---
book: 5
number: 4
id: "V.4"
kind: "theorem"
uses: ["[[book-5/proposition-3]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:5.prop.4"
license: "CC-BY-SA-4.0"
---

# V.4

*If a first magnitude have to a second the same ratio as a third to a fourth*, *any equimultiples whatever of the first and third will also have the same ratio to any equimultiples whatever of the second and fourth respectively*, *taken in corresponding order*.

## Proof

For let a first magnitude *A* have to a second *B* the same ratio as a third *C* to a fourth *D*; and let equimultiples *E*, *F* be taken of *A*, *C*, and *G*, *H* other, chance, equimultiples of *B*, *D*; I say that, as *E* is to *G*, so is *F* to *H*.

For let equimultiples *K*, *L* be taken of *E*, *F*, and other, chance, equimultiples *M*, *N* of *G*, *H*.

Since *E* is the same multiple of *A* that *F* is of *C*, and equimultiples *K*, *L* of *E*, *F* have been taken, therefore *K* is the same multiple of *A* that *L* is of *C*. [[book-5/proposition-3|V. 3]]

For the same reason *M* is the same multiple of *B* that *N* is of *D*.

And, since, as *A* is to *B*, so is *C* to *D*, and of *A*, *C* equimultiples *K*, *L* have been taken, and of *B*, *D* other, chance, equimultiples *M*, *N*, therefore, if *K* is in excess of *M*, *L* also is in excess of *N*, if it is equal, equal, and if less, less. [[book-5/definitions#Definition 5|V. Def. 5]]

And *K*, *L* are equimultiples of *E*, *F*, and *M*, *N* other, chance, equimultiples of *G*, *H*; therefore, as *E* is to *G*, so is *F* to *H*. [[book-5/definitions#Definition 5|V. Def. 5]]

Therefore etc. Q. E. D.