V.24

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.

Proof

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. V. 22

And, since the magnitudes are proportional separando, they will also be proportional componendo; V. 18 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. V. 22

Therefore etc. Q. E. D.