Non-Euclidean geometry
Branch · 1820s-1860s · Mathematics, stage 2: Map the branches
For two thousand years mathematicians tried to prove Euclid's parallel postulate from his other assumptions. In the 1820s and 1830s Lobachevsky and Bolyai independently developed a geometry in which it fails and found no contradiction; Gauss had reached the same conclusion earlier and did not publish. Riemann then generalised the whole framework in 1854, treating geometry as the study of spaces with a curvature that can vary from point to point, which turned out to be the mathematics general relativity needed sixty years later.
What it claims
- The parallel postulate does not follow from Euclid's other assumptions.
- Replacing it produces geometries that are internally consistent and can be studied in their own right.
- Which geometry describes physical space is a question for measurement, not for pure reasoning.
Key ideas
People
Sources
- Non-Euclidean geometry MacTutor History of Mathematics
- Nineteenth Century Geometry Stanford Encyclopedia of Philosophy
- Non-Euclidean geometry Wikipedia
