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Pythagoras

Pythagoras


Mathematician and Philosopher
c. 582 b.c.e.c. 500 b.c.e.

Considered a mathematician, but foremost a philosopher, Pythagoras was a very important figure in mathematics, astronomy, musical theory, and in the world's history. However, little in the way of reliable record is known about his life and accomplishments. The accounts of Pythagoras inventing the musical scale, performing miracles, and announcing prophecies are probably only legend, and appear to have little historical foundation. Scholars generally agree only upon the main events in his life, and usually combine together discoveries by Pythagoras with those by his band of loyal followers.

Pythagoras established in what is now the southeastern coast of Italy a philosophical, political, and religious society whose members believed that the world could be explained using mathematics as based upon whole numbers and their ratios. Their motto was "All is number." Even the words philosophy (or "love of wisdom") and mathematics (or "that which is learned") is believed to have been first used (and defined) by the Pythagoreans.

Many Pythagorean beliefs (such as secrecy, vegetarianism, periods of food abstinence and silence, refusal to eat beans, refusal to wear animal skins, celibacy, self-examination, immortality, and reincarnation) were directed as "rules of life." The main focus of Pythagorean thought was ethics, developed primarily within philosophy, mathematics, music, and gymnastics. The beliefs of the society were that reality is mathematical; philosophy is used for spiritual purification; the soul is divine; and certain symbols possess mystical significance. Both men and women were permitted to become members. In fact, several female Pythagoreans became noted philosophers.*

*Aesara of Lucania was a Pythagorean philosopher known for her theory of the tripart soul, which she believed consisted of the mind, spiritedness, and desire.

How Pythagoreans Conceptualized Numbers

Pythagoreans believed that all relationships could be reduced to numbers in order to account for geometrical properties. This generalization originated from the observation that whenever the ratios of lengths of strings were whole numbers, harmonious tones were produced when these strings were vibrated.

The society studied properties of numbers that are familiar to modern mathematicians, such as even and odd numbers, prime and square numbers. From this viewpoint, the Pythagoreans developed the concept of number, which became their dominant principle of all proportion, order, and harmony in the universe.

The society also believed in such numerical properties as masculine or feminine, perfect or incomplete, and beautiful or ugly. These opposites, they believed, were found everywhere in nature, and the combination of them brought about the harmony of the world.

The primary belief of Pythagoreans in the sole existence of whole numbers was later challenged by their own findings, which proved the existence of "incommensurables," known today as irrational numbers . What is commonly called the "first crisis in mathematics" caused a scandal within the society, so serious that some members tried to suppress the knowledge of the existence of incommensurables.

The Pythagorean philosophy was dominated by the ideal that numbers were not only symbols of reality, but also were the final substance of real things, known as "number mysticism." They held, for example, that one is the point, two the line, three the surface, and four the solid. Seven was considered the destiny that dominates human life because infancy ends there, and also because the number was associated with the seven wandering stars. Moreover, Pythagoreans believed that maturity began at age 14, marriage occurred in the twenty-first year, and 70 years was the normal life span. Ten was identified as the "perfect number" because it was the sum of one, two, three, and four.

Pythagorean Contributions to Mathematics

The formalization of mathematics with the use of axiomatic systems was the most profound contribution that the Pythagorean society made to mathematics. Pythagoreans developed this significant concept by showing that arbitrary laws of empirical geometry could be proved as logical conclusions from a small number of axioms, or postulates. Typical of the developed axioms was "A straight line is the shortest distance between two points."

From these axioms, a number of theorems about the properties of points, lines, angles, curves, and planes could be logically deduced. These theorems include the famous Pythagorean theorem, which states that "the square of the hypotenuse of a right-angled triangle is equal to the sum of the squares of the other two sides." Another theorem states that the sum of the interior angles of any triangle is equal to the sum of two right angles.

The Pythagorean Theorem

The Pythagoreans knew that any triangle whose sides were in the ratio 3:4:5 was a right-angled triangle. Their desire to find the mathematical harmonies of all things led them to prove the geometric theorem, today named for Pythagoras. The earlier Egyptians stated this theorem as an empirical relationship and, as far as is known today, the Pythagoreans were the first to prove it.

The Pythagorean (hypotenuse) theorem states that the square of the hypotenuse of a right-angle triangle (c ) is equal to the sum of the squares of the other two sides (a and b ), shown as c 2 = a 2 + b 2. The numbers 3, 4, and 5 are called Pythagorean numbers (52 = 32 + 42, or 25 = 9 + 16). However, the Pythagoreans did not consider the square on the hypotenuse to be that number (c ) multiplied by itself (c 2). Instead, it was conceptualized as a geometrical square (C ) constructed on the side of the hypotenuse, and that the sum of the areas of the two squares (A and B ) is equal to the area of the third square (C ), as shown below.

Astronomy and the Pythagoreans

In astronomy, the Pythagoreans produced important advances in ancient scientific thought. They were the first to consider the Earth as a sphere revolving with the other planets and the Sun around a universal "central fire." Ten planets were believed to exist in order to produce the "magical" number of 10. This arrangement was explained as the harmonious arrangement of bodies in a complete sphere of reality based on a numerical pattern, calling it a "harmony of sphere." The Pythagoreans also recognized that the orbit of the Moon was inclined to the equator of the Earth, and were one of the first to accept that Venus was both the evening star and the morning star.

Even though much of the Pythagorean doctrine consisted of numerology and number mysticism, their influence in developing the idea that nature could be understood through mathematics and science was extremely important for studying and understanding the world in which we live.

see also Numbers: Abundant, Deficient, Perfect, and Amicable; Numbers, Forbidden and Superstitious; Numbers, Irrational; Numbers, Rational; Numbers, Whole; Triangle.

William Arthur Atkins with

Philip Edward Koth

Bibliography

Boyer, Carl B. A History of Mathematics, 2nd ed., New York: John Wiley & Sons, 1991.

O'Meara, Dominic J. Pythagoras Revived: Mathematics and Philosophy in Late Antiquity. New York: Clarendon Press, 1990.

Philip, James A. Pythagoras and Early Pythagoreanism. Toronto: University of Toronto Press, 1966.


MAGIC OVER MATHEMATICS

During the time of Pythagoras, most people either believed that the world could only be explained by magic or that it could not be explained at all. Thus, many people did not attempt to understand mathematics.


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Pythagoras

Pythagoras

The Greek philosopher, scientist, and religious teacher Pythagoras (ca. 575-ca. 495 B.C.) evolved a school of thought that accepted the transmigration of souls and established number as the principle in the universe.

Born on the island of Samos, Pythagoras was the son of Mnesarchus. He fled to southern Italy to escape the tyranny of Polycrates, who came to power about 538, and he is said to have traveled to Egypt and Babylon. He and his followers became politically powerful in Croton in southern Italy, where Pythagoras had established a school for his newly formed sect. It is probable that the Pythagoreans took positions in the local government in order to lead men to the pure life which their teachings set forth. Eventually, however, a rival faction launched an attack on the Pythagoreans at a gathering of the sect, and the group was almost completely annihilated. Pythagoras either had been banished from Croton or had left voluntarily shortly before this attack. He died in Metapontum early in the 5th century.

Religious Teachings

Pythagoras and his followers were important for their contributions to both religion and science. His religious teachings were based on the doctrine of metempsychosis, which held that the soul was immortal and was destined to a cycle of rebirths until it could liberate itself from the cycle through the purity of its life. A number of precepts were drawn up as inviolable rules by which initiates must live.

Pythagoreanism differed from the other philosophical systems of its time in being not merely an intellectual search for truth but a whole way of life which would lead to salvation. In this respect it had more in common with the mystery religions than with philosophy. Several taboos and mystical beliefs were taught which sprang from a variety of primitive sources such as folk taboo, ritual, and sympathetic magic and were examples of the traditional beliefs that the Greeks continued to hold while developing highly imaginative and rational scientific systems.

An important underlying tenet of Pythagoreanism was the kinship of all life. A universal life spirit was thought to be present in animal and vegetable life, although there is no evidence to show that Pythagoras believed that the soul could be born in the form of a plant. It could be born, however, in the body of an animal, and Pythagoras claimed to have heard the voice of a dead friend in the howl of a dog being beaten.

The number of lives which the soul had to live before being liberated from the cycle is uncertain. Its liberation came through an ascetic life of high moral and ethical standards and strict adherence to the teachings and practices of the sect. Pythagoras himself claimed to remember four different lives. Followers of the sect were enjoined to secrecy, although the discussions of Pythagoras's teachings in other writers proved that the injunction was not faithfully observed.

Mathematical Teachings

The Pythagoreans posited the dualism between Limited and Unlimited. It was probably Pythagoras himself who declared that number was the principle in the universe, limiting and giving shape to matter. His study of musical intervals, leading to the discovery that the chief intervals can be expressed in numerical ratios between the first four integers, also led to the theory that the number 10, the sum of the first four integers, embraced the whole nature of number.

So great was the Pythagoreans' veneration for the "Tetractys of the Decad" (the sum of 1 + 2 + 3 + 4) that they swore their oaths by it rather than by the gods, as was conventional. Pythagoras may have discovered the theorem which still bears his name (in right triangles, the square on the hypotenuse equals the sum of the squares on the other sides), although this proposition has been discovered on a tablet dating from the time of the Babylonian king Hammurabi. Regardless of their sources, the Pythagoreans did important work in systematizing and extending the body of mathematical knowledge.

As a more general scheme, the Pythagoreans posited the two contraries, Limited and Unlimited, as ultimate principles. Numerical oddness and evenness are equated with Limited and Unlimited, as are one and plurality, right and left, male and female, motionlessness and movement, straight and crooked, light and darkness, good and bad, and square and oblong. It is not clear whether an ultimate One, or Monad, was posited as the cause of the two categories.

Cosmological Views

As a result of their religious beliefs and their careful study of mathematics, the Pythagoreans developed a cosmology which differed in some important respects from the world views of their contemporaries, the most important of which was their view of the earth as a sphere which circled the center of the universe. The center of this system was fire, which was invisible to man because his side of the earth was turned from it. The sun reflected that fire; there was a counterearth closer to the center, and the other five planets were farther away and followed longer courses around the center. It is not known how much of this theory was attributable to Pythagoras himself. Later writers ascribe much of it to Philolaos (active 400 B.C.), although it circulated as a view of the school as a whole.

The systematization of mathematical knowledge carried out by Pythagoras and his followers would have sufficed to make him an important figure in the history of Western thought. However, his religious sect and the asceticism which he taught, embracing as it did a vast number of ancient beliefs, make him one of the great teachers of religion in the ancient Greek world.

Further Reading

Pythagoras left no written works. A first-rate technical book, J. A. Philip, Pythagoras and Early Pythagoreanism (1966), separates the valid from the spurious among the legends that surround Pythagoras and his views. An excellent and thorough treatment of the evidence for his life and teachings is in W. K. C. Guthrie, A History of Greek Philosophy (3 vols., 1962-1969). A good account of Pythagoras and his followers is in Kathleen Freeman, The Pre-Socratic Philosophers (1946; 3d ed. 1953), and G. S. Kirk and J. E. Raven, The Presocratic Philosophers (1962). Briefer treatments of the Pythagoreans and the intellectual currents of their time are in the standard histories of Greek literature, such as Albin Lesky, A History of Greek Literature (trans. 1966), or in accounts of Greek philosophy, such as John Burnet, Greek Philosophy (1914). □

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Pythagoras

Pythagoras

Born: c. 575 b.c.e.
Samos, Greece
Died: c. 495 b.c.e.
Metapontum

Greek philosopher, scientist, and religious scholar

The Greek philosopher, scientist, and religious teacher Pythagoras developed a school of thought that accepted the passage of the soul into another body and established many influential mathematical and philosophical theories.

Early life

Born on the island of Samos, off Greece, in the Mediterranean Sea, Pythagoras was the son of Mnesarchus. Little is known about his early life. After studying in Greece, he fled to southern Italy to escape the harsh rule of Polycrates (died c. 522 b.c.e.), who came to power about 538 b.c.e. Pythagoras is said to have traveled to Egypt and Babylon during this time.

Pythagoras and his followers became politically powerful in Croton in southern Italy, where Pythagoras had established a school for his newly formed sect, or group of followers. It is probable that the Pythagoreans took positions in the local government in order to lead men to the pure life that was directed by their teachings. Eventually, however, a rival group launched an attack on the Pythagoreans at a gathering of the sect, and the group was almost completely destroyed. Pythagoras either had been forced to leave Croton or had left voluntarily shortly before this attack. He died in Metapontum early in the fifth century b.c.e.

Religious teachings

Pythagoras and his followers were important for their contributions to both religion and science. His religious teachings were based on the doctrine (teaching) of metempsychosis, which teaches that the soul never dies and is destined to a cycle of rebirths until it is able to free itself from the cycle through the purity of its life.

Pythagoreanism differed from the other philosophical systems of its time in being not merely an intellectual search for truth but a whole way of life which would lead to salvation, or to be delivered from sin. An important part of Pythagoreanism was the relationship of all life. A universal life spirit was thought to be present in animal and vegetable life, although there is no evidence to show that Pythagoras believed that the soul could be born in the form of a plant. It could be born, however, in the body of an animal, and Pythagoras claimed to have heard the voice of a dead friend in the howl of a dog being beaten.

Mathematical teachings

The Pythagoreans presented as fact the dualism (that life is controlled by opposite forces) between Limited and Unlimited. It was probably Pythagoras who declared that numbers could uncover the secrets of the universe, limiting and giving shape to matter (anything that takes up space). His study of musical intervals, leading to the discovery that the chief intervals can be expressed in numerical ratios (relationships between numbers) between the first four integers (positive whole numbers), also led to the theory that the number ten, the sum of the first four integers, embraced the whole nature of number.

So great was the Pythagoreans' respect for the "Tetractys of the Decad" (the sum of 1 + 2 + 3 + 4) that they swore their oaths (promises) by it rather than by the gods, as was normal during his day. Pythagoras may have discovered the theorem which still bears his name (in right triangles [triangle with one angle equal to 90 degrees], the square on the hypotenuse equals the sum of the squares on the other sides), although this proposal has been discovered on a writing stone dating from the time of the Babylonian king Hammurabi (died c. 1750 b.c.e.). Regardless of their sources, the Pythagoreans did important work in extending the body of mathematical knowledge.

As a more general outline, the Pythagoreans presented the two contraries (opposites), Limited and Unlimited, as ultimate principles, or truths. Numerical oddness and evenness are equated with Limited and Unlimited, as are one and plurality (many), right and left, male and female, motionlessness and movement, straight and crooked, light and darkness, and good and bad. It is not clear whether an ultimate One, or Monad, was presented as the cause of the two categories.

Cosmological views

The Pythagoreans, as a result of their religious beliefs and careful study of mathematics, developed a cosmology (dealing with the structures of the universe) which differed in some important respects from the world views at the time, the most important of which was their view of the Earth as a sphere which circled the center of the universe. It is not known how much of this theory was credited to Pythagoras himself.

The mathematical knowledge carried out by Pythagoras and his followers would have been enough to make him an important figure in the history of Western thought. However, his religious sect and the self-discipline and dedication which he taught, embracing as it did a vast number of ancient beliefs, make him one of the great teachers of religion in the ancient Greek world.

For More Information

Fey, James. Looking for Pythagoras: The Pythagorean Theorem. White Plains, NY: Dale Seymour Publications, 1997.

Philip, J. A. Pythagoras and Early Pythagoreanism. Toronto: University of Toronto Press, 1966.

Strohmeyer, John, and Peter Westbrook. Divine Harmony: The Life and Teachings of Pythagoras. Berkeley, CA: Berkeley Hills Books, 1999.

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Pythagoras

PYTHAGORAS

PYTHAGORAS. Pythagoras (c. 580c. 580 B.C.E.) was a Greek mathematician, philosopher, and mystic. He wrote nothing himself, so his ideas survive through the writings of others, including Aristotle. Many people are familiar with him as the mathematician who formulated the Pythagorean theorem in geometry that relates the lengths of the sides in a right triangle. Others know him as a mystic and the first person known to be motivated by moral and philosophical concerns to adopt a vegetarian diet.

The schools and societies Pythagoras founded in the southern Italian area of Magna Graecia flourished for a while, and they developed and spread many of his concepts, which were later adopted and expanded by others. These concepts include bodily humors (evident in modern descriptions of melancholic and phlegmatic personalities), a tripartite soul, reincarnation, and the numerical ratios that determine the concordant intervals of the musical scales. Permeating all of his thoughts was the idea that all things are numbers. Numbers (individuals, groups, and series) were imbued with mystical properties that were carefully guarded and only shared among initiates to the Pythagorean schools founded by him or his disciples.

Pythagoras and his followers practiced one of the first recorded diets known as vegetarianism. He advocated a diet devoid of the flesh of slaughtered animals partially because he felt food influenced the distribution of the bodily humors and thereby the health of the individual and partially because it would prevent the killing of a reincarnated individual and its transmigrated soul. Up until the late nineteenth century nonmeat eaters were generally known as "Pythagoreans."

Pythagoras is also alleged to have admonished his disciples to abstain from eating beans. Ancient and medieval writers ingeniously ascribed this pronouncement to the belief that beans contained or transmitted souls. The Greek phrase supporting this gastronomic recommendation, however, could also be construed to imply that his followers should avoid politics. Black and white beans were used as counters in voting in Magna Graecia. The school Pythagoras founded there became actively involved in the populist political views that gained ascendancy in the town of Kroton, where he lived for many years. Later an opposing aristocratic party gained control of the city and banished him and his followers for their political views and activism. Pythagoras died in exile. His supposed warning to "abstain from beans" is therefore thought to have meant "avoid politics." Alternatively he may have realized that eating undercooked broad (fave) beans (Vica faba vulgaris), a common food of the Mediterranean region, produced a severe hemolytic anemia (favism) in some people. Interestingly the same mutant gene that makes people sensitive to favism also increases their resistance to the malarial parasite, possibly accounting for the widespread presence of the mutant gene in regions with endemic malaria.

See also Greece, Ancient ; Vegetarianism .

BIBLIOGRAPHY

Bamford, Christopher, ed. Homage to Pythagoras. Hudson, N.Y.: Lindisfarne Press, 1994.

Gorman, Peter. Pythagoras. London: Routledge and Kegan Paul, 1979.

Spencer, Colin. The Heretic's Feast: A History of Vegetarianism. London: Fourth Estate, 1993.

Walters, Kerry S., and Lisa Portmess, eds. Ethical Vegetarianism: From Pythagoras to Peter Singer. Albany: State University of New York Press, 1999.

Mikal E. Saltveit

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Pythagoras

Pythagoras (pĬthăg´ərəs), c.582–c.507 BC, pre-Socratic Greek philosopher, founder of the Pythagorean school. He migrated from his native Samos to Crotona and established a secret religious society or order similar to, and possibly influenced by, the earlier Orphic cult. We know little of his life and nothing of his writings. Since his disciples came to worship him as a demigod and to attribute all the doctrines of their order to its founder, it is virtually impossible to distinguish his teachings from those of his followers. The Pythagoreans are best known for two teachings: the transmigration of souls and the theory that numbers constitute the true nature of things. The believers performed purification rites and followed moral, ascetic, and dietary rules to enable their souls to achieve a higher rank in their subsequent lives and thus eventually be liberated from the "wheel of birth." This belief also led them to regard the sexes as equal, to treat slaves humanely, and to respect animals. The highest purification was "philosophy," and tradition credits Pythagoras with the first use of the term. Beginning with the discovery that the relationship between musical notes could be expressed in numerical ratios (see Greek music), the Pythagoreans elaborated a theory of numbers, the exact meaning of which is still disputed by scholars. Briefly, they taught that all things were numbers, meaning that the essence of things was number, and that all relationships—even abstract ethical concepts like justice—could be expressed numerically. They held that numbers set a limit to the unlimited—thus foreshadowing the distinction between form and matter that plays a key role in all later philosophy. The Pythagoreans were influential mathematicians and geometricians, and the theorem that bears their name is witness to their influence on the initial part of Euclidian geometry. They made important contributions to medicine and astronomy and were among the first to teach that the earth was a spherical planet, revolving about a fixed point. At the end of the 5th cent. BC the Pythagoreans were forced to flee Magna Graecia when people grew enraged at their interference with traditional religious customs; many were killed. A short-lived Neo-Pythagoreanism developed at the beginning of the Christian era; it borrowed some elements from Jewish and Hellenistic thought and greatly emphasized the mystical element in Pythagorean ideas.

See biographies by P. Gorman (1978) and T. Stanley (1988); D. J. O'Meara, Pythagoras Revived: Mathematics and Philosophy in Late Antiquity (1989).

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Pythagoras

Pythagoras (c.580–500 bc), Greek philosopher; known as Pythagoras of Samos. Pythagoras sought to interpret the entire physical world in terms of numbers, and founded their systematic and mystical study; he is best known for the theorem of the right-angled triangle. His analysis of the courses of the sun, moon, and stars into circular motions was not set aside until the 17th century.

Pythagoras also founded a secret religious, political, and scientific sect in Italy: the Pythagoreans held that the soul is condemned to a cycle of reincarnation, from which it may escape by attaining a state of purity.
Pythagoras' theorem the theorem attributed to Pythagoras that the square on the hypotenuse of a right-angled triangle is equal in area to the sum of the squares on the other two sides.
Pythagorean letter the Greek letter Y, used by Pythagoras as a symbol of the two divergent paths of virtue and of vice.
Pythagorean system the system of astronomy proposed by Pythagoras, in which all celestial bodies, including the earth, were held to revolve around a central fire (not the sun, but presumably identified with the sun, resulting in the system being assumed identical with the Copernican system).

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Pythagoras of Rhegium

Pythagoras of Rhegium (pĬthăg´ərəs rē´jəm), fl. 5th cent. BC, Greek sculptor. In a signature on a pedestal at Olympia he declares himself a Samian, but the period of his training and work belongs to Rhegium, Italy. As no works are known that can with certainty be identified as his, his fame depends upon the statements of those who saw his statues and named them. They were mainly of athletes and mark a step in the transition between the archaic and the classical styles. He is said to have been the first to represent hair, veins, and muscles naturally and the first to aim at rhythm and symmetry in sculpture. Among his statues were a portrait of the boxer Euthymus, a figure of a man singing to a lyre, and one of Apollo shooting the Python with his arrows.

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Pythagoras

Pythagoras (c.580–500 bc) Greek philosopher and founder of the Pythagorean school. The Pythagoreans were bound to their teacher by rigid vows and were ascetic in their way of life. They believed in the transmigration of souls and that numbers and their interrelationships constitute the true nature of the universe. Pythagoras is credited with advances in mathematics and geometry, medicine and philosophy. The famous theorem, that the square of the hypotenuse of a right-angled triangle equals the sum of the squares of the other two sides, is named after him, but was already known to the Egyptians and Babylonians. His school was suppressed at the end of the 6th century, but the Romans revived its doctrines c.500 years later.

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Pythagoras

PythagorasArras, embarrass, harass •gynandrous, polyandrous •Pancras • charas • Tatras • disastrous •ferrous • leprous • ambidextrous •Carreras, mayoress •scabrous •cirrus, Pyrrhus •chivalrous •citrous, citrus •ludicrous • tenebrous •Cyrus, Epirus, papyrus, virus •fibrous • hydrous • Cyprus •retrovirus • monstrous •brachiosaurus, brontosaurus, canorous, chorus, Epidaurus, Horus, megalosaurus, pelorus, porous, sorus, stegosaurus, Taurus, thesaurus, torus, tyrannosaurus •walrus •ochrous (US ocherous) •cumbrous • wondrous • lustrous •Algeciras, Severus •desirous •Arcturus, Epicurus, Honduras •barbarous • tuberous • slumberous •Cerberus • rapturous •lecherous, treacherous •torturous • vulturous • Pandarus •slanderous • ponderous •malodorous, odorous •thunderous • murderous •carboniferous, coniferous, cruciferous, melliferous, odoriferous, pestiferous, somniferous, splendiferous, umbelliferous, vociferous •phosphorous, phosphorus •sulphurous (US sulfurous) •Anaxagoras, Pythagoras •clangorous, languorous •rigorous, vigorous •dangerous • verdurous •cankerous, cantankerous, rancorous •decorous • Icarus • valorous •dolorous • idolatrous •amorous, clamorous, glamorous •timorous •humerus, humorous, numerous •murmurous • generous • sonorous •onerous • obstreperous • Hesperus •vaporous • viviparous • viperous •Bosporus, prosperous •stuporous • cancerous •Monoceros, rhinoceros •sorcerous • adventurous • Tartarus •nectarous • dexterous • traitorous •preposterous • slaughterous •boisterous, roisterous •uterus • adulterous • stertorous •cadaverous • feverous •carnivorous, herbivorous, insectivorous, omnivorous •Lazarus

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Pythagoras

PYTHAGORAS

PYTHAGORAS . The ancient tradition presents different images of Pythagoras (c. 570 bcec. 500 bce) that hardly square with one another: philosopher and initiator of rational inquiry, scientist and mathematician, politician and lawgiver, and religious wonderworker and leader of a sect of initiates. Surely he was an extraordinary personality and a charismatic chief, venerated by his followers and desecrated by his opponents. Soon he became a legend, whose historical nucleus is difficult to ascertain. The very nature of the association he founded is consequently controversial: it is mainly described as a philosophical school where scientific inquiries were practiced, as a political party, or as a religious confra-ternity.

The main sources on Pythagoras, while plentiful, are late and rarely impartial; for the most part they are cast in the distorting light of hostile polemic or religious veneration. Whether Pythagoras left any writing was in ancient times already controversial and is still debatable. Original works by him, if there were any, were soon lost. In addition, there are no extant writings from ancient Pythagoreans. Pythagorean material is mainly constituted by reports whose reliability is uncertain and by apocryphal writings, which were composed beginning in the Hellenistic age and gradually increased until a remarkable amount existed.

Few details of Pythagoras's life are definitively known. He was born to Pythais and Mnemarchos (or Mnesarchos), a gem-engraver or merchant, on the Ionian island of Samos in 571 or 570 bce. He lived there until 532 or 531, when he migrated, perhaps to escape the tyranny of Policrates, to the Achaean city of Croton in Magna Graecia (southern Italy), soon after the defeat of the city by its neighbors, the Locrians. By his teaching, which Pythagoras gave to citizens through public speeches, he is said to have converted the city from luxury to temperance. In Croton he founded an association, some of whose members came to exercise a leading role in the government of the city. During this period Croton extended its power over many cities of southern Italy, defeating the rival Sibaris in 510. There followed a period of internal struggle; anti-Pythagorean movements culminated in a burning of the houses of some Pythagoreans, where Pythagoras himself perished. Other sources have him dying, probably at about the age of eighty, in Metapontum, where he had retired having predicted the events.

Pythagoras's image as wonderworker is variously attested. His followers, who avoided pronouncing his name, considered him a god, or at least a semidivine person ("among rational beings one is god, another one man, the third like Pythagoras"), while some of his detractors depicted him as "chief of swindlers" and a charlatan. The ancient sources connect him with the Orient and its wisdom. He journeyed in Egypt, where he was told the secret lore of the priests; he also had contacts with the Phoenicians, the Chaldeans, and the Magi in Babylonia, and was initiated into their mysteries. Among his teachers were Pherecydes of Syros and Zaratas (Zoroaster), by whom he is said to have been purified and instructed in cosmology. The connections of Pythagoras with Apollo are basic; he was called the Hyperborean Apollo by the Crotoniates, and he revealed his golden thigh to the Hyperborean Abaris, a priest of Apollo, who identified him as the god. According to other reports Pythagoras was born from Apollo and Pythais and was the god's prophet among humans.

Pythagoras was credited with supernatural faculties and extraordinary mental powers, as is shown by miraculous tales that attribute to him the capacity of predicting future events, healing diseases, being simultaneously in different places, and taming animals. Still more impressive than his magical relationship with the natural world are his connections to the underworld and the afterlifehe could remember his past lives and journeyed to Hades. Hence the much discussed interpretation that sees him mainly as a sort of shaman.

The sources unanimously ascribe to Pythagoras the belief in transmigration and reincarnation of the soul (metempsychosis), whose origin can be traced back to Indo-Iranian cultures. Though most details remain unknown, it is safe to assume that for Pythagoras the soul (psyche ) was immortal, being not merely the life or spirit of the body, but independent, and in many ways an opposing force to the physical self; it enters other human bodies and certain species of animals, thereby experiencing the cycle of punishments and rewards that stems from one's conduct in life. The metempsychosis was in fact connected with an eschatology, whose traces survive in the so-called akousmata (things heard) or in symbols.

It seems certain that Pythagoras developed a practical way of life (bios ) based upon such oral maxims and instructions. Perhaps Aristotle originated a threefold distinction of them: what is ("what is the oracle of Delphi? the tetractys, that is, the harmony in which the Sirens sing"); what most ("what is the wisest thing? number"); and what should or should not be done (e.g., one should not travel by the main roads). Some symbols seem bizarre and difficult to explain: do not stir the fire with a knife; do not wear the image of a god on a ring; do not urinate towards the sun. For this reason, in the fourth century bce some authors provided an allegorical interpretation, which purported to explain their hidden meaning (e.g., do not stir the fire would mean "do not excite an angry man"; abstinence from beans would mean "the prohibition of sexual intercourse"). The original connection is, however, with sacrificial ritual and related purity: do not eat the heart, do not sacrifice the white cock, do not dip the hand into holy water. Some of the symbols are undoubtedly related to mystery cults, including those that command silence and fasting; prescribe the practices of sacrifice, entering the temple barefoot, and wearing a linen garment; or forbid picking up food that has fallen to the earth. Other symbols exhibit connections to an eschatology whose details are little known: what, for example, are the isles of the blest? sun and moon. Bread is not to be broken, because it contributes to the judgment of Hades. Symbols are also passwords for recognition of the initiates by their fellows and by the gods.

Reports about dietetic instructions are also variegated and sometimes self-contradictory. A logical corollary of metempsychosis was total abstinence from meat (including some fishes). Such a radical prohibition clearly contrasted with official cults of Greek religion, where animal sacrifice was central, and was thus incompatible with political offices. The original prohibition may have come to be restricted to animals into which the souls of human beings were supposed to enter, and members of the society may have been allowed to eat animals that could be sacrificed. Other reports limit the prohibition against eating meat to particular parts of the animal, such as the heart or womb. Some sources have Pythagoras making only inanimate offerings and worshiping at Delos at the altar of Apollo genetor, where sacrificing animals was forbidden. Most notorious is the beans taboo (vicia faba ), which has been variously explained, though the main connection is, once again, with metempsychosis: beans represent the gates of Hades, and through beans souls return to earth for reincarnation. Some explanations point to supposed similarities between humans and beans.

The connections with Orphism are important with regard to eschatology. Some traditions made Pythagoras a direct disciple of Orpheus, attributing to him similar powers, such as mastery over animals. Both movements were concerned with salvation of the soul and the afterlife; yet, while Orphism seemingly assured salvation by simple offerings and vows, Pythagoreanism, as a sort of puritanism, was centered on an irreprehensible conduct of life. It is highly probable that Pythagoras and Pythagoreans composed or commented on Orphic literature, where theogonies and cosmogonies were interpreted and further developed.

Ancient reports describe the internal organization of the association in terms of a monastic brotherhood, whose adherents shared their goods and followed strict rules of community life. New disciples underwent a rigid process of admission with a probationary period of five years of silence before final initiation. Close fraternal ties bound the members to each other ("common are the things of friends"). Much discussed is the distinction made by later sources between acusmatics (members who received only basic, undemonstrated tenets) and mathematics (truly philosophizing members). The distinction points not to different degrees of membership but to a deepened interest in scientific inquiry by later groups compared to the former, prescientific wisdom based upon the akousmata. Many of these tracts, as with the community of gods, are possibly later projections of a monastic ideal of life.

Nevertheless, it is undeniable that the Pythagorean society was something more than an ordinary political club (hetairia ) or a philosophical group; it was rather an association whose adherents were tied to the way of life expressed by the symbols. Being a Pythagorean meant not so much professing a definite philosophical doctrine or practicing scientific inquiry as following a certain sort of life. This will also explain Pythagoras's image as a lawgiver and reformer, a founder of a politically oriented educational program, and a promoter of social concord and moral authority, which had a strong influence on the political life of many cities in southern Italy. Pythagoras's actions in Croton, in fact, seem to have consisted in political advice rather than in direct involvement in the government; his supposed activity as a lawgiver has left no concrete traces, though his influence on politics is undeniable.

According to Diogenes Laertios (third century ce) Pythagoras was the originator of Italian philosophy, the first to use the term philosophy, and the first to call himself a philosopher, although these are probably later projections of the ideal of contemplative life. More questionable is whether Pythagoras can be considered a philosopher at all. Plato's Academy played a fundamental role in the transmission of Pythagorean philosophy, though in a profoundly transfigured form. Academics attributed to ancient Pythagoreanism doctrines they themselves had worked out, as that of two principles (one and indefinite dyad). The most reliable source for Pythagorean philosophy remains therefore, despite possible distortions, Aristotle, who strives to avoid confusion between Platonism and Pythagoreanism. Based on a written source, Aristotle sketches the philosophy of the "so-called Pythagoreans" (he dealt with Pythagoras's mirabilia in the lost works), which is a later development, maybe due to Philolaos. In any case, it cannot be ruled out that Pythagoras himself had worked out some general philosophical ideas about number as principle and cosmic harmony. Plato alludes to him, recalling a novel Prometheus that gave to humankind a divine doctrine: all things consist of oneness and multiplicity, limitedness and unlimitedness, in close analogy with the system described by Aristotle.

Both Plato and the Pythagoreans regard number as the principle of reality; for the latter, however, numbers are not separate entities, but the things themselves. Having applied mathematics, they discovered affinities between numbers and existing things and assumed that the elements of numbers were the elements of the things; heaven itself is harmony and number. Elements of the number are the even (unlimited) and the odd (limited). Limited and unlimited are then the ultimate principles; from them rises the odd-even "one," which is a principle of number. Here cosmogony and arithmogony are intermingled: once the "one" came into being, the unlimited, which was outside, was breathed in as void by the limited, thereby separating things from one another. The Pythagorean cosmos includes, along with fire at the center, the invisible "counter-earth," the earth, the moon, the sun, the five planets, and the fixed stars. Cosmic harmony is explained by numerical relations that determine the concordant intervals of the scale (2:1, the octave; 3:2, the fifth; 4:3, the fourth). Of paramount importance is, among other "sacred" numbers, the tetraktys, or decad, containing the basic ratios (1+2+3+4=10). Other Pythagoreans listed ten basic principles in a table of oppositions (limited/unlimited, oneness/multiplicity, odd/even, square/oblong, good/bad, male/female, right/left, at rest/moving, straight/crooked, light/darkness). The representation of numbers by arrangements of pebbles was based on the correspondence of odd and limited.

Pythagoras emerges from many reports as a mathematician and a scientist. He is credited with the discovery of the celebrated theorem of musical harmony and its basic intervals, together with the construction of musical instruments. Testimonies that attribute to him the theoretical study of geometry or the discovery of irrational magnitudes are not reliable and are deeply influenced by the tradition of Platonism. Yet it is reasonable that, having come of age in the Ionia of the sixth century bce, Pythagoras could not ignore the major scientific achievements of his time, which possibly are mirrored in his doctrines.

Pythagoreanism soon spread outside Croton. Iamblichus's catalogue, which may date back to Aristoxenos in the fourth century bce, lists 235 Pythagoreans from different cities, with Croton, Metaponto, Locri, and Tarent playing prominent roles. After the anti-Pythagorean strife the only center in Magna Graecia where Pythagoreanism survived was Tarent, where Archytas held the office of strategos. After its extinction in the fourth century bce, Pythagoreanism survived as a philosophy, inspiring individual personalities who continued to lead a Pythagorean way of life. Beggars-Pythagorists appear in the Middle Comedy (fourth century bce); they live ascetically, practicing silence and following such dietetic rules as meat abstinence or intensive fasting. However, the existence of Pythagorean groups in Greece cannot be clearly documented.

Much debated is whether Pythagorean communities survived elsewhere during the Hellenistic age. An interest in Pythagoreanism of a literary or antiquarian nature is well attested, which explains the production in the third to second centuries bce of apocryphal writings attributed to Pythagoras (the Golden Verses being the best known) or ancient Pythagoreans, whose place of origin remains controversial (candidates are Rome, Alexandria, and southern Italy). A revival of Pythagoreanism is attested in the first century bce in Rome, where Nigidius Figulus attempted to revive the antica disciplina. There is also somewhat questionable evidence in Rome of the existence of circles and religious sects that referred to Pythagoreanism. In addition, a renewed philosophical interest became visible in Alexandria of Egypt, where the circle of the Pythagorean Platonic Eudoros possibly contributed to the production of apocrypha. Between the first century bce and the second century ce, a number of authors arose who explicitly defined themselves or came to be defined as Pythagoreans. They include Moderatus, Nicomachus, and Numenius, all of whom profess doctrines that are substantially Platonic, and some of whom are supposed to have adopted a Pythagorean bios. At this point, boundaries between Platonism and Pythagoreanism become very unclear. More clear is the case of Apollonios of Tiana, a wonderworker who explicitly purported to revive the Pythagorean life and presented himself as Pythagoras redivivus. Such Neoplatonic writers as Iamblichus and Porphyry accomplished in their biographies the apotheosis of Pythagoras, until Pythagoreanism completely merged into Neoplatonism.

See Also

Neoplatonism; Orpheus; Plato; Platonism.

Bibliography

The best sourcebook in English containing texts and fragments related to Pythagoras is Cornelia J. de Vogel's Greek Philosophy: A Collection of Texts, vol. 1, Thales to Plato (Leiden, 1950). The most complete collection of testimonies is M. Timpanaro Cardini, I Pitagorici: Testimonianze e frammenti, 3 vols. (Florence, Italy, 19581964). Two excellent background works, which place Pythagoras within the context of Greek religious thought, are E. R. Dodds's The Greeks and the Irrational (Berkeley, Calif., 1951) and Walter Burkert's Greek Religion (Cambridge, Mass., 1985). Edwin L. Minar's Early Pythagorean Politics in Practice and Theory (Baltimore, 1942) offers a summary of the Pythagorean hetaireia. Francis M. Cornford's "Mysticism and Science in the Pythagorean Tradition," Classical Quarterly 16 (1922): 137150, is very good on defining the religious vision of Pythagoras, less so on Pythagorean atomism. J. E. Raven gives a controversial account of the development of Pythagorean thought in the fifth century in Pythagoreans and Eleatics (Cambridge, U.K., 1948). W. K. C. Guthrie's account of Pythagoras in A History of Greek Philosophy, vol. 1, The Earlier Presocratics and Pythagoreans (Cambridge, U.K., 1962), can be considered the standard general assessment on the subject.

Other important works on Pythagoras are James A. Philip's Pythagoras and Early Pythagoreanism (Toronto, 1966) and above all Walter Burkert's Lore and Science in Ancient Pythagoreanism (Cambridge, Mass., 1972). Philip sees little evidence for a religious organization in early Pythagoreanism. Burkert, conversely, offers the most detailed analysis of the Pythagorean religious understanding and practices. Charles H. Kahn's "Pythagorean Philosophy before Plato," in The Pre-Socratics, edited by Alexander P. D. Mourelatos (Garden City, N.Y., 1974), pp. 161185, provides a balanced overview of Pythagoras's religious and scientific holdings. Leonid Zhmud, Wissenschaft, Philosophie, und Religion im frühen Pythagoreismus (Berlin, 1997), while reappraising the image of Pythagoras as scientist and philosopher, is a strong critic of the "shamanistic" interpretation. Critical surveys of the whole phenomenon of Pythagoreanism are Bruno Centrone, Introduzione ai Pitagorici (Rome, 1996) and Christoph Riedweg, Pythagoras: Leben, Lehre, Nachwirkung (Munich, 2002). A complete collection of pseudopythagorica is Holger Thesleff, ed., The Pythagorean Texts of the Hellenistic Period (Åbo, Finland, 1965). See also Thesleff's An Introduction to the Pythagorean Writings of the Hellenistic Period (Åbo, Finland, 1961). Constantinos Macris's "Pythagore, un maître de sagesse charismatique de la fin de la période archaïque," in Carisma profetico: Fattore di innovazione religiosa, edited by Giovanni Filoramo (Brescia, Italy, 2003), pp. 243289, uses the tools of the sociology of religion with historical and philological accuracy to interpret Pythagoras as a charismatic "master of wisdom" (a rich and well-chosen bibliography is included).

Bruno Centrone (2005)

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Pythagoras

Pythagoras

Circa 580-Circa 500 b.c.e

Philosopher and mathematician

Sources

Croton . The details of Pythagoras’s life are lost in a blur of legends invented by his disciples. Pythagoras was born sometime around 580 b.c.e. in the eastern Aegean, and after travels in Egypt and Babylonia reportedly settled in the Greek city of Croton in southern Italy. There he lived as the head of a religious and philosophical community rumored to be nearly three hundred in number. It eventually gained wide political influence in Croton, but after twenty years of virtual rule the group devolved into factions. In the ensuing violence many members were killed while others fled into exile. Pythagoras sought refuge in a temple in the neighboring city of Metapontum, and there he starved to death.

Contributions. Although none of his writings have survived, Pythagoras is credited with the theory of the functional significance of numbers. Nonetheless other discoveries attributed to him such as the incommensurability of the side and diagonal of a square and the theorem for right triangles were probably developed by some of his later followers. Overall, his contribution to Greek intellectual tradition has more to do with mystical wisdom than it does with scientific scholarship.

Listeners and Learners . Upon his death, the cult split into two groups: the Akousmatikoi (Listeners), who cultivated his religious, mystical, and ethical teachings, and the Mathêmatikoi (Learners), who continued his work in mathematics, astronomy, and music. The former were all sworn to observe strict rules of purity and abstinence, along with rigid dietary prohibitions that seem to have included vegetarianism, based on a belief in the reincarnation of the soul after death (the meat a person eats might once have been a man). According to a roughly contemporary anecdote, Pythagoras once stopped someone from beating a small dog because he recognized in its barking the voice of a dead friend, who had obviously been reborn as the puppy.

Beliefs. Some of the rules of behavior of the Akousmatikoi cult reportedly included never stir a fire with something made of iron and always put your right shoe on first, but always wash your left foot before you wash the right one. Meanwhile, the Mathêmatikoi believed that the fundamental principles of the universe were arranged in ten contrary pairs: limit and unlimited; odd and even; one and many; right and left; male and female; rest and motion; straight and curved; light and dark; good and bad; and square and oblong.

Sources

Walter Burkert, Lore and Science in Ancient Pythagoreanism (Cambridge, Mass.: Harvard University Press, 1972).

Peter Kingsley, Ancient Philosophy, Mystery, and Magic: Empedocles and Pythagorean Tradition (Oxford: Clarendon Press; New York: Oxford University Press, 1995).

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Pythagoras

Pythagoras

c. 550-480 b.c.

Greek philosopher and mathematician who founded a philosophical and religious movement based on the belief that all natural systems are determined by mystical numerical relationships. He and his followers established the close relationship between science and mathematics and strongly influenced Plato. The Pythagoreans attempted to apply their principles to all areas of the natural world; it has been suggested that they may have written the Hippocratic Oath.

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Pythagoras

Pythagoras

569?-475? b.c.

Greek Philosopher and Mathematician

Despite the fact that he is famous for the discovery of the theorem that bears his name, Pythagoras did not view himself primarily as a mathematician; nor did the members of the society he founded, whose principles addressed nonscientific subjects such as reincarnation, or metempsychosis. Yet Pythagoras and his followers, the Pythagoreans, viewed the concept of the number as fundamental to the universe, and with their focus on numerical properties virtually inaugurated the serious study of mathematics in the West.

The son of Mnesarchus, a merchant from Tyre (now in Lebanon), and his wife Pythias, Pythagoras grew up in Samos, Ionia, on what is now the western part of Turkey. His father's profession gave him reason to travel widely, and apparently the boy accompanied him on trips as far away as Italy. During his youth, Pythagoras fell under the influence of several great teachers, most notably Thales (625?-547? b.c.).

Around the year 535 b.c., Pythagoras visited Egypt, where he became interested in various mystical rites while studying at the temple of Diospolis. Following the Persian invasion of Egypt in 525 b.c. he was taken back to Babylon as a prisoner, which placed him in that city alongside the Israelites during the Captivity. Like the Israelites, who adopted the idea of Satan from Zoroastrian scriptures, Pythagoras came under the influence of Zoroastrianism and the much older religion of the Magi.

By 520 b.c., he had made his way back to Samos, where he founded a school based on his emerging mystical worldview. It appears that Samian students were not interested in his rather unusual, Egyptian-influenced teaching style, and in an effort to avoid being forced into a life of public service in his hometown, Pythagoras used this lack of interest as an excuse to move to Italy. In 518 b.c. he settled in Croton, at the eastern tip of the peninsula's boot heel, where he established the Pythagorean society.

His followers, a group that grew steadily after his arrival in Croton, called themselves mathematikoi. They believed that mathematics was at the heart of reality, and that symbols possessed a mystical significance that drew the human closer to the divine. One of the world's first secret societies, the Pythagoreans lived communally, practiced vegetarianism, and practiced vows of secrecy and loyalty. Aside from all their other unusual qualities, they stood apart from much of the ancient world in that women were allowed to enjoy full participation in their society, and did so as full intellectual equals of men.

From a mathematical standpoint, one of the most significant contributions made by Pythagoras was his treatment of number as an abstract entity separable from all specifics. Perhaps for the first time, 2 was just two—not two pebbles or two horses or two ships. As for his famous theorem—that the square of a right triangle's hypotenuse is equal to the sum of the squares of its other two sides—it appears that the Babylonians a millennium before him recognized this principle, and that Pythagoras was simply the first to prove the theorem. He and his followers also showed that the sum of the angles in a triangle is equal to two right angles.

The Pythagoreans—and specifically Hippasus of Metapontum (fl. c. 500 b.c.)—are also credited with the discovery of irrational numbers, or infinite decimals with no indefinitely repeating digits. The idea of an irrational, however, went against Pythagorean precepts, which maintained at all things can be expressed in terms of whole numbers, or the ratios of whole numbers. Similar reasoning led to the Pythagorean rejection of concepts such as the infinite and the infinitesimal. These positions highlight the fact that though he and his followers made many mathematical discoveries, Pythagoras—who believed that each number had a "personality"—was at heart a mystic and not a mathematician.

In 508 b.c., a noble named Cylon tried to force his way into the society, and Pythagoras rejected him because he did not regard Cylon as having a pure interest in mathematics for its own sake. Cylon then set out to destroy the society, and Pythagoras fled to the Italian city of Metapontum, where according to some accounts he committed suicide rather than allow Cylon to take over the society he had founded. In later years, the Pythagoreans became a powerful force in southern Italy, so much so that in the mid-fifth century b.c. they came under severe attack from enemies, and had to flee to Thebes and other cities in Greece.

JUDSON KNIGHT

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Pythagoras

Pythagoras

c. 580 b.c.e.–c. 500 b.c.e.

Philosopher
Scientist
Mathematician

Studied Mathematics, Science, and Philosophy.

Pythagoras, son of Mnesarchus, was born in Samos in the mid-sixth century b.c.e., and was said to have died as a refugee in Metapontum, Italy. One of the most influential figures in Greek intellectual history, Pythagoras was both a philosopher of religion and a scientist, yet very little is known about the man himself; there are no written records. It is therefore impossible to tell how much of the Pythagorean tradition in mathematics, music, and science can be traced back to the man himself and his early followers, called Pythagoreans. As a philosopher, Pythagoras is said to have introduced the "doctrine of transmigration of souls"; as a mathematician, Pythagoras is credited with, among other discoveries, the "Pythagorean Theorem" in geometry.

Discovered Musical Consonances.

He also discovered the musical consonances, represented by the mathematical ratios of 2:1, 3:2, and 4:3 (the octave, perfect fifth, and perfect fourth). According to Pythagoras, the consonances of a fourth, fifth, and octave were models of harmonia (harmony); sounds and rhythms, which were ordered by numbers, exemplified and corresponded to the fitting-together (harmony) of the cosmos. Thus, the ratios, which were displayed in the tetractys (an equilateral triangle composed of ten dots), carried religious, as well as scientific, significance for early followers. The scientific application of Pythagorean mathematics appears early, in the Sectio Canonis (Division of the Canon), dating to fourth–third century b.c.e., and the acoustic notions of the Pythagoreans—that the same numerical laws that governed the universe also governed music and, by extension, the soul—profoundly influenced Plato, Aristotle, and the later Greek and Latin music theorists. The treatment of Pythagorean theories of consonance and harmonics in the Manuale harmonices of Nicomachus (fl. 100–150 c.e.) and the Harmonika of Claudius Ptolemy (fl. 127–148 c.e.) represents the persistence of the Pythagorean tradition in later Greek music theory.

sources

Andrew Barker, ed., Greek Musical Writings (Cambridge: Cambridge University Press, 1984–1989).

Walter Burkert, Lore and Science in Ancient Pythagoreanism. Trans. Edwin L. Minar Jr. (Cambridge: Harvard University Press, 1972).

Charles H. Kahn, Pythagoras and the Pythagoreans: A Brief History (Indianapolis, Ind.: Hackett, 2001). Thomas J. Mathiesen, Apollo's Lyre: Greek Music and Music

Theory in Antiquity and the Middle Ages (Lincoln: University of Nebraska Press, 1999).

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Pythagoras

Pythagoras

Pythagoras , Greek philosopher and mathematician; b. Samos, c.580 B.C.; d. Metapontum, c.500 B.C. His doctrines on the musical ratios are preserved in the WRITINGS of his followers, no books by Pythagoras having come down to us. The Pythagoreans (Archytas, Didymos, Eratosthenes, Euclid, Ptolemy, etc.) reckoned only the 5th and octave as pure consonances (the 4th being the 5th below); their system recognized only intervals reached by successive skips of pure 5ths, the major 3rd being the 4th 5th above (ratio 64:81, instead of the modern 64:80, or 4:5), their minor 3rd the 3rd 5thbelow; etc. Their 3rds and 6ths were, consequently, dissonant intervals.

—Nicolas Slonimsky/Laura Kuhn/Dennis McIntire

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