History of geometry: Difference between revisions

====Euclid====

[[Image:EuclidStatueOxford.jpg|thumb|Statue of Euclid in the [[Oxford University Museum of Natural History]]]]

====Non-Euclidean geometry====

The very old problem of proving Euclid's Fifth Postulate, the "[[Parallel Postulate]]", from his first four postulates had never been forgotten. Beginning not long after Euclid, many attempted demonstrations were given, but all were later found to be faulty, through allowing into the reasoning some principle which itself had not been proved from the first four postulates. Though Omar Khayyám was also unsuccessful in proving the parallel postulate, his criticisms of Euclid's theories of parallels and his proof of properties of figures in non-Euclidean geometries contributed to the eventual development of [[non-Euclidean geometry]]. By 1700 a great deal had been discovered about what can be proved from the first four, and what the pitfalls were in attempting to prove the fifth. [[Saccheri]], [[Johann Heinrich Lambert|Lambert]], and [[Adrien-Marie Legendre|Legendre]] each did excellent work on the problem in the 18th century, but still fell short of success. In the early 19th century, [[Carl Friedrich Gauss|Gauss]], [[János Bolyai|Johann Bolyai]], and [[Nikolai Lobachevsky|Lobachevsky]], each independently, took a different approach. Beginning to suspect that it was impossible to prove the Parallel Postulate, they set out to develop a self-consistent geometry in which that postulate was false. In this they were successful, thus creating the first non-Euclidean geometry. By 1854, [[Bernhard Riemann]], a student of Gauss, had applied methods of calculus in a ground-breaking study of the intrinsic (self-contained) geometry of all smooth surfaces, and thereby found a different non-Euclidean geometry. This work of Riemann later became fundamental for [[Albert Einstein|Einstein]]'s [[theory of relativity]].

[[Image:Newton-WilliamBlake.jpg|thumb|left|[[William Blake]]'s "Newton" is a demonstration of his opposition to the 'single-vision' of [[scientific materialism]]; here, [[Isaac Newton]] is shown as 'divine geometer' (1795).]]