# How does Euclid's Fifth postulate not hold here?

If look at the Latitude lines - blue circles below - and pick one, each point $$P$$ not in one selected circle/line, will only lie in one other blue circle, circle here taken as line, and will follow the exact definition of postulate 5.

So we have also the same notion of a parallel line.

Is this incorrect, and why ?

See the blue lines I imagine just below: • With this choice, you do not have a line through every pair of points. Sep 18 at 20:56
• But do we need to ? I'm asking out of curiosity. Just trying to see where it leads as well. And then, this way it wouldn't contradict the fifth postulate, but some other postulate, correct ? @MoisheKohan Sep 18 at 20:59
• Well, it depends on your objective. Your geometry will contradict the 1st postulate. Sep 18 at 21:02
• Thank you, I would like to read about what is derived from a geometry that looks like that, but don't know whether something simple will appear. Thanks for the comment though. If you make an answer of it, I think it is enough, just adding some references. Previously people shower this with downvotes. I think this is part of understanding something, idk. @MoisheKohan Sep 18 at 21:04
• Incidentally: You are referring to the Playfaire postulate, not Euclid's 5th postulate. The latter becomes essentially meaningless once the 1st postulate is violated as in your case (simply because there are no transversals). Sep 18 at 21:11

It appears that your proposed model geometry is the unit sphere in $${\mathbb R}^3$$ and "lines" in this geometry are the longitudes. I also suppose that the north and south poles also qualify as "lines" in this geometry (otherwise, Playfaire's Axiom will be violated). Such a geometry indeed will satisfy Playfaire's Axiom. However, it will violate other axioms/postulates of Euclidean geometry, e.g. the 1st postulate. (Arguably, it will also violate the 2nd postulate and, depending on your interpretation, the 3rd one as well. Even the 4th one is suspect since it is very unclear what the "right angle" means in this geometry, where there are no transversal lines and where some points are also lines!) By the way, I prefer Hilbert's axioms since they provide for a clean and complete treatment of the subject.