---
book: 5
number: 3
id: "V.3"
kind: "theorem"
uses: ["[[book-5/proposition-2]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:5.prop.3"
license: "CC-BY-SA-4.0"
---

# V.3

*If a first magnitude be the same multiple of a second that a third is of a fourth*, *and if equimultiples be taken of the first and third*, *then also* ex aequali *the magnitudes taken will be equimultiples respectively*, *the one of the second and the other of the fourth*.

## Proof

Let a first magnitude *A* be the same multiple of a second *B* that a third *C* is of a fourth *D*, and let equimultiples *EF*, *GH* be taken of *A*, *C*; I say that *EF* is the same multiple of *B* that *GH* is of *D*.

For, since *EF* is the same multiple of *A* that *GH* is of *C*, therefore, as many magnitudes as there are in *EF* equal to *A*, so many also are there in *GH* equal to *C*.

Let *EF* be divided into the magnitudes *EK*, *KF* equal to *A*, and *GH* into the magnitudes *GL*, *LH* equal to *C*; then the multitude of the magnitudes *EK*, *KF* will be equal to the multitude of the magnitudes *GL*, *LH*.

And, since *A* is the same multiple of *B* that *C* is of *D*, while *EK* is equal to *A*, and *GL* to *C*, therefore *EK* is the same multiple of *B* that *GL* is of *D*.

For the same reason

*KF* is the same multiple of *B* that *LH* is of *D*.

Since, then, a first magnitude *EK* is the same multiple of a second *B* that a third *GL* is of a fourth *D*, and a fifth *KF* is also the same multiple of the second *B* that a sixth *LH* is of the fourth *D*, therefore the sum of the first and fifth, *EF*, is also the same multiple of the second *B* that the sum of the third and sixth, *GH*, is of the fourth *D*. [[book-5/proposition-2|V. 2]]

Therefore etc. Q. E. D.