---
book: 5
number: 22
id: "V.22"
kind: "theorem"
uses: ["[[book-5/proposition-4]]", "[[book-5/proposition-20]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:5.prop.22"
license: "CC-BY-SA-4.0"
---

# V.22

*If there be any number of magnitudes whatever*, *and others equal to them in multitude*, *which taken two and two together are in the same ratio*, *they will also be in the same ratio* ex aequali.

## Proof

Let there be any number of magnitudes *A*, *B*, *C*, and others *D*, *E*, *F* equal to them in multitude, which taken two and two together are in the same ratio, so that, as *A* is to *B*, so is *D* to *E*, and, as *B* is to *C*, so is *E* to *F*; I say that they will also be in the same ratio *ex aequali*, ltthat is, as *A* is to *C*, so is *D* to *F*gt.

For of *A*, *D* let equimultiples *G*, *H* be taken, and of *B*, *E* other, chance, equimultiples *K*, *L*; and, further, of *C*, *F* other, chance, equimultiples *M*, *N*.

Then, since, as *A* is to *B*, so is *D* to *E*, and of *A*, *D* equimultiples *G*, *H* have been taken, and of *B*, *E* other, chance, equimultiples *K*, *L*, therefore, as *G* is to *K*, so is *H* to *L*. [[book-5/proposition-4|V. 4]]

For the same reason also, as *K* is to *M*, so is *L* to *N*.

Since, then, there are three magnitudes *G*, *K*, *M*, and others *H*, *L*, *N* equal to them in multitude, which taken two and two together are in the same ratio, therefore, ex aequali, if *G* is in excess of *M*, *H* is also in excess of *N*; if equal, equal; and if less, less. [[book-5/proposition-20|V. 20]]

And *G*, *H* are equimultiples of *A*, *D*, and *M*, *N* other, chance, equimultiples of *C*, *F*.

Therefore, as *A* is to *C*, so is *D* to *F*. [[book-5/definitions#Definition 5|V. Def. 5]]

Therefore etc. Q. E. D.