---
book: 7
number: 13
id: "VII.13"
kind: "theorem"
uses: ["[[book-7/proposition-10]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:7.prop.13"
license: "CC-BY-SA-4.0"
---

# VII.13

*If four numbers be proportional, they will also be proportional alternately.*

## Proof

Let the four numbers *A*, *B*, *C*, *D* be proportional, so that, as *A* is to *B*, so is *C* to *D*; I say that they will also be proportional alternately, so that, as *A* is to *C*, so will *B* be to *D*.

For since, as *A* is to *B*, so is *C* to *D*, therefore, whatever part or parts *A* is of *B*, the same part or the same parts is *C* of *D* also. [[book-7/definitions#Definition 20|VII. Def. 20]]

Therefore, alternately, whatever part or parts *A* is of *C*, the same part or the same parts is *B* of *D* also. [[book-7/proposition-10|VII. 10]]

Therefore, as *A* is to *C*, so is *B* to *D*. [[book-7/definitions#Definition 20|VII. Def. 20]] Q. E. D.