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

# VII.9

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

## Proof

For let the number *A* be a part of the number *BC*, and another, *D*, the same part of another, *EF*, that *A* is of *BC*; I say that, alternately also, whatever part or parts *A* is of *D*, the same part or parts is *BC* of *EF* also.

For since, whatever part *A* is of *BC*, the same part also is *D* of *EF*, therefore, as many numbers as there are in *BC* equal to *A*, so many also are there in *EF* equal to *D*.

Let *BC* be divided into the numbers equal to *A*, namely *BG*, *GC*, and *EF* into those equal to *D*, namely *EH*, *HF*; thus the multitude of *BG*, *GC* will be equal to the multitude of *EH*, *HF*.

Now, since the numbers *BG*, *GC* are equal to one another, and the numbers *EH*, *HF* are also equal to one another, while the multitude of *BG*, *GC* is equal to the multitude of *EH*, *HF*, therefore, whatever part or parts *BG* is of *EH*, the same part or the same parts is *GC* of *HF* also; so that, in addition, whatever part or parts *BG* is of *EH*, the same part also, or the same parts, is the sum *BC* of the sum *EF*. [[book-7/proposition-5|VII. 5, 6]]

But *BG* is equal to *A*, and *EH* to *D*; therefore, whatever part or parts *A* is of *D*, the same part or the same parts is *BC* of *EF* also. Q. E. D.