http://SaturnianCosmology.Org/ mirrored file For complete access to all the files of this collection see http://SaturnianCosmology.org/search.php ========================================================== Pythagoras ------------------------------------------------------------------------ Greek philosopher and mathematician; b. *580 BC *(Samos, Ionia, Greece), d. *500 BC* (Metapontum, Lucania) ------------------------------------------------------------------------ Although Pythagoras was one of the most important personalities of ancient Greece, reliable information about him is very limited. He is commonly thought of as a mathematician, but it is almost certain that he saw himself as a religious philosopher. According to some records Pythagoras was the son of the merchant Mnesarchus of Tyre, who had moved to Samos and married Pythais, a local girl. He received a good education, which included how to play the lyre. In about 550 BC Pythagoras spent some time in Egypt and became accustomed with aspects of Egyptian priesthood such as secrecy, refusal to eat beans, refusal to wear clothes made from animal skins, and striving for purity, ideals that played an important role in his later life. Some time around 532 Pythagoras migrated to Croton (today's Crotone) in southern Italy, where he founded a religious order that became known later as "the Pythagoreans". Dedicated followers were vegetarians, had no personal belongings and generally followed the principles learnt in Egypt. All members of the order adhered to the five principles that * at its deepest level, reality is mathematical in nature, * certain symbols have a mystical significance, * philosophy can be used for spiritual purification, * the soul can rise to union with the divine, and * all members of the order should observe strict loyalty and secrecy. The last of these rules is of some consequence for historians, since it makes it virtually impossible to separate the achievements of Pythagoras from those of other members of the order. The radical principles of the order - it despised personal possessions and allowed women as members - brought it into conflict with wealthy influential members of the establishment, and Pythagoras had to seek refuge in Metapontum near Tarentum, where he stayed until his death. The origin of the Pythagoreans' fame as mathematicians has to be seen in the combination of their desire to unite with the mystery of the universe and the ancient Greek belief that the universe follows simple and beautiful rules. The "sacred decad" or /tetraktys/ in particular signified cosmic harmony to them: 1+2+3+4=10 or, in the form of the "perfect triangle", o o o o o o o o o o The belief in cosmic harmony stimulated them to investigate relationships between numbers and produce outstanding contributions to number theory. The most widely known of such relationships is the Pythagorean theorem for right triangles, illustrated on the left through a Greek postage stamp of 1955, issued to commemorate the 2500th anniversary of the founding of the order: The square of the longest side of a right triangle equals the sum of the squares of the other two sides (in the postage stamp example 25=9+16.) The Babyloneans had known the theorem for some 1000 years already, but the Pythagoreans are believed to have developed the first proof. Among other mathematical theorems discovered or proven by the Pythagoreans is the rule that the sum of the angles of a triangle is equal to two right angles, and its generalisation that the sum of interior angles of a polygon with /n/ sides is (2/n/ - 4) right angles and the sum of its exterior angles is four right angles. The Pythagoreans discovered irrational numbers and developed elementary ways to handle them. A rigorous method to calculate irrational numbers was developed by Eudoxus, who lived about 400 - 350 BC. Significant results were also achieved in the area of music theory. The Pythagoreans introduced the concept of expressing musical intervals as proportions between numbers, such as 1:2 for the octave, 3:2 for the fifth and 4:3 for the fourth. They calculated the diatonic scale, based on the ratio 9:8 for the single tone interval. Archytas of Tarentum, a Pythagorean who lived around 400 - 350 BC, developed the numerical intervals for the chromatic and enharmonic scales. Not long after Pythagoras' death the order encountered violent suppression and had to hold its meetings in secret. A report from Croton mentions that around 450 BC a meeting of 50 or 60 Pythagoreans was surprised and nearly all persons present were killed. Later in the 5th century BC the order split into two directions. The /akousmatikoi/ ("teachers of esoteric beliefs") stressed its religious character and teachings, such as the interaction of opposites as the main driving mechanism for the world and successive reincarnation of the individual until eventual purification. The /mathematikoi/ ("teachers of science") concentrated on number relationships and their applications in acoustics, geometry and astronomy. Pythagoreanism as a religious order lived only for some 200 years; it came to an end during the 4th century BC. Its influence in philosophy and mathematics continued well into medieval Europe, when aspects of its cosmic harmony entered aesthetics and the visual arts. The Renaissance and the Enlightenment considered Pythagoras the founder of exact science; Kepler considered his work to be in his tradition, Galilei was seen as a Pythagorean, and Leibnitz saw his work as part of the Pythagorean tradition. ------------------------------------------------------------------------ (Some sources give Pythagoras' life dates as c. 569 - c. 475 BC and date the major events in his life correspondingly later.) Portrait bust: Vatican Museum, Rome; public domain /(Wikipedia)/ ------------------------------------------------------------------------ home <../index.html>