---
book: 5
number: 2
id: "V.2"
kind: "theorem"
uses: []
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:5.prop.2"
license: "CC-BY-SA-4.0"
---

# V.2

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

## Proof

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*, also be the same multiple of the second, *C*, that a sixth, *EH*, is of the fourth *F*; I say that the sum of the first and fifth, *AG*, will be the same multiple of the second, *C*, that the sum of the third and sixth, *DH*, is of the fourth, *F*.

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

For the same reason also, as many as there are in *BG* equal to *C*, so many are there also in *EH* equal to *F*; therefore, as many as there are in the whole *AG* equal to *C*, so many also are there in the whole *DH* equal to *F*.

Therefore, whatever multiple *AG* is of *C*, that multiple also is *DH* of *F*.

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

Therefore etc. Q.E.D.