(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 $F$.
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 \(\alpha,\beta,\gamma,\delta,\epsilon,\zeta\).
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