Magnitudes with Irrational Ratio are Incommensurable

Proof
Let $A$ and $B$ be magnitudes which do not have to one another the ratio which a number has to a number.

Suppose $A$ and $B$ are commensurable.

Then from Ratio of Commensurable Magnitudes it follows that $A$ and $B$ have to one another the ratio which a number has to a number.

From this contradiction follows the result.