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Proposition: 5.11: Equality of Ratios is Transitive

(Proposition 11 from Book 5 of Euclid's “Elements”)

(Ratios which are) the same with the same ratio are also the same with one another.

• For let it be that as $A$ (is) to $B$, so $C$ (is) to $D$, and as $C$ (is) to $D$, so $E$ (is) to $F$.
• I say that as $A$ is to $B$, so $E$ (is) to $F$.

Modern Formulation

In modern notation, this proposition reads that if \[\frac\alpha\beta=\frac\gamma\delta\text{ and }\frac\gamma\delta=\frac\epsilon\zeta\] then \[\frac\alpha\beta=\frac\epsilon\zeta\]

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

Generalized Formulation

see any equality is an equivalence relation (and in particular transitive)

see also common notion 1.1. The above proposition is true for all real numbers with \(\beta\neq 0, \delta\neq 0, \zeta\neq 0\).

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 27 28 29 30 31 32 33 34
Sections: 35

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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