# Commensurable Magnitudes are Incommensurable with Same Magnitude

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## 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.*

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

## 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*.

## Sources

- 1926: Sir Thomas L. Heath:
*Euclid: The Thirteen Books of The Elements: Volume 3*(2nd ed.) ... (previous) ... (next): Book $\text{X}$. Propositions