Concept
Non-Euclidean geometry
Overview
In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geometry arises by either replacing the parallel postulate with an alternative, or consideration of quadratic forms other than the definite quadratic forms associated with metric geometry. In the former case, one obtains hyperbolic geometry and elliptic geometry, the traditional non-Euclidean geometries. When isotropic quadratic forms are admitted, then there are affine planes associated with the planar algebras, which give rise to kinematic geometries that have also been called non-Euclidean geometry.
Overview adapted from Wikipedia (CC BY-SA 4.0; see the article history for authors).
Possible passages
Found by search and not yet checked by a person; they may only mention the idea in passing.
Congruence was to refer to geometrical bodies, with none of the properties of ordinary bodies except impenetrability (Erdmann, Axiome der Geometrie, p. 26). A straight line is defined as determined by two of its points[14], and a plane as determined by three. These premisses, with a slight exception as to the straight line, we shall hereafter find essential to every Geometry.
Bertrand Russell, An essay on the foundations of geometry
States what Non-Euclidean geometry is; the author is tied to Non-Euclidean geometry in the knowledge graph.
The reasons for this will concern us later. What concerns us now is the way in which non-Euclidean geometry results from a generalization of the work of Gauss.
Bertrand Russell, The A B C of Relativity
The author is tied to Non-Euclidean geometry in the knowledge graph; names Non-Euclidean geometry twice.
1. _Introduction_ In the articles I have heretofore devoted to space I have above all emphasized the problems raised by non-Euclidean geometry, while leaving almost completely aside other questions more difficult of approach, such as those which pertain to the number of dimensions. All the geometries I considered had thus a common basis, that tridimensional continuum which was the same for all and which…
Henri Poincaré, The Foundations of Science
Names Non-Euclidean geometry twice; opens on Non-Euclidean geometry.
Philosophers who discussed Non-Euclidean geometry
- Moritz Schlick
- William Kingdon Clifford
- Willard Van Orman Quine
- Bertrand Russell
- Gaston Bachelard
- Edmund Husserl
- Stephen F. Barker
Related ideas
- Garrett Birkhoff
- Book of Optics
- Galen
- Hungary
- H. P. Lovecraft
- Infinite set
- History of science
- Immanuel Kant
- Ancient Greek mathematics
- Negation
- Ibn al-Haytham
- Mathematics
- Infinity
- Manifold
- Reductio ad absurdum
- Fantasy
- Ptolemy
- Euclid's Elements
- Paradigm shift
- William Kingdon Clifford
- Russia
- Rome
- Academic Press
- Robert M. Pirsig
Place in the tradition
Part of Bertrand Russell.