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## Euclid's 5th postulate

Is Euclid's 5th postulate the basic thing which, if valid or not, makes a geometry Euclidean or non-Euclidean?
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 Recognitions: Gold Member Science Advisor Staff Emeritus Fundamentally, yes. There is also the axiom (I confess I don't rember the number- it might be #2!) that asserts that there exist exactly one line between any two points. "Hyperbolic geometry", in particular, the geometry on the surface of a sphere, does not satisfy that but generally speaking, the distinction between Euclidean geometry and "elliptic geometry"- normally thought of as "non-Euclidean" geometry is the requirement that, through a given point, there exist exactly one line parallel to a given point (known as "Playfair's axiom). While elliptic geometry allows that there exist more than one axiom, hyperbolic geometry requires exist exactly one such line.

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