Corollary: 5.07: Ratios of Equal Magnitudes

(related to Proposition: 5.07: Ratios of Equal Magnitudes)

(Corollary to Proposition 7 from Book 5 of Euclid's “Elements”)

So (it is) clear, from this, that if some magnitudes are proportional then they will also be proportional inversely. (Which is) the very thing it was required to show.

Modern Formulation

In modern notation, this corollary reads that if \[\frac\alpha\beta=\frac\gamma\delta\] then \[\frac\beta\alpha=\frac\delta\gamma\]

for all positive rational number \(\alpha,\beta,\gamma,\delta\).

Generalized Formulation

The above corollary even for all real numbers with \(\alpha\neq0, \beta\neq 0, \gamma\neq 0,\delta\neq 0\). Algebraically, 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
Sections: 8

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Adapted from (subject to copyright, with kind permission)

  1. Fitzpatrick, Richard: Euclid's "Elements of Geometry"