Nature - Euclid's Parallel Postulate: its Nature, Validity, and Place in Geometrical Systems. Skip to main content. Thank you for visiting nature.com.

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.

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 ...

In non-Euclidean geometries, this ‘parallels postulate’ is dropped. Two main possibilities then arise: one is spherical geometry, in which parallel lines can ... In Euclidean space, staring at ...

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 ...