(related to Proposition: Prop. 10.013: Commensurable Magnitudes are Incommensurable with Same Magnitude)

- Let $A$ and $B$ be two commensurable magnitudes, and let one of them, $A$, be incommensurable with some other (magnitude), $C$.
- I say that the remaining (magnitude), $B$, is also incommensurable with $C$.

- For if $B$ is commensurable with $C$, but $A$ is also commensurable with $B$, $A$ is thus also commensurable with $C$ [Prop. 10.12].
- But, (it is) also incommensurable (with $C$).
- The very thing (is) impossible.
- Thus, $B$ is not commensurable with $C$.
- Thus, (it is) incommensurable.
- Thus, if two magnitudes are commensurable, and so on ....∎

**Fitzpatrick, Richard**: Euclid's "Elements of Geometry"