83
edits
Chapter 2: An ancient theorem and a modern question: Difference between revisions
Chapter 2: An ancient theorem and a modern question (edit)
Revision as of 01:58, 15 June 2020
, 15 June 2020→Euclidian Geometry
| Line 58: | Line 58: | ||
Euclid's fifth postulate cannot be proven as a theorem (by assuming only the first four), although this was attempted by many people. Euclid himself used only the first four postulates ("absolute geometry") for the first 28 propositions of the Elements, but was forced to invoke the parallel postulate on the 29th. In 1823, Janos Bolyai and Nicolai Lobachevsky independently realized that entirely self-consistent "non-Euclidean geometries" could be created in which the parallel postulate did not hold. (Gauss had also discovered but suppressed the existence of non-Euclidean geometries.) | Euclid's fifth postulate cannot be proven as a theorem (by assuming only the first four), although this was attempted by many people. Euclid himself used only the first four postulates ("absolute geometry") for the first 28 propositions of the Elements, but was forced to invoke the parallel postulate on the 29th. In 1823, Janos Bolyai and Nicolai Lobachevsky independently realized that entirely self-consistent "non-Euclidean geometries" could be created in which the parallel postulate did not hold. (Gauss had also discovered but suppressed the existence of non-Euclidean geometries.) | ||
[[File:Euclid-woodcut-1584.jpg|thumb | [[File:Euclid-woodcut-1584.jpg|thumb|Euclid, coloured woodcut, 1584.]] | ||
=== Radians and <math> \pi </math> === | === Radians and <math> \pi </math> === | ||