If a first (magnitude) has to a second the same ratio that third (has) to a fourth, and a fifth (magnitude) also has to the second the same ratio that a sixth (has) to the fourth, then the first (magnitude) and the fifth, added together, will also have the same ratio to the second that the third (magnitude) and sixth (added together, have) to the fourth. * For let a first (magnitude) $AB$ have the same ratio to a second $C$ that a third $DE$ (has) to a fourth $F$. * And let a fifth (magnitude) $BG$ also have the same ratio to the second $C$ that a sixth $EH$ (has) to the fourth $F$. * I say that the first (magnitude) and the fifth, added together, $AG$, will also have the same ratio to the second $C$ that the third (magnitude) and the sixth, (added together), $DH$, (has) to the fourth $F$.
In modern notation, this proposition reads that if for positive real numbers \(\alpha,\beta,\gamma,\delta\) \[\frac\alpha\beta=\frac\gamma\delta\text{ and }\frac\epsilon\beta=\frac\zeta\delta,\] then \[\alpha+\frac\epsilon\beta=\gamma+\frac\zeta\delta.\]
Proofs: 1