[Folio 0189 | p. 183]

PROPOSITION 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.

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.

[Folio 0190 | p. 184]

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,

[V. 22]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; [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;

[V. 22]therefore, ex aequali, as AG is to C, so is DH to F.

Therefore etc. Q. E. D.

Dependency Graph

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See Commentary on Proposition V.24.