In geometry, a nonagon or enneagon is a nine-sided polygon or 9-gon. The name "nonagon" is a prefix hybrid formation, from Latin, used equivalently, attested in the 16th century in French nonogone and in English from the 17th century; the name "enneagon" comes from Greek enneagonon, is arguably more correct, though less common than "nonagon". A regular nonagon is represented by Schläfli symbol and has internal angles of 140°; the area of a regular nonagon of side length a is given by A = 9 4 a 2 cot π 9 = a r = 9 r 2 tan = R 2 sin ≃ 6.18182 a 2, where the radius r of the inscribed circle of the regular nonagon is r = cot and where R is the radius of its circumscribed circle: R = 2 + r 2 = r sec . Although a regular nonagon is not constructible with compass and straightedge, there are old methods of construction that produce close approximations, it can be constructed using neusis, or by allowing the use of an angle trisector. Accuracy: 10-6The following is an approximate construction of a nonagon using a straightedge and compass.
Example to illustrate the error, when the constructed central angle is 39.99906°: At a circumscribed circle radius r = 100 m, the absolute error of the 1st side would be 1.6 mm. Accuracy: 10-10Downsize the angle JMK with four bisections of angle and make a thirds of circular arc MON with an approximate solution between bisections of angle w3 and w4. Straight auxiliary line g aims over the point O to the point N, between O and N, therefore no auxiliary line. Thus, the circular arc MON is accessible for the intersection point R. ∠ RMK = 40.0000000052441...°360° ÷ 9 = 40° ∠ RMK - 40° = 5.2... E-9°Example to illustrate the error:At a circumscribed circle radiusr = 100,000 km, the absolute error of the 1st side would be 8.6 mm. See the calculation; the regular enneagon has Dih9 symmetry, order 18. There are 2 subgroup dihedral symmetries: Dih3 and Dih1, 3 cyclic group symmetries: Z9, Z3, Z1; these 6 symmetries can be seen in 6 distinct symmetries on the enneagon. John Conway labels these by a group order.
Full symmetry of the regular form is r18 and no symmetry is labeled a1. The dihedral symmetries are divided depending on whether they pass through vertices or edges, i when reflection lines path through both edges and vertices. Cyclic symmetries in the middle column are labeled as g for their central gyration orders; each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the g9 subgroup has no degrees of freedom; the regular enneagon can tessellate the euclidean tiling with gaps. These gaps can be filled with isosceles triangles. In the notation of symmetrohedron this tiling is called H with H representing *632 hexagonal symmetry in the plane; the K9 complete graph is drawn as a regular enneagon with all 36 edges connected. This graph represents an orthographic projection of the 9 vertices and 36 edges of the 8-simplex, they Might Be Giants have a song entitled "Nonagon" on their children's album Here Come the 123s. It refers to both an attendee at a party at which "everybody in the party is a many-sided polygon" and a dance they perform at this party.
Slipknot's logo is a version of a nonagon, being a nine-pointed star made of three triangles, referring to the nine members. King Gizzard & the Lizard Wizard have an album titled'Nonagon Infinity', the album art featuring a nonagonal complete graph. Temples of the Baha'i Faith are required to be nonagonal; the U. S. Steel Tower is an irregular nonagon. Enneagram Trisection of the angle 60°, Proximity construction Properties of a Nonagon
George Ivanovich Gurdjieff was a mystic, spiritual teacher, composer of Armenian and Greek descent, born in Alexandrapol, Armenia. Gurdjieff taught that most humans do not possess a unified consciousness and thus live their lives in a state of hypnotic "waking sleep", but that it is possible to awaken to a higher state of consciousness and achieve full human potential. Gurdjieff described a method attempting to do so, calling the discipline "The Work" or "the Method". According to his principles and instructions, Gurdjieff's method for awakening one's consciousness unites the methods of the fakir and yogi, thus he referred to it as the "Fourth Way". Gurdjieff was born to a Caucasus Greek father, Ἰωάνης Γεωργιάδης, an Armenian mother, Evdokia, in Alexandropol, Armenia part of the Russian Empire in the Transcaucasus; the name Gurdjieff represents a Russified form of the Pontic Greek surname "Georgiades". The exact year of his birth remains unknown; some authors argue for 1866. Both Olga de Hartmann, the woman Gurdjieff called "the first friend of my inner life", Louise Goepfert March, Gurdjieff's secretary in the early 1930s, believed that Gurdjieff was born in 1872.
A passport gave a birthdate of November 28, 1877, but he once stated that he was born at the stroke of midnight at the beginning of New Year's Day. Although the dates of his birth vary, the year of 1872 is inscribed in a plate on the gravemarker in the cemetery of Avon, Seine-et-Marne, where his body was buried. Gurdjieff spent his childhood in Kars, from 1878 to 1918, was the administrative capital of the Russian-ruled Transcaucasus province of Kars Oblast, a border region captured from the Ottoman Empire, it contained extensive grassy plateau-steppe and high mountains, was inhabited by a multi-ethnic and multi-confessional population that had a history of respect for travelling mystics and holy men, for religious syncretism and conversion. Both the city of Kars and the surrounding territory were home to an diverse population: although part of the Armenian Plateau, Kars Oblast was home to Armenians, Caucasus Greeks, Turks and smaller numbers of Christian communities from eastern and central Europe such as Caucasus Germans and Russian sectarian communities like the Molokans, Doukhobors and Subbotniki.
Gurdjieff makes particular mention of the Yazidi community. Growing up in a multi-ethnic society, Gurdjieff became fluent in Armenian, Pontic Greek and Turkish, speaking the last in a mixture of elegant Osmanlı and some dialect, he acquired "a working facility with several European languages". Early influences on him included his father, a carpenter and amateur ashik or bardic poet, the priest of the town's Russian church, Dean Borsh, a family friend; the young Gurdjieff avidly read Russian-language scientific literature. Influenced by these writings, having witnessed a number of phenomena that he could not explain, he formed the conviction that there existed a hidden truth not to be found in science or in mainstream religion. In early adulthood, according to his own account Gurdjieff's curiosity led him to travel to Central Asia, Iran, India and Rome before he returned to Russia for a few years in 1912, he was always unforthcoming about the source of his teachings. The only account of his wanderings appears in his book Meetings with Remarkable Men.
Most commentators leave his background unexplained, it is not considered to be a reliable or straightforward autobiography. Each chapter is named after an individual "remarkable man". After Gurdjieff's death, J. G. Bennett researched his sources extensively and suggested that these characters were symbolic of the three types of people to whom Gurdjieff referred: No. 1 centred in their physical body. He asserts that he has encounters with dervishes and descendants of the extinct Essenes, whose teaching had been, he claimed, conserved at a monastery in Sarmoung; the book has an overarching quest narrative involving a map of "pre-sand Egypt" and culminating in an encounter with the "Sarmoung Brotherhood". Gurdjieff wrote. On his reappearance, as far as the historical record is concerned, the ragged wanderer had transformed into a well-heeled businessman, his only autobiographical writing concerning this period is Herald of Coming Good. In it, he mentions acting as hypnotherapist specialising in the cure of addictions and using people as guinea pigs for his methods.
It is speculated that during his travels, he was engaged in a certain amount of political activity, as part of The Great Game. From 1913 to 1949, the chronology appears to be based on material that can be confirmed by primary documents, independent witnesses, cross-references and reasonable inference. On New Year's Day in 1912, Gurdjieff arrived in Moscow and attracted his first students, including his cousin, the sculptor Sergey Merkurov, the eccentric Rachmilievitch. In the same year, he married the Polish Julia Ostrowska in Saint Petersburg. In 1914, Gurdjieff advertised his ballet, The Struggle of the Magicians, he
Pragmatism is a philosophical tradition that began in the United States around 1870. Its origins are attributed to the philosophers William James, John Dewey, Charles Sanders Peirce. Peirce described it in his pragmatic maxim: "Consider the practical effects of the objects of your conception. Your conception of those effects is the whole of your conception of the object."Pragmatism considers words and thought as tools and instruments for prediction, problem solving and action, rejects the idea that the function of thought is to describe, represent, or mirror reality. Pragmatists contend that most philosophical topics—such as the nature of knowledge, concepts, meaning and science—are all best viewed in terms of their practical uses and successes; the philosophy of pragmatism "emphasizes the practical application of ideas by acting on them to test them in human experiences". Pragmatism focuses on a "changing universe rather than an unchanging one as the Idealists and Thomists had claimed". Pragmatism as a philosophical movement began in the United States in the 1870s.
Charles Sanders Peirce is given credit for its development, along with twentieth century contributors, William James and John Dewey. Its direction was determined by The Metaphysical Club members Charles Sanders Peirce, William James, Chauncey Wright, as well as John Dewey and George Herbert Mead; the first use in print of the name pragmatism was in 1898 by James, who credited Peirce with coining the term during the early 1870s. James regarded Peirce's "Illustrations of the Logic of Science" series as the foundation of pragmatism. Peirce in turn wrote in 1906 that Nicholas St. John Green had been instrumental by emphasizing the importance of applying Alexander Bain's definition of belief, "that upon which a man is prepared to act". Peirce wrote. John Shook has said, "Chauncey Wright deserves considerable credit, for as both Peirce and James recall, it was Wright who demanded a phenomenalist and fallibilist empiricism as an alternative to rationalistic speculation."Peirce developed the idea that inquiry depends on real doubt, not mere verbal or hyperbolic doubt, said, in order to understand a conception in a fruitful way, "Consider the practical effects of the objects of your conception.
Your conception of those effects is the whole of your conception of the object", which he called the pragmatic maxim. It equates any conception of an object to the general extent of the conceivable implications for informed practice of that object's effects; this is the heart of his pragmatism as a method of experimentational mental reflection arriving at conceptions in terms of conceivable confirmatory and disconfirmatory circumstances—a method hospitable to the generation of explanatory hypotheses, conducive to the employment and improvement of verification. Typical of Peirce is his concern with inference to explanatory hypotheses as outside the usual foundational alternative between deductivist rationalism and inductivist empiricism, although he was a mathematical logician and a founder of statistics. Peirce further wrote on pragmatism to make clear his own interpretation. While framing a conception's meaning in terms of conceivable tests, Peirce emphasized that, since a conception is general, its meaning, its intellectual purport, equates to its acceptance's implications for general practice, rather than to any definite set of real effects.
Peirce in 1905 coined the new name pragmaticism "for the precise purpose of expressing the original definition", saying that "all went happily" with James's and Schiller's variant uses of the old name "pragmatism" and that he nonetheless coined the new name because of the old name's growing use in "literary journals, where it gets abused". Yet in a 1906 manuscript he cited as causes his differences with Schiller. And, in a 1908 publication, his differences with James as well as literary author Giovanni Papini. Peirce in any case regarded his views that truth is immutable and infinity is real, as being opposed by the other pragmatists, but he remained allied with them on other issues. Pragmatism enjoyed renewed attention after Willard Van Orman Quine and Wilfrid Sellars used a revised pragmatism to criticize logical positivism in the 1960s. Inspired by the work of Quine and Sellars, a brand of pragmatism known sometimes as neopragmatism gained influence through Richard Rorty, the most influential of the late twentieth century pragmatists along with Hilary Putnam and Robert Brandom.
Contemporary pragmatism may be broadly divided into a strict analytic tradition and a "neo-classical" pragmatism that adheres to the work of Peirce and Dewey. Inspiration for various pragmatists included: Francis Bacon who coined the saying ipsa scientia potestas est David Hume for his naturalistic account of knowledge and action Thomas Reid, for his direct realism Immanuel Kant, for his idealism and from whom Peirce derives the name "pragmatism" G. W. F. Hegel who introduced temporality into philosophy J. S. Mill for his nominalism and empiricism George Berkeley for his project to eliminate all unclear concepts from philosophy Henri Bergson who influenced William James to renounce intellectualism and logical methods A few of the various but interrelated positions characteristic
Bertrand Arthur William Russell, 3rd Earl Russell, was a British philosopher, mathematician, writer, social critic, political activist, Nobel laureate. At various points in his life, Russell considered himself a liberal, a socialist and a pacifist, although he confessed that his skeptical nature had led him to feel that he had "never been any of these things, in any profound sense." Russell was born in Monmouthshire into one of the most prominent aristocratic families in the United Kingdom. In the early 20th century, Russell led the British "revolt against idealism", he is considered one of the founders of analytic philosophy along with his predecessor Gottlob Frege, colleague G. E. Moore and protégé Ludwig Wittgenstein, he is held to be one of the 20th century's premier logicians. With A. N. Whitehead he wrote Principia Mathematica, an attempt to create a logical basis for mathematics, the quintessential work of classical logic, his philosophical essay "On Denoting" has been considered a "paradigm of philosophy".
His work has had a considerable influence on mathematics, set theory, artificial intelligence, cognitive science, computer science and philosophy the philosophy of language and metaphysics. Russell was a prominent anti-war activist and he championed anti-imperialism, he advocated preventive nuclear war, before the opportunity provided by the atomic monopoly had passed and "welcomed with enthusiasm" world government. He went to prison for his pacifism during World War I. Russell concluded that war against Adolf Hitler's Nazi Germany was a necessary "lesser of two evils" and criticised Stalinist totalitarianism, attacked the involvement of the United States in the Vietnam War and was an outspoken proponent of nuclear disarmament. In 1950, Russell was awarded the Nobel Prize in Literature "in recognition of his varied and significant writings in which he champions humanitarian ideals and freedom of thought". Bertrand Russell was born on 18 May 1872 at Ravenscroft, Monmouthshire, into an influential and liberal family of the British aristocracy.
His parents and Viscountess Amberley, were radical for their times. Lord Amberley consented to his wife's affair with their children's tutor, the biologist Douglas Spalding. Both were early advocates of birth control at a time. Lord Amberley was an atheist and his atheism was evident when he asked the philosopher John Stuart Mill to act as Russell's secular godfather. Mill died the year after Russell's birth, his paternal grandfather, the Earl Russell, had been asked twice by Queen Victoria to form a government, serving her as Prime Minister in the 1840s and 1860s. The Russells had been prominent in England for several centuries before this, coming to power and the peerage with the rise of the Tudor dynasty, they established themselves as one of the leading British Whig families, participated in every great political event from the Dissolution of the Monasteries in 1536–1540 to the Glorious Revolution in 1688–1689 and the Great Reform Act in 1832. Lady Amberley was Lady Stanley of Alderley. Russell feared the ridicule of his maternal grandmother, one of the campaigners for education of women.
Russell had two siblings: brother Frank, sister Rachel. In June 1874 Russell's mother died followed shortly by Rachel's death. In January 1876, his father died of bronchitis following a long period of depression. Frank and Bertrand were placed in the care of their staunchly Victorian paternal grandparents, who lived at Pembroke Lodge in Richmond Park, his grandfather, former Prime Minister Earl Russell, died in 1878, was remembered by Russell as a kindly old man in a wheelchair. His grandmother, the Countess Russell, was the dominant family figure for the rest of Russell's childhood and youth; the countess was from a Scottish Presbyterian family, petitioned the Court of Chancery to set aside a provision in Amberley's will requiring the children to be raised as agnostics. Despite her religious conservatism, she held progressive views in other areas, her influence on Bertrand Russell's outlook on social justice and standing up for principle remained with him throughout his life, her favourite Bible verse, became his motto.
The atmosphere at Pembroke Lodge was one of frequent prayer, emotional repression, formality. Russell's adolescence was lonely, he contemplated suicide, he remarked in his autobiography that his keenest interests were in religion and mathematics, that only his wish to know more mathematics kept him from suicide. He was educated at home by a series of tutors; when Russell was eleven years old, his brother Frank introduced him to the work of Euclid, which he described in his autobiography as "one of the great events of my life, as dazzling as first love."During these formative years he discovered the works of Percy Bysshe Shelley. Russell wrote: "I spent all my spare time reading him, learning him by heart, knowing no one to whom I could speak of what I thought or felt, I used to reflect how wonderful it would have been to know Shelley, to wonder whether
Alfred North Whitehead
Alfred North Whitehead was an English mathematician and philosopher. He is best known as the defining figure of the philosophical school known as process philosophy, which today has found application to a wide variety of disciplines, including ecology, education, biology and psychology, among other areas. In his early career Whitehead wrote on mathematics and physics, his most notable work in these fields is the three-volume Principia Mathematica, which he wrote with former student Bertrand Russell. Principia Mathematica is considered one of the twentieth century's most important works in mathematical logic, placed 23rd in a list of the top 100 English-language nonfiction books of the twentieth century by Modern Library. Beginning in the late 1910s and early 1920s, Whitehead turned his attention from mathematics to philosophy of science, to metaphysics, he developed a comprehensive metaphysical system which radically departed from most of western philosophy. Whitehead argued that reality consists of processes rather than material objects, that processes are best defined by their relations with other processes, thus rejecting the theory that reality is fundamentally constructed by bits of matter that exist independently of one another.
Today Whitehead's philosophical works – Process and Reality – are regarded as the foundational texts of process philosophy. Whitehead's process philosophy argues that "there is urgency in coming to see the world as a web of interrelated processes of which we are integral parts, so that all of our choices and actions have consequences for the world around us." For this reason, one of the most promising applications of Whitehead's thought in recent years has been in the area of ecological civilization and environmental ethics pioneered by John B. Cobb Jr. Alfred North Whitehead was born in Ramsgate, England, in 1861, his father, Alfred Whitehead, was a minister and schoolmaster of Chatham House Academy, a school for boys established by Thomas Whitehead, Alfred North's grandfather. Whitehead himself recalled both of them as being successful schools, but that his grandfather was the more extraordinary man. Whitehead's mother was Maria Sarah Whitehead Maria Sarah Buckmaster. Whitehead was not close with his mother, as he never mentioned her in any of his writings, there is evidence that Whitehead's wife, had a low opinion of her.
Whitehead was educated at Sherborne School, one of the best public schools in the country. His childhood was described as over-protected, but when at school he excelled in sports and mathematics and was head prefect of his class. In 1880, Whitehead began attending Trinity College and studied mathematics, his academic advisor was Edward John Routh. He earned his BA from Trinity in 1884, graduated as fourth wrangler. Elected a fellow of Trinity in 1884, Whitehead would teach and write on mathematics and physics at the college until 1910, spending the 1890s writing his Treatise on Universal Algebra, the 1900s collaborating with his former pupil, Bertrand Russell, on the first edition of Principia Mathematica, he was a Cambridge Apostle. In 1890, Whitehead married an Irish woman raised in France. Eric Whitehead died in action at the age of 19, while serving in the Royal Flying Corps during World War I. Alfred's brother Henry became Bishop of Madras, wrote a observed ethnographic account of the Village Gods of South-India, still of value today.
In 1910, Whitehead resigned his senior lectureship in mathematics at Trinity and moved to London without first lining up another job. After being unemployed for a year, Whitehead accepted a position as lecturer in applied mathematics and mechanics at University College London, but was passed over a year for the Goldsmid Chair of Applied Mathematics and Mechanics, a position for which he had hoped to be considered. In 1914 Whitehead accepted a position as professor of applied mathematics at the newly chartered Imperial College London, where his old friend Andrew Forsyth had been appointed chief professor of mathematics. In 1918 Whitehead's academic responsibilities began to expand as he accepted a number of high administrative positions within the University of London system, of which Imperial College London was a member at the time, he was elected dean of the Faculty of Science at the University of London in late 1918, a member of the University of London's Senate in 1919, chairman of the Senate's Academic Council in 1920, a post which he held until he departed for America in 1924.
Whitehead was able to exert his newfound influence to lobby for a new history of science department, help establish a Bachelor of Science degree, make the school more accessible to less wealthy students. Toward the end of his time in England, Whitehead turned his attention to philosophy. Though he had no advanced training in philosophy, his philosophical work soon became regarded. After publishing The Concept of Nature in 1920, he served as president of the Aristotelian Society from 1922 to 1923. In 1924, Henry Osborn Taylor invited the 63-year-old Whitehead to join the faculty at Harvard University as a professor of philosophy. During his time at Harvard, Whitehead produced his most important philosophical contributions. In 1925, he wrote Science and the Modern World, hailed as an alternative to the Cartesian dualism that plagued popular scien
Ludwig von Bertalanffy
Karl Ludwig von Bertalanffy was an Austrian biologist known as one of the founders of general systems theory, the "conceptual part" of, first introduced by Alexander Bogdanov. This is an interdisciplinary practice that describes systems with interacting components, applicable to biology and other fields. Bertalanffy proposed that the classical laws of thermodynamics applied to closed systems, but not to "open systems" such as living things, his mathematical model of an organism's growth over time, published in 1934, is still in use today. Bertalanffy grew up in Austria and subsequently worked in Vienna, London and the United States. Ludwig von Bertalanffy grew up in the little village of Atzgersdorf near Vienna; the Bertalanffy family had roots in the 16th century nobility of Hungary which included several scholars and court officials. His grandfather Charles Joseph von Bertalanffy had settled in Austria and was a state theatre director in Klagenfurt and Vienna, which were important sites in imperial Austria.
Ludwig's father Gustav. On his mother's side Ludwig's grandfather Joseph Vogel was an imperial counsellor and a wealthy Vienna publisher. Ludwig's mother Charlotte Vogel was seventeen, they divorced when Ludwig was ten, both remarried outside the Catholic Church in civil ceremonies. Ludwig von Bertalanffy grew up as an only child educated at home by private tutors; when he arrived at his Gymnasium he was well habituated in learning by reading, he continued to study on his own. His neighbour, the famous biologist Paul Kammerer, became a mentor and an example to the young Ludwig. In 1918, Bertalanffy started his studies at the university level in philosophy and art history, first at the University of Innsbruck and at the University of Vienna. Bertalanffy had to make a choice between studying philosophy of science and biology. In 1926 he finished his PhD thesis on philosopher Gustav Theodor Fechner. For the next six years he concentrated on a project of "theoretical biology" which focused on the philosophy of biology.
He received his habilitation in 1934 in "theoretical biology". Bertalanffy was appointed Privatdozent at the University of Vienna in 1934; the post yielded little income, Bertalanffy faced continuing financial difficulties. He applied for promotion to the status of associate professor, but funding from the Rockefeller Foundation enabled him to make a trip to Chicago in 1937 to work with Nicolas Rashevsky, he was able to visit the Marine Biological Laboratory in Massachusetts. Bertalanffy was still in the US when he heard of the Anschluss in March 1938. However, his attempts to remain in the US failed, he returned to Vienna in October of that year. Within a month of his return, he joined the Nazi Party, which facilitated his promotion to professor at the University of Vienna in 1940. During the Second World War, he linked his "organismic" philosophy of biology to the dominant Nazi ideology, principally that of the Führerprinzip. Following the defeat of Nazism, Bertalanffy found denazification problematic and left Vienna in 1948.
He moved to the University of London. In 1972, he died from a heart attack. Bertalanffy met Maria, in April 1924 in the Austrian Alps, they were hardly apart for the next forty-eight years. She never did, instead devoting her life to Bertalanffy's career. In Canada, she would work both for him and with him in his career, after his death she compiled two of Bertalanffy's last works, they had one child, a son who followed in his father's footsteps by making his profession in the field of cancer research. Today, Bertalanffy is considered to be a founder and one of the principal authors of the interdisciplinary school of thought known as general systems theory. According to Weckowicz, he "occupies an important position in the intellectual history of the twentieth century, his contributions went beyond biology, extended into cybernetics, history, psychiatry and sociology. Some of his admirers believe that this theory will one day provide a conceptual framework for all these disciplines". Spending most of his life in semi-obscurity, Ludwig von Bertalanffy may well be the least known intellectual titan of the twentieth century.
The individual growth model published by Ludwig von Bertalanffy in 1934 is used in biological models and exists in a number of permutations. In its simplest version the so-called Bertalanffy growth equation is expressed as a differential equation of length over time: L ′ = r B when r B is the Bertalanffy growth rate and L ∞ the ultimate length of the individual; this model was proposed earlier by August Friedrich Robert Pūtter, writ
Pythagoreanism originated in the 6th century BC, based on the teachings and beliefs held by Pythagoras and his followers, the Pythagoreans. Pythagoras established the first Pythagorean community in Italy. Early-Pythagorean communities lived throughout Magna Graecia. Espousing a rigorous life of the intellect and strict rules on diet and behavior comprised a cult of following Pythagorean's Code. For example, the Code's diet prohibits the consumption or touching any sort of bean or legume. Pythagoras’ death and disputes about his teachings led to the development of two philosophical traditions within Pythagoreanism; the practitioners of akousmatikoi were superseded in the 4th century BC as a significant mendicant school of philosophy by the Cynics. The Pythagorean mathēmatikoi philosophers were in the 4th century BC absorbed into the Platonic school. Following the political instability in the Magna Graecia, some Pythagorean philosophers fled to mainland Greece while others regrouped in Rhegium. By about 400 BC the majority of Pythagorean philosophers had left Italy.
Pythagorean ideas exercised a marked influence on Plato and through him, on all of Western philosophy. Many of the surviving sources on Pythagoras originate with Aristotle and the philosophers of the Peripatetic school; as a philosophic tradition, Pythagoreanism was revived in the 1st century BC, giving rise to Neopythagoreanism. The worship of Pythagoras continued in Italy and as a religious community Pythagoreans appear to have survived as part of, or influenced, the Bacchic cults and Orphism. Pythagoras was in ancient times well known for the mathematical achievement of the Pythagorean theorem. Pythagoras had discovered that "in a right-angled triangle the square of the hypotenuse is equal to the squares of the other two sides". In ancient times Pythagoras was noted for his discovery that music had mathematical foundations. Antique sources that credit Pythagoras as the philosopher who first discovered music intervals credit him as the inventor of the monochord, a straight rod on which a string and a movable bridge could be used to demonstrate the relationship of musical intervals.
Much of the surviving sources on Pythagoras originate with Aristotle and the philosophers of the Peripatetic school, which founded histographical academic traditions such as biography and the history of science. The surviving 5th century BC sources on Pythagoras and early Pythagoreanism are void of supernatural elements. While surviving 4th century BC sources on Pythagoreas' teachings introduced legend and fable. Philosophers who discussed Pythagoreanism, such as Anaximander, Andron of Ephesus and Neanthes had access to historical written sources as well as the oral tradition about Pythagoreanism, which by the 4th century BC was in decline. Neopythagorean philosophers, who authored many of the surviving sources on Pythagoreanism, continued the tradition of legend and fantasy; the earliest surviving ancient source on Pythagoras and his followers is a satire by Xenophanes, on the Pythagorean beliefs on the transmigration of souls. Xenophanes wrote of Pythagoras that: Once they say that he was passing by when a puppy was being whipped, And he took pity and said: "Stop!
Do not beat it! For it is the soul of a friend That I recognized when I heard it giving tongue." In a surviving fragment from Heraclitus and his followers are described as follows: Pythagoras, the son of Mnesarchus, practised inquiry beyond all other men and selecting of these writings made for himself a wisdom or made a wisdom of his own: a polymathy, an imposture. Two other surviving fragments of ancient sources on Pythagoras are by Ion of Empedocles. Both were born after Pythagoras' death. By that time he was known as a sage and his fame had spread throughout Greece. According to Ion, Pythagoras was:... distinguished for his many virtue and modesty in death has a life, pleasing to his soul, if Pythagoras the wise achieved knowledge and understanding beyond that of all men. Empedocles described Pythagoras as "a man of surpassing knowledge, master of all kinds of wise works, who had acquired the upmost wealth of understanding." In the 4th century BC the Sophist Alcidamas wrote that Pythagoras was honored by Italians.
Today scholars distinguish two periods of Pythagoreanism: early-Pythagoreanism, from the 6th till the 5th century BC, late-Pythagoreanism, from the 4th till the 3rd century BC. The Spartan colony of Taranto in Italy became the home for many practitioners of Pythagoreanism and for Neopythagorean philosophers. Pythagoras had lived in Crotone and Metaponto, both were Achaean colonies. Early-Pythagorean sects lived throughout Magna Graecia, they espoused to a rigorous life of the intellect and strict rules on diet and behavior. Their burial rites were tied to their belief in the immortality of the soul. Early-Pythagorean sects were closed societies and new Pythagoreans were chosen based on merit and discipline. Ancient sources record that early-Pythagoreans underwent a five year initiation period of listening to the teachings in silence. Initiates could through a test become members of the inner circle. However, Pythagoreans could leave the community if they wished. Iamblichus listed 235 Pythagoreans by name, among them 17 women who he described as the "most famous" women practitioners of Pythagoreanism.
It was customary that family members became Pythagoreans, as Pythagoreanism developed into a philosophic traditions that entailed rules for everyday life and Pythagoreans were bound by secrets. The home of a Pythagorean was known as the site of mysteries. Pythagoras had been born on the island of Samos at around 570 BC and left his homeland at around 530 BC in opposition