Editing Chapter 2: An ancient theorem and a modern question
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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|Euclid, coloured woodcut, 1584.]] | [[File:Euclid-woodcut-1584.jpg|thumb}|Euclid, coloured woodcut, 1584.]] | ||
=== Radians and <math>\pi</math> === | === Radians and <math>\pi</math> === | ||