Pythagoras and The Pythagoreans. Pythagoras (572 BC 497 BC) made significant contributions to the study of

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1 Tyree 1 Makayla M. Tyree Dr. Shanyu Ji History of Mathematics June 11, 2017 Pythagoras and The Pythagoreans Pythagoras (572 BC 497 BC) made significant contributions to the study of mathematics. Pythagoras, born on the island of Samos, studied with Thales in Miletus, who suggested that he travel to Egypt and Babylon to study mathematics. After completing his studies, Pythagoras settled in Croton in southern Italy. It was here that he founded his school. Pythagoras pupils, known as the Pythagoreans, lived and studied in a communal society. Pythagoras and the Pythagoreans studied many fields, including philosophy and cosmology, and they also influenced Greek politics and the study of mathematics all while following a strict code of conduct. Pythagoras Image:

2 Tyree 2 Pythagoras and his students made such mathematical discoveries as the Pythagorean theorem, which states that the square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides. 1 They also found that the internal angles of a right triangle are equal to two right angles. 2 Pythagoras was also the first to teach that the earth was spherical. 3 In the Pythagorean model of the universe, the earth and all astronomical bodies were perfect spheres, and those spheres moved around a central fire. 4 There were ten objects that revolved around that fire: a sphere of fixed stars, the moon, the sun, five planets, the earth, and a counterearth, which was used to explain eclipses. 5 The Pythagoreans also described the harmony of the spheres, which states that there are particular frequencies that resonate from each planetary body. 6 The Pythagoreans also studied music theory, which lead to their belief that music, number, and the cosmos were all intimately related. 7 Pythagoras model of the universe

3 Tyree 3 The Pythagoreans were divided into two classes: the neophytes and the masters. 8 The neophytes were taught only the most basic and general concepts, whereas the masters were invited to live and study communally. 9 Pythagoras pupils followed a strict, though at times unreasonable, code of conduct. The following were a few of the more unusual rules of the Pythagoreans 10 : Abstain from the consumption of beans Do not pick up that which has fallen Do not stir the fire with iron Do not look in a mirror beside a light The Pythagoreans also followed a vegetarian diet due to their belief in the transmigration of souls. 11 The Pythagorean s badge of identity was the regular pentagonal star. 12 Image: Kay, David C College Geometry: A Discovery Approach. Addison Wesley Longman, Inc.: Boston, MD.

4 Tyree 4 The school of Pythagoras dictum was All is number, and so the Pythagoreans studied science, religion, and philosophy, and attempted to describe each of these fields using mathematics. The Pythagoreans believed that the universe could be described by whole numbers. They associated physical figures with whole numbers; for example, one was associated with a point, two was associated with a line, three was associated with a surface, and four was associated with a solid. 13 The sum of these values, ten, was considered sacred. 14 The Pythagoreans were polytheistic, as they worshipped the numbers one through ten as gods. 15 They believed that numbers were alive and that one could communicate telepathically with the numbers if one meditated. 16 Pythagoras even described himself as semi-divine, saying that there are men, gods, and men like Pythagoras. 17 The School of Athens by Raphael. Pythagoras is pictured in the lower left corner, holding a book. Image: Pickover, Clifford A The Math Book. Barnes & Noble, Inc.: New York, NY. 16 Pickover, Clifford A The Math Book. Barnes & Noble, Inc.: New York, NY. 17

5 Tyree 5 In addition to their mathematical and philosophical studies, the Pythagoreans were influential in Greek politics. The Greek author Dicaearchus reported that when Pythagoras arrived in Croton, he gave a speech to the city s leaders and also spoke to the citizens. 18 The Pythagoreans were said to be the leading citizens of many cities and had a significant impact on society in southern Italy. 19 Pythagoras spread the seeds of political liberty to locations such as Crotona, Metapontum, and Rhegium. 20 The Pythagoreans were not a political party but a fraternal organization that influenced politics through individual members that gained political power in the city-states of southern Italy. 21 Two famous legislators, Charondas and Zaleucus, were influenced by Pythagoras and his followers. 22 The cause of Pythagoras death is unknown. 23 In one account, he perished in a fire at the Pythagorean s meeting place. There are also reports of Pythagoras being one of the few who survived the fire and, after fleeing to Metapontum, Pythagoras was unwilling to walk through a sacred bean field and was then killed by his attackers. 24 Although Pythagoras died in approximately 497 BC, the work of the Pythagoreans influenced literature, art, science, and philosophy into the Middle Ages and Renaissance. 25 While the most well-known discovery made by Pythagoras and his followers is the Pythagorean theorem, Pythagoras influenced mathematics in many ways. Because of Pythagoras recognition of abstractions in mathematics, he is credited with creating the study of pure mathematics. In the words of Bertrand Russell, It is to this gentleman that we owe pure mathematics. The contemplative ideal -- since it led to pure Naifeh, CiCi. n.d. Pythagoras. 24 Naifeh, CiCi. n.d. Pythagoras. 25 Huffman, Carl Pythagoreanism.

6 Tyree 6 mathematics -- was the source of a useful activity. This increased its prestige and gave it a success in theology, in ethics, and in philosophy

7 Tyree 7 References [Online] [Cited: June 7, 2017] [Online] [Cited: June 7, 2017] [Online] [Cited: June 7, 2017] Huffman, Carl Pythagoreanism. [Online] [Cited: June 7, 2017] Kay, David C College Geometry: A Discovery Approach. Addison Wesley Longman, Inc.: Boston, MD. Naifeh, CiCi. n.d. Pythagoras. [Online] [Cited: June 7, 2017] Pickover, Clifford A The Math Book. Barnes & Noble, Inc.: New York, NY. [Online] [Cited: June 7, 2017]