---
book: 7
number: 6
id: "VII.6"
kind: "theorem"
uses: ["[[book-7/proposition-5]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:7.prop.6"
license: "CC-BY-SA-4.0"
---

# VII.6

*If a number be parts of a number, and another be the same parts of another, the sum will also be the same parts of the sum that the one is of the one.*

## Proof

For let the number *AB* be parts of the number *C*, and another, *DE*, the same parts of another, *F*, that *AB* is of *C*; I say that the sum of *AB*, *DE* is also the same parts of the sum of *C*, *F* that *AB* is of *C*.

For since, whatever parts *AB* is of *C*, *DE* is also the same parts of *F*, therefore, as many parts of *C* as there are in *AB*, so many parts of *F* are there also in *DE*.

Let *AB* be divided into the parts of *C*, namely *AG*, *GB*, and *DE* into the parts of *F*, namely *DH*, *HE*; thus the multitude of *AG*, *GB* will be equal to the multitude of *DH*, *HE*.

And since, whatever part *AG* is of *C*, the same part is *DH* of *F* also, therefore, whatever part *AG* is of *C*, the same part also is the sum of *AG*, *DH* of the sum of *C*, *F*. [[book-7/proposition-5|VII. 5]]

For the same reason, whatever part *GB* is of *C*, the same part also is the sum of *GB*, *HE* of the sum of *C*, *F*.

Therefore, whatever parts *AB* is of *C*, the same parts also is the sum of *AB*, *DE* of the sum of *C*, *F*. Q. E. D.