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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.

If we accept it, we get Euclidean geometry, but if we abandon it, other geometries become possible, most famously the hyperbolic variety. Here’s how the parallel postulate works.

A postulate is supposed to be self-evident, but this business about parallel lines struck a nerve with mathematicians. It didn’t seem as intuitively convincing — Euclid didn’t even invoke the Fifth ...

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, analytic and ...

The parallel postulate was essential to Euclid in many of his proofs. But he could have chosen another axiom. For example, he could have started, as Thales did, ...

Schoolchildren now study Euclidean geometry, all thanks to Greek mathematician Euclid who was born in approx. 300 BC. His main work is “The Elements,” in which he described the five main ...