Refresher - Famous Geometers
Euclid of Alexandria (325-265 BC) is best known for his 13-book treatise Elements (circa 300 BC), in which he collected the theorems of Pythagoras, Hippocrates, Theaetetus, Eudoxus, and other predecessors into a logically connected whole.
Other famous Geometers include J. Henri Poincare, Hypatia of Alexandria, Omar Khayyam, Blaise Pascal, János Bolyai, Nikolay Lobachevsky, René Descartes, Pierre de Fermat, and Bernhard Riemann.
Euclid's Elements is purported to be the most brilliant, famous, significant, and widely read textbook ever. It systematized the works of generations of mathematicians before Euclid and was preserved by later generations, passing through the Greek, Roman, Byzantine, and Islamic empires, across North Africa to Spain, to medieval European universities, to modern civilizations.
Elements started with 23 definitions and five postulates, and systematically built the rest of plane and solid geometry upon this foundation. Euclidean geometry is the study of geometry that satisfies Euclid's five postulates.
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Euclid and many other mathematicians attempted, without success, to prove the fifth postulate from the other four, and thus make this into a theorem. Finally, in the 19th century, several mathematicians working independently explored the possibility of rejecting Euclid's fifth postulate.
Two mathematicians decided that if the fifth postulate was provable by the other four they should be able to "replace" the fifth postulate with a contrary one, and reach a contradiction somewhere. Bolyai and Lobachevsky purported that there was more than one line parallel to a given line from a point outside, and Riemann purported that there was no such thing as parallel lines.
To their surprise, no contradictions were found. The result was that they had invented new systems of geometry, and cleverly called these Non-Euclidean geometries. In a Non-Euclidean geometry such as spherical geometry, two lines can be parallel and still intersect. To visualize this, check out a globe and look at the lines of longitude. At certain places, say between 20º W and 30º W, these lines appear parallel. However at the polls they appear to intersect. In Non-Euclidean geometries triangles do not have angle sums of 180º.