(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 ....∎

**Fitzpatrick, Richard**: Euclid's "Elements of Geometry"