---
book: 10
number: 12
id: "X.12"
kind: "theorem"
uses: ["[[book-10/proposition-5]]", "[[book-8/proposition-4]]", "[[book-5/proposition-11]]", "[[book-5/proposition-22]]", "[[book-10/proposition-6]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:10.prop_1.12"
license: "CC-BY-SA-4.0"
---

# X.12

*Magnitudes commensurable with the same magnitude are commensurable with one another also*.

## Proof

For let each of the magnitudes *A*, *B* be commensurable with *C*; I say that *A* is also commensurable with *B*.

For, since *A* is commensurable with *C*, therefore *A* has to *C* the ratio which a number has to a number. [[book-10/proposition-5|X. 5]]

Let it have the ratio which *D* has to *E*.

Again, since *C* is commensurable with *B*, therefore *C* has to *B* the ratio which a number has to a number. [[book-10/proposition-5|X. 5]]

Let it have the ratio which *F* has to *G*.

And, given any number of ratios we please, namely the ratio which *D* has to *E* and that which *F* has to *G*, let the numbers *H*, *K*, *L* be taken continuously in the given ratios; cf. [[book-8/proposition-4|VIII. 4]] so that, as *D* is to *E*, so is *H* to *K*, and, as *F* is to *G*, so is *K* to *L*.

Since, then, as *A* is to *C*, so is *D* to *E*, while, as *D* is to *E*, so is *H* to *K*, therefore also, as *A* is to *C*, so is *H* to *K*. [[book-5/proposition-11|V. 11]]

Again, since, as *C* is to *B*, so is *F* to *G*, while, as *F* is to *G*, so is *K* to *L*, therefore also, as *C* is to *B*, so is *K* to *L*. [[book-5/proposition-11|V. 11]]

But also, as *A* is to *C*, so is *H* to *K*; therefore, ex aequali, as *A* is to *B*, so is *H* to *L*. [[book-5/proposition-22|V. 22]]

Therefore *A* has to *B* the ratio which a number has to a number; therefore *A* is commensurable with *B*. [[book-10/proposition-6|X. 6]]

Therefore etc. Q. E. D.