Pythagoras was born on the island of Samos, off the coast of Asia Minor. He was born to Pythais (a native of Samos) and Mnesarchus (a merchant from Tyre). As a young man, he left his native city for Croton in Southern Italy, to escape the tyrannical government of Polycrates. Many writers credit him with visiting the sages of Egypt and Babylon before going west; such travels feature in the biographies of many Greek sages. Upon his migration from Samos to Croton, Pythagoras established a secret religious society very similar to (and possibly influenced by) the earlier Orphic cult. Pythagoras undertook a reform of the cultural life of Croton, urging the citizens to follow virtue and form an elite circle of followers around himself. Very strict rules of conduct governed this cultural center. He opened his school to men and women students alike. Those who joined the inner circle of Pythagoras' society called themselves the Mathematikoi. They lived at the school, owned no personal possessions and were required to assume a vegetarian diet. Other students who lived in neighboring areas were also permitted to attend Pythagoras' school. Known as Akousmatics, these students were permitted to eat meat and own personal belongings. According to Iamblichus, the Pythagoreans followed a structured life of religious teaching, common meals, exercise, reading and philosophical study. Music featured as an essential organizing factor of this life: the disciples would sing hymns to Apollo together regularly; they used the lyre to cure illness of the soul or body; poetry recitations occurred before and after sleep to aid the memory. The Pythagorean theorem that bears his name was known earlier in Mesopotamia, Egypt and India. For a chronology of the theorem and its proofs, see the article on the Pythagorean theorem. Whether Pythagoras himself proved this theorem is not known, as it was common in the ancient world to credit to a famous teacher the discoveries of his students. The earliest known mention of Pythagoras's name in connection with the theorem occurred five centuries after his death, in the writings of Cicero and Plutarch. The Golden Ratio The golden ratio, also known as the golden proportion, golden mean, golden section, golden number, divine proportion or sectio divina, is an irrational number, approximately 1.618 033 988 749 894 848, that possesses many interesting properties. Shapes proportioned according to the golden ratio have long been considered aesthetically pleasing in Western cultures, and the golden ratio is still used frequently in art and design, suggesting a natural balance between symmetry and asymmetry. The ancient Pythagoreans, who defined numbers as expressions of ratios (and not as units as is common today), believed that reality is numerical and that the golden ratio expressed an underlying truth about existence. The golden ratio was first studied by ancient mathematicians because of its frequent appearance in geometry and may have even been understood and used as far back in history as the Egyptians. It is also believed that after tracing the path of Venus in the sky, they found that the ratio of the length of the long arm of the pentagon shape to the length of the shorter arm was 1.618 ... ... More commonly, however, the discovery of the golden ratio is ascribed to the ancient Greeks, and is usually attributed to Pythagoras (or to the Pythagoreans, notably Theodorus) or to Hippasus of Metapontum. Euclid spoke of the "golden mean" this way, "A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the lesser". The golden ratio is represented by the Greek letter (phi, after Phidias, a sculptor who commonly employed it) or less commonly by τ (tau). Pythagoreans Pythagoras' followers were commonly called "Pythagoreans." For the most part we remember them 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. No texts by Pythagoras survive, although forgeries under his name — a few of which remain extant — did circulate in antiquity. Critical ancient sources like Aristotle and Aristoxenus cast doubt on these writings. And ancient Pythagoreans usually quoted their master's doctrines with the phrase autos ephe ("he himself said") — emphasizing the essentially oral nature of his teaching. Pythagoras appears as a character in the last book of Ovid's Metamorphoses , where Ovid has him expound upon his philosophical viewpoints. Some consider Pythagoras the pupil of Anaximander and some ancient sources tell of his visiting, in his twenties, the philosopher Thales, just before the death of the latter. No account exists of the specifics of the meeting, other than the report that Thales recommended that Pythagoras travel to Egypt in order to further his philosophical and mathematical training. He is also supposed to have been a disciple of Pherekydes. This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Pythagoras" and Wikipedia article "Golden Ratio". |
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