Description: Incidence geometry in planes and space, separation in planes and space, angular measure, neutral geometry, history of the parallel postulate, Euclidean geometry of the plane and space, ...

Such a geometry has no parallel lines, yet it obeys his first four postulates. The manuscript hints that Euclid also explored the curious geometry that arises from the existence of multiple parallels.

The fifth and final axiom of Euclid’s system – the so-called parallel postulate – turns out not to be correct. Or at least not necessarily so. If we accept it, we get Euclidean geometry ...

As an example, we could take Euclid’s first four axioms as the base, omitting the fifth, the parallel postulate. The word geometry means measurement of the Earth, and the Greeks strove to ...