If a first magnitude is the same multiple of a second that a third is of a fourth, and a fifth also is the same multiple of the second that a sixth is of the fourth, then the sum of the first and fifth also is the same multiple of the second that the sum of the third and sixth is of the fourth.

Let a first magnitude *AB* be the same multiple of a second *C* that a third *DE* is of a fourth *F,* and let a fifth *BG* be the same multiple of the second *C* that a sixth *EH* is of the fourth *F.*

I say that the sum *AG* of the first and fifth is the same multiple of the second, *C,* that the sum *DH* of the third and sixth is of the fourth, *F.*

Since *AB* is the same multiple of *C* that *DE* is of *F,* therefore there are as many magnitudes in *AB* equal to *C* as there are in *DE* equal to *F.*

For the same reason there are as many magnitudes in *BG* equal to *C* as there are in *EH* equal to *F.* Therefore, there are as many magnitudes in the whole *AG* equal to *C* as there are in the whole *DH* equal to *F.*

Therefore, *AG* is the same multiple of *C* that *DH* is of *F.*

Therefore the sum *AG* of the first and fifth is the same multiple of the second, *C,* that the sum *DH* of the third and sixth is of the fourth, *F.*

Therefore, *if a first magnitude is the same multiple of a second that a third is of a fourth, and a fifth also is the same multiple of the second that a sixth is of the fourth, then the sum of the first and fifth also is the same multiple of the second that the sum of the third and sixth is of the fourth.*

Q.E.D.

Note that the magnitudes don’t all have to be of the same kind. Different colors are used in the figures here to indicate different kinds of magnitudes.