---
book: 10
number: 11
id: "X.11"
kind: "theorem"
uses: ["[[book-10/proposition-5]]", "[[book-10/proposition-6]]", "[[book-10/proposition-7]]", "[[book-10/proposition-8]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:10.prop_1.11"
license: "CC-BY-SA-4.0"
---

# X.11

*If four magnitudes be proportional, and the first be commensurable with the second, the third will also be commensurable with the fourth; and, if the first be incommensurable with the second, the third will also be incommensurable with the fourth*.

## Proof

Let *A*, *B*, *C*, *D* be four magnitudes in proportion, so that, as *A* is to *B*, so is *C* to *D*, and let *A* be commensurable with *B*; I say that *C* will also be commensurable with *D*.

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

And, as *A* is to *B*, so is *C* to *D*; therefore *C* also has to *D* the ratio which a number has to a number; therefore *C* is commensurable with *D*. [[book-10/proposition-6|X. 6]]

Next, let *A* be incommensurable with *B*; I say that *C* will also be incommensurable with *D*.

For, since *A* is incommensurable with *B*, therefore *A* has not to *B* the ratio which a number has to a number. [[book-10/proposition-7|X. 7]]

And, as *A* is to *B*, so is *C* to *D*; therefore neither has *C* to *D* the ratio which a number has to a number; therefore *C* is incommensurable with *D*. [[book-10/proposition-8|X. 8]]

Therefore etc.