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