Proposition: 5.09: Magnitudes with Same Ratios are Equal

(Proposition 9 from Book 5 of Euclid's “Elements”)

(Magnitudes) having the same ratio to the same (magnitude) are equal to one another. And those (magnitudes) to which the same (magnitude) has the same ratio are equal.



Modern Formulation

In modern notation, this proposition reads that if \(\alpha =\beta\) then \[\frac\alpha\gamma = \frac\beta\gamma\] and if \[\frac\gamma\alpha = \frac\gamma\beta\] then \(\alpha =\beta\) for all positive real numbers \(\alpha,\beta,\gamma\).

Generalized Formulation

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

Proofs: 1

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

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