Proposition: 5.07: Ratios of Equal Magnitudes

Euclid's Formulation

Equal (magnitudes) have the same ratio to the same (magnitude), and the latter (magnitude has the same ratio) to the equal (magnitudes).


Modern Formulation

In modern notation, this proposition reads that if \(\alpha=\beta\) and \[\frac\alpha\gamma=\frac\beta\gamma\] then \[\frac\gamma\alpha=\frac\gamma\beta\]

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

Generalized Formulation

The above proposition is even true for all real numbers with \(\alpha\neq 0, \beta\neq 0, \gamma\neq 0\). Algebraically, it follows from the existence and uniqueness of inverse real numbers with respect to multiplication.

Proofs: 1 Corollaries: 1

Proofs: 1 2 3 4 5 6 7 8 9 10
Sections: 11

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