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 ....
∎
Thank you to the contributors under CC BY-SA 4.0! 
- Github:
-
- non-Github:
- @Fitzpatrick
References
Adapted from (subject to copyright, with kind permission)
- Fitzpatrick, Richard: Euclid's "Elements of Geometry"
