If composed magnitudes are proportional then they will also be proportional (when) separated. * Let $AB$, $BE$, $CD$, and $DF$ be composed magnitudes (which are) proportional, (so that) as $AB$ (is) to $BE$, so $CD$ (is) to $DF$. * I say that they will also be proportional (when) separated, (so that) as $AE$ (is) to $EB$, so $CF$ (is) to $DF$.
In modern notation, this proposition reads that if \[\frac{\alpha+\beta}\beta=\frac{\gamma+\delta}\delta,\] then \[\frac\alpha\beta=\frac\gamma\delta,\]
for all positive real numbers \(\alpha,\beta,\gamma,\delta\).
Proofs: 1