Commensurable Magnitudes are Incommensurable with Same Magnitude

From ProofWiki
Jump to navigation Jump to search


In the words of Euclid:

If two magnitudes be commensurable, and the one of them be incommensurable with any magnitude, the remaining will also be incommensurable with the same.

(The Elements: Book $\text{X}$: Proposition $13$)


Let $A$ and $B$ be magnitudes which are commensurable with each other.

Let $A$ be incommensurable with any other magnitude $C$.

Suppose $B$ is commensurable with $C$.

But $A$ is commensurable with $B$.

So from Commensurability is Transitive Relation it follows that $A$ is commensurable with $C$.

From that contradiction it follows that $B$ cannot be commensurable with $C$.

Hence the result.


Historical Note

This proof is Proposition $13$ of Book $\text{X}$ of Euclid's The Elements.