# Commensurable Magnitudes are Incommensurable with Same Magnitude

## Theorem

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.

## Proof

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.

$\blacksquare$

## Historical Note

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