---
book: 5
number: 20
id: "V.20"
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.20"
license: "CC-BY-SA-4.0"
---

# V.20

*If there be three magnitudes*, *and others equal to them in multitude*, *which taken two and two are in the same ratio*, *and if* ex aequali *the first be 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, so that, as *A* is to *B*, so is *D* to *E*, and as *B* is to *C*, so is *E* to *F*; 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, and the greater has to the same a greater ratio than the less has, [[book-5/proposition-8|V. 8]] therefore *A* has to *B* a greater ratio than *C* has to *B*.

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

But, of magnitudes which have a ratio to the same, that which has a greater ratio is greater; [[book-5/proposition-10|V. 10]] 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.