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). * Let $A$ and $B$ be equal magnitudes, and $C$ some other random magnitude. * I say that $A$ and $B$ each have the same ratio to $C$, and (that) $C$ (has the same ratio) to each of $A$ and $B$.

fig07e

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


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References

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", https://proofwiki.org/wiki/Main_Page, 2016