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The 2500-Year-Old Pythagorean Theorem

When you see a paper with "Pythagorean theorem" in its title, you might say "I know this stuff" and skip it. But I think it's still worthwhile thinking about the "good old Pythagorean theorem," to which this paper is devoted. Pythagoras was born about 570 B.C. on the island of Samos and died about 490 B.C. Many other well-known philosophers lived and worked around the same time, but in other civilizations. Let us mention only Gautama Buddha in eastern Asia, Confucius (or Kung Fu-tse) in China, Zoroaster (or Zarathustra) in Persia, and the prophet Isaac (or Iitzak) in Judea. Was this simultaneous flowering of philosophy a mere accident? Pythagoras was well educated, learning to play the lyre and to recite Homer. Most important among his teachers was Thales of Miletus (ca. 624-548 B.C.), who introduced Pythagoras to mathematical ideas and to astronomy, and sent him to Egypt to learn more of these subjects. Pythagoras returned to Samos, but soon left the island to escape the tyranny of Polycrates, settling in Croton in southern Italy, where he founded a philosophical and religious school that had many followers. Pythagoras led the Society, with an inner circle of followers known as mathematikoi. The mathematikoi lived permanently with the Society, had no personal possessions, and were vegetarians. They were taught by Pythagoras himself and obeyed strict rules. Among the beliefs of Pythagoras were these: (1) reality is fundamentally mathematical; (2) philosophy can lead to spiritual purification; (3) the soul can rise to union with the divine; (4) certain symbols have mystical significance; and (5) all members of the order should observe strict loyalty and secrecy. Pythagoras and the mathematikoi studied mathematics, but they were less interested in formulating and solving problems than in the principles of mathematics, the concepts of number, triangle, and space form, and the abstract idea of proof. Pythagoras believed that all of nature and its order could be reduced to numerical relations. He studied properties of even, odd, triangular, and perfect numbers, and he assigned to each number its own "personality." Numbers might be masculine or feminine, perfect or incomplete, beautiful or ugly, etc. Ten was regarded as perfect: it contained the first four integers (1 + 2 + 3 + 4 = 10) and when written in dot notation formed a perfect triangle. The Society at Croton was not unaffected by political events, despite Pythagoras's desire to avoid politics. After the Society was attacked, Pythagoras escaped to Metapontium. It is believed that he died there, though some authors claim that he committed suicide after the attack on his Society. The Society expanded rapidly after 500 B.C., became political in nature and then split into factions. About 460 B.C. the