# 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 $$\beta\neq 0, \delta\neq 0, \zeta\neq 0$$.

Proofs: 1

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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