Proposition: 5.22: Equality of Ratios Ex Aequali

(Proposition 22 from Book 5 of Euclid's “Elements”)

If there are any number of magnitudes whatsoever, and (some) other (magnitudes) of equal number to them, (which are) also in the same ratio taken two by two, then they will also be in the same ratio via equality.


Modern Formulation

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

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

Proofs: 1

Definitions: 1
Proofs: 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
Sections: 19

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