(related to Proposition: 5.19: Proportional Magnitudes have Proportional Remainders)
And since it was shown (that) as $AB$ (is) to $CD$, so $EB$ (is) to $FD$, (it is) also (the case), alternately, (that) as $AB$ (is) to $BE$, so $CD$ (is) to $FD$. * Thus, composed magnitudes are proportional. * And it was shown (that) as $BA$ (is) to $AE$, so $DC$ (is) to $CF$. * And (the latter) is converted (from the former).
In modern notation, this corollary reads that if \[\frac\alpha\beta=\frac\gamma\delta,\] then \[\frac\alpha{\alpha-\beta}=\frac\gamma{\gamma-\delta,}\] for all positive real numbers \(\alpha,\beta,\gamma,\delta\) with \(\alpha > \beta\) and \(\gamma > \delta\).
Proofs: 1