---
book: 7
number: 10
id: "VII.10"
kind: "theorem"
uses: ["[[book-7/proposition-9]]", "[[book-7/proposition-5]]", "[[book-7/proposition-6]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:7.prop.10"
license: "CC-BY-SA-4.0"
---

# VII.10

*If a number be parts of a number, and another be the same parts of another, alternately also, whatever parts or part the first is of the third, the same parts or the same part will the second also be of the fourth.*

## Proof

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

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

Now since, whatever part *AG* is of *C*, the same part also is *DH* of *F*, alternately also, whatever part or parts *AG* is of *DH*, the same part or the same parts is *C* of *F* also. [[book-7/proposition-9|VII. 9]]

For the same reason also, whatever part or parts *GB* is of *HE*, the same part or the same parts is *C* of *F* also; so that, in addition, whatever parts or part *AB* is of *DE*, the same parts also, or the same part, is *C* of *F*. [[book-7/proposition-5|VII. 5, 6]] Q. E. D.