---
book: 10
number: 2
id: "X.2"
kind: "theorem"
uses: []
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:10.prop_1.2"
license: "CC-BY-SA-4.0"
---

# X.2

*If, when the less of two unequal magnitudes is continually subtracted in turn from the greater, that which is left never measures the one before it, the magnitudes will be incommensurable.*

## Proof

For, there being two unequal magnitudes *AB*, *CD*, and *AB* being the less, when the less is continually subtracted in turn from the greater, let that which is left over never measure the one before it; I say that the magnitudes *AB*, *CD* are incommensurable.

For, if they are commensurable, some magnitude will measure them.

Let a magnitude measure them, if possible, and let it be *E*; let *AB*, measuring *FD*, leave *CF* less than itself, let *CF* measuring *BG*, leave *AG* less than itself, and let this process be repeated continually, until there is left some magnitude which is less than *E*.

Suppose this done, and let there be left *AG* less than *E*.

Then, since *E* measures *AB*, while *AB* measures *DF*, therefore *E* will also measure *FD*.

But it measures the whole *CD* also; therefore it will also measure the remainder *CF*.

But *CF* measures *BG*; therefore *E* also measures *BG*.

But it measures the whole *AB* also; therefore it will also measure the remainder *AG*, the greater the less: which is impossible.

Therefore no magnitude will measure the magnitudes *AB*, *CD*; therefore the magnitudes *AB*, *CD* are incommensurable.

Therefore etc. [[book-10/definitions#Definition 1|X. Def. 1]]