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Proof: By Euclid

(related to Proposition: Prop. 10.007: Incommensurable Magnitudes Have Irrational Ratio)

• Let $A$ and $B$ be incommensurable magnitudes.
• I say that $A$ does not have to $B$ the ratio which (some) number (has) to (some) number.

• For if $A$ has to $B$ the ratio which (some) number (has) to (some) number then $A$ will be commensurable with $B$ [Prop. 10.6].
• But it is not.
• Thus, $A$ does not have to $B$ the ratio which (some) number (has) to (some) number.
• Thus, incommensurable numbers do not have to one another, and so on ....
  ∎

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References

Adapted from (subject to copyright, with kind permission)

1. Fitzpatrick, Richard: Euclid's "Elements of Geometry"