---
book: 10
number: 5
id: "X.5"
kind: "theorem"
uses: ["[[book-5/proposition-7]]", "[[book-5/proposition-22]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:10.prop_1.5"
license: "CC-BY-SA-4.0"
---

# X.5

*Commensurable magnitudes have to one another the ratio which a number has to a number*.

## Proof

Let *A*, *B* be commensurable magnitudes; I say that *A* has to *B* the ratio which a number has to a number.

For, since *A*, *B* are commensurable, some magnitude will measure them.

Let it measure them, and let it be *C*.

And, as many times as *C* measures *A*, so many units let there be in *D*; and, as many times as *C* measures *B*, so many units let there be in *E*.

Since then *C* measures *A* according to the units in *D*, while the unit also measures *D* according to the units in it, therefore the unit measures the number *D* the same number of times as the magnitude *C* measures *A*; therefore, as *C* is to *A*, so is the unit to *D*; [[book-7/definitions#Definition 20|VII. Def. 20]] therefore, inversely, as *A* is to *C*, so is *D* to the unit. cf. [[book-5/proposition-7|V. 7, Por.]]

Again, since *C* measures *B* according to the units in *E*, while the unit also measures *E* according to the units in it, therefore the unit measures *E* the same number of times as *C* measures *B*; therefore, as *C* is to *B*, so is the unit to *E*.

But it was also proved that, as *A* is to *C*, so is *D* to the unit; therefore, ex aequali, as *A* is to *B*, so is the number *D* to *E*. [[book-5/proposition-22|V. 22]]

Therefore the commensurable magnitudes *A*, *B* have to one another the ratio which the number *D* has to the number *E*. Q. E. D.