Denying or going beyond Euclidean principles in geometry, especially in contravening the postulate that only one line through a given point can be parallel to a given line.
- ‘Beltrami in this 1868 paper did not set out to prove the consistency of non-Euclidean geometry or the independence of the Euclidean parallel postulate.’
- ‘A modern realist would say that what mathematicians hadn't realised is that, as well as Euclidean geometry, there also exists non-Euclidean geometry.’
- ‘In 1915, Albert Einstein found it more convenient, the conventionalist would say, to develop his theory of general relativity using non-Euclidean rather than Euclidean geometry.’
- ‘By chapter two he is proving Pythagoras' theorem from first principles and introducing non-Euclidean geometry.’
- ‘It can be shown that if Euclidean geometry is internally consistent, then the alternative non-Euclidean geometries are consistent as well.’