[seanhunter]

Thank you for the correction I actually didn't realise that so have learned something.

Specifically for people who are interested it seems you have to replace the parallel postulate with a postulate that says every point is a saddle point (which is like the centre point of a pringle if you know what that looks like).

https://en.wikipedia.org/wiki/Pringles

[lmm]

You can also replace it with a different postulate and get projective geometry, IIRC.