---
book: 5
number: 21
id: "V.21"
kind: "theorem"
uses: ["[[book-5/proposition-8]]", "[[book-5/proposition-13]]", "[[book-5/proposition-10]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:5.prop.21"
license: "CC-BY-SA-4.0"
---

# V.21

*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*, *then*, *if* ex aequali *the first magnitude is greater than the third*, *the fourth will also be greater than the sixth; if equal*, *equal; and if less*, *less*.

## Proof

Let there be three magnitudes *A*, *B*, *C*, and others *D*, *E*, *F* equal to them in multitude, which taken two and two are in the same ratio, 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*, and let *A* be greater than *C* ex aequali; I say that *D* will also be greater than *F*; if *A* is equal to *C*, equal; and if less, less.

For, since *A* is greater than *C*, and *B* is some other magnitude, therefore *A* has to *B* a greater ratio than *C* has to *B*. [[book-5/proposition-8|V. 8]]

But, as *A* is to *B*, so is *E* to *F*, and, as *C* is to *B*, inversely, so is *E* to *D*. Therefore also *E* has to *F* a greater ratio than *E* has to *D*. [[book-5/proposition-13|V. 13]]

But that to which the same has a greater ratio is less; [[book-5/proposition-10|V. 10]] therefore *F* is less than *D*; therefore *D* is greater than *F*.

Similarly we can prove that, if *A* be equal to *C*, *D* will also be equal to *F*; and if less, less.

Therefore etc. Q. E. D.