Proposition: 5.16: Proportional Magnitudes are Proportional Alternately
(Proposition 16 from Book 5 of Euclid's “Elements”)
If four magnitudes are proportional then they will also be proportional alternately.
- Let $A$, $B$, $C$ and $D$ be four proportional magnitudes, (such that) as $A$ (is) to $B$, so $C$ (is) to $D$.
- I say that they will also be [proportional] alternately, (so that) as $A$ (is) to $C$, so $B$ (is) to $D$.
Modern Formulation
In modern notation, this proposition reads that if \[\frac\alpha\beta=\frac\gamma\delta,\] then \[\frac\alpha\gamma=\frac\beta\delta,\]
for all positive real numbers \(\alpha,\beta,\gamma,\delta\).
Generalized Modern Formulation
This is a special case of the fact that the multiplication of real numbers is cancellative. Since for $\frac\beta\gamma\neq 0$:
$$\frac\alpha\beta=\frac\gamma\delta\Leftrightarrow \frac \beta\gamma \frac\alpha\beta=\frac \beta\gamma \frac\gamma\delta.$$
Table of Contents
Proofs: 1
Mentioned in:
Proofs: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
Sections: 27
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References
Adapted from (subject to copyright, with kind permission)
- Fitzpatrick, Richard: Euclid's "Elements of Geometry"
Adapted from CC BY-SA 3.0 Sources:
- Prime.mover and others: "Pr∞fWiki", https://proofwiki.org/wiki/Main_Page, 2016
