frac\delta\zeta,\]

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

Proofs: 1

Sections: 1

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