(Magnitudes) having the same ratio to the same (magnitude) are equal to one another. And those (magnitudes) to which the same (magnitude) has the same ratio are equal. * For let $A$ and $B$ each have the same ratio to $C$. * I say that $A$ is equal to $B$. * So, again, let $C$ have the same ratio to each of $A$ and $B$. * I say that $A$ is equal to $B$.
In modern notation, this proposition reads that if \(\alpha =\beta\) then \[\frac\alpha\gamma = \frac\beta\gamma\] and if \[\frac\gamma\alpha = \frac\gamma\beta\] then \(\alpha =\beta\) for all positive real numbers \(\alpha,\beta,\gamma\).
The above proposition is true for all real numbers with \(\alpha\neq 0, \beta\neq 0, \gamma\neq 0\), since 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 11 12 13 14 15 16