Proposition: 5.24: Sum of Antecedents of Proportion

(Proposition 24 from Book 5 of Euclid's “Elements”)

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


Modern Formulation

In modern notation, this proposition reads that if for positive real numbers \(\alpha,\beta,\gamma,\delta\) \[\frac\alpha\beta=\frac\gamma\delta\text{ and }\frac\epsilon\beta=\frac\zeta\delta,\] then \[\alpha+\frac\epsilon\beta=\gamma+\frac\zeta\delta.\]

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