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

# X.8

*If two magnitudes have not to one another the ratio which a number has to a number*, *the magnitudes will be incommensurable*.

## Proof

For let the two magnitudes *A*, *B* not have to one another the ratio which a number has to a number; I say that the magnitudes *A*, *B* are incommensurable.

For, if they are commensurable, *A* will have to *B* the ratio which a number has to a number. [[book-10/proposition-5|X. 5]]

But it has not; therefore the magnitudes *A*, *B* are incommensurable.

Therefore etc.