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Pythagoras: Pythagorean Theorem
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Pythagoras is most famous for his Pythagorean Theory relating the lengths of the sides of a right triangle. Although it seems that the Babylonians knew of the theory, the Pythagoreans are credited with being the first to prove it. Also, the theory was not proved until after Pythagoras had died.
Pythagoras started a secret society called the Pythagorean brotherhood devoted to the study of mathematics. This had a great effect on future esoteric traditions, such as Rosicrucianism and Freemasonry, both of which were occult groups dedicated to the study of mathematics and both of which claimed to have evolved out of the Pythagorean brotherhood. The mystical and occult qualities of Pythagorean mathematics are discussed in a chapter of Manly P. Hall's The Secret Teachings of All Ages entitled "Pythagorean Mathematics".
The Pythagoreans (followers of Pythagoras) branched out. Those who adhered strictly to his precepts were known as the Akousmatikoi 'Hearers'; the others were called the Mathematikoi 'the learned'. Among the unusual rules the "Hearers" followed were injunctions against eating beans, letting the mark of a pot stand in its ashes or letting the impression of the human body remain in the bedclothes.
Thales was about 50 years senior to Pythagoras, and Miletus and Samos are very close geographically, so it may be that Pythagoras actually studied under Thales. Whether this was the case or not, most agree that Pythagoras was aware of, and built on the discoveries and teaching of Thales. It is ... generally accepted that the Pythagoreans contributed the bulk of the first two books of The Elements (on angles, triangles, areas, and geometric algebra) together with some of books IV (polygons), VI (proportions) and XIII (polyhedrons).
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Pythagoras's followers were commonly called "Pythagoreans". They are generally accepted as philosophical mathematicians who had an influence on the beginning of axiomatic geometry, which after two hundred years of development was written down by Euclid in The Elements.
For all this period, the disciple was regarded as a "novice" and was not allowed to present himself before Pythagoras. If after all this the pupil was considered unsuitable, he was expelled and his property returned. The Pythagoreans lived a common life together in one building and were divided into groups. Some of these studied geometry and some astronomy.
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