Proof: By Euclid
(related to Proposition: Prop. 10.008: Magnitudes with Irrational Ratio are Incommensurable)
- For let the two magnitudes $A$ and $B$ not have to one another the ratio which (some) number (has) to (some) number.
I say that the magnitudes $A$ and $B$ are incommensurable.
For if they are commensurable, $A$ will have to $B$ the ratio which (some) number (has) to (some) number [Prop. 10.5].
- But it does not have (such a ratio).
- Thus, the magnitudes $A$ and $B$ are incommensurable.
- Thus, if two magnitudes ... to one another, and so on ....
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Adapted from (subject to copyright, with kind permission)
- Fitzpatrick, Richard: Euclid's "Elements of Geometry"