---
book: 5
number: 23
id: "V.23"
kind: "theorem"
uses: ["[[book-5/proposition-15]]", "[[book-5/proposition-11]]", "[[book-5/proposition-16]]", "[[book-5/proposition-21]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:5.prop.23"
license: "CC-BY-SA-4.0"
---

# V.23

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

## Proof

Let there be three magnitudes *A*, *B*, *C*, and others equal to them in multitude, which, taken two and two together, are in the same proportion, namely *D*, *E*, *F*; and let the proportion of them be perturbed, so that, as *A* is to *B*, so is *E* to *F*, and, as *B* is to *C*, so is *D* to *E*; I say that, as *A* is to *C*, so is *D* to *F*.

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

Then, since *G*, *H* are equimultiples of *A*, *B*, and parts have the same ratio as the same multiples of them, [[book-5/proposition-15|V. 15]] therefore, as *A* is to *B*, so is *G* to *H*.

For the same reason also, as *E* is to *F*, so is *M* to *N*. And, as *A* is to *B*, so is *E* to *F*; therefore also, as *G* is to *H*, so is *M* to *N*. [[book-5/proposition-11|V. 11]]

Next, since, as *B* is to *C*, so is *D* to *E*, alternately, also, as *B* is to *D*, so is *C* to *E*. [[book-5/proposition-16|V. 16]]

And, since *H*, *K* are equimultiples of *B*, *D*, and parts have the same ratio as their equimultiples, therefore, as *B* is to *D*, so is *H* to *K*. [[book-5/proposition-15|V. 15]]

But, as *B* is to *D*, so is *C* to *E*; therefore also, as *H* is to *K*, so is *C* to *E*. [[book-5/proposition-11|V. 11]]

Again, since *L*, *M* are equimultiples of *C*, *E*, therefore, as *C* is to *E*, so is *L* to *M*. [[book-5/proposition-15|V. 15]]

But, as *C* is to *E*, so is *H* to *K*; therefore also, as *H* is to *K*, so is *L* to *M*, [[book-5/proposition-11|V. 11]] and, alternately, as *H* is to *L*, so is *K* to *M*. [[book-5/proposition-16|V. 16]]

But it was also proved that, as *G* is to *H*, so is *M* to *N*.

Since, then, there are three magnitudes *G*, *H*, *L*, and others equal to them in multitude *K*, *M*, *N*, which taken two and two together are in the same ratio, and the proportion of them is perturbed, therefore, ex aequali, if *G* is in excess of *L*, *K* is also in excess of *N*; if equal, equal; and if less, less. [[book-5/proposition-21|V. 21]]

And *G*, *K* are equimultiples of *A*, *D*, and *L*, *N* of *C*, *F*.

Therefore, as *A* is to *C*, so is *D* to *F*.

Therefore etc. Q. E. D.