If a first magnitude have to a second the same ratio as a third has to a fourth, and also a fifth have to the second the same ratio as a sixth to the fourth, the first and fifth added together will have to the second the same ratio as the third and sixth have to the fourth.
Let a first magnitude AB have to a second C the same ratio as a third DE has
to a fourth F;
and let also a fifth BG have to the second C the same ratio as a sixth EH has
to the fourth F;
I say that the first and fifth added together, AG, will have to the second C
the same ratio as the third and sixth, DH, has to the fourth F.
For since, as BG is to C, so is EH to F, inversely, as C is to BG, so is F to
EH.
Since, then, as AB is to C, so is DE to F, and, as C is to BG, so is F to EH,
therefore, ex aequali, as AB is to BG, so is DE to EH.
And, since the magnitudes are proportional separando, they will also be
proportional componendo; therefore, as AG is to GB, so is DH to HE.
But also, as BG is to C, so is EH to F; therefore, ex aequali, as AG is to C,
so is DH to F.
Therefore etc.
Q.E.D