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 $B$.
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 \(\alpha,\beta,\gamma\).
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 2 3 4 5 6 7 8 9 10