Pythagoras - Greek Mathematics - The Story of Mathematics The Story of Mathematics GREEK MATHEMATICS - PYTHAGORAS Pythagoras of Samos (c.570-495 BCE) It is sometimes claimed that we owe pure mathematics to Pythagoras, and he is often called the first "true" mathematician. But, although his contribution was clearly important, he nevertheless remains a controversial figure. He left no mathematical writings himself, and much of what we know about Pythagorean thought comes to us from the writings of Philolaus and other later Pythagorean scholars. Indeed, it is by no means clear whether many (or indeed any) of the theorems ascribed to him were in fact solved by Pythagoras personally or by his followers. The school he established at Croton in southern Italy around 530 BCE was the nucleus of a rather bizarre Pythagorean sect.

Although Pythagorean thought was largely dominated by mathematics, it was also profoundly mystical, and Pythagoras imposed his quasi-religious philosophies, strict vegetarianism, communal living, secret rites and odd rules on all the members of his school (including bizarre and apparently random edicts about never urinating towards the sun, never marrying a woman who wears gold jewellery, never passing an ass lying in the street, never eating or even touching black fava beans, etc) . The members were divided into the "mathematikoi" (or "learners"), who extended and developed the more mathematical and scientific work that Pythagoras himself began, and the "akousmatikoi" (or "listeners"), who focused on the more religious and ritualistic aspects of his teachings. There was always a certain amount of friction between the two groups and eventually the sect became caught up in some fierce local fighting and ultimately dispersed. Resentment built up against the secrecy and exclusiveness of the Pythagoreans and, in 460 BCE, all their meeting places were burned and destroyed, with at least 50 members killed in Croton alone.

The over-riding dictum of Pythagoras's school was �All is number� or �God is number�, and the Pythagoreans effectively practised a kind of numerology or number-worship, and considered each number to have its own character and meaning. For example, the number one was the generator of all numbers; two represented opinion; three, harmony; four, justice; five, marriage; six, creation; seven, the seven planets or �wandering stars�; etc. Odd numbers were thought of as female and even numbers as male. The Pythagorean Tetractys The holiest number of all was "tetractys" or ten, a triangular number composed of the sum of one, two, three and four. It is a great tribute to the Pythagoreans' intellectual achievements that they deduced the special place of the number 10 from an abstract mathematical argument rather than from something as mundane as counting the fingers on two hands. However, Pythagoras and his school - as well as a handful of other mathematicians of ancient Greece - was largely responsible for introducing a more rigorous mathematics than what had gone before, building from first principles using axioms and logic.

Before Pythagoras, for example, geometry had been merely a collection of rules derived by empirical measurement. Pythagoras discovered that a complete system of mathematics could be constructed, where geometric elements corresponded with numbers, and where integers and their ratios were all that was necessary to establish an entire system of logic and truth. He is mainly remembered for what has become known as Pythagoras� Theorem (or the Pythagorean Theorem): that, for any right-angled triangle, the square of the length of the hypotenuse (the longest side, opposite the right angle) is equal to the sum of the square of the other two sides (or �legs�). Written as an equation: a2 + b2 = c2. What Pythagoras and his followers did not realize is that this also works for any shape: thus, the area of a pentagon on the hypotenuse is equal to the sum of the pentagons on the other two sides, as it does for a semi-circle or any other regular (or even irregular( shape.

Pythagoras' (Pythagorean) Theorem The simplest and most commonly quoted example of a Pythagorean triangle is one with sides of 3, 4 and 5 units (32 + 42 = 52, as can be seen by drawing a grid of unit squares on each side as in the diagram at right), but there are a potentially infinite number of other integer �Pythagorean triples�, starting with (5, 12 13), (6, 8, 10), (7, 24, 25), (8, 15, 17), (9, 40, 41), etc. It should be noted, however that (6, 8, 10) is not what is known as a �primitive� Pythagorean triple, because it is just a multiple of (3, 4, 5). Pythagoras� Theorem and the properties of right-angled triangles seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry, and it was touched on in some of the most ancient mathematical texts from Babylon and Egypt , dating from over a thousand years earlier.

One of the simplest proofs comes from ancient China , and probably dates from well before Pythagoras' birth. It was Pythagoras, though, who gave the theorem its definitive form, although it is not clear whether Pythagoras himself definitively proved it or merely described it. Either way, it has become one of the best-known of all mathematical theorems, and as many as 400 different proofs now exist, some geometrical, some algebraic, some involving advanced differential equations, etc. It soom became apparent, though, that non-integer solutions were also possible, so that an isosceles triangle with sides 1, 1 and √2, for example, also has a right angle, as the Babylonians had discovered centuries earlier. However, when Pythagoras�s student Hippasus tried to calculate the value of √2, he found that it was not possible to express it as a fraction, thereby indicating the potential existence of a whole new world of numbers, the irrational numbers (numbers that can not be expressed as simple fractions of integers).

This discovery rather shattered the elegant mathematical world built up by Pythagoras and his followers, and the existence of a number that could not be expressed as the ratio of two of God's creations (which is how they thought of the integers) jeopardized the cult's entire belief system. Poor Hippasus was apparently drowned by the secretive Pythagoreans for broadcasting this important discovery to the outside world. But the replacement of the idea of the divinity of the integers by the richer concept of the continuum, was an essential development in mathematics. It marked the real birth of Greek geometry, which deals with lines and planes and angles, all of which are continuous and not discrete. Among his other achievements in geometry, Pythagoras (or at least his followers, the Pythagoreans) also realized that the sum of the angles of a triangle is equal to two right angles (180�), and probably also the generalization which states that the sum of the interior angles of a polygon with n sides is equal to (2n - 4) right angles, and that the sum of its exterior angles equals 4 right angles.

They were able to construct figures of a given area, and to use simple geometrical algebra, for example to solve equations such as a(a - x) = x2 by geometrical means. The Pythagoreans also established the foundations of number theory, with their investigations of triangular, square and also perfect numbers (numbers that are the sum of their divisors). They discovered several new properties of square numbers, such as that the square of a number n is equal to the sum of the first n odd numbers (e.g. 42 = 16 = 1 + 3 + 5 + 7). They also discovered at least the first pair of amicable numbers, 220 and 284 (amicable numbers are pairs of numbers for which the sum of the divisors of one number equals the other number, e.g.

the proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110, of which the sum is 284; and the proper divisors of 284 are 1, 2, 4, 71, and 142, of which the sum is 220). Pythagoras is credited with the discovery of the ratios between harmonious musical tones Pythagoras is also credited with the discovery that the intervals between harmonious musical notes always have whole number ratios. For instance, playing half a length of a guitar string gives the same note as the open string, but an octave higher; a third of a length gives a different but harmonious note; etc. Non-whole number ratios, on the other hand, tend to give dissonant sounds.

In this way, Pythagoras described the first four overtones which create the common intervals which have become the primary building blocks of musical harmony: the octave (1:1), the perfect fifth (3:2), the perfect fourth (4:3) and the major third (5:4). The oldest way of tuning the 12-note chromatic scale is known as Pythagorean tuning, and it is based on a stack of perfect fifths, each tuned in the ratio 3:2. The mystical Pythagoras was so excited by this discovery that he became convinced that the whole universe was based on numbers, and that the planets and stars moved according to mathematical equations, which corresponded to musical notes, and thus produced a kind of symphony, the �Musical Universalis� or �Music of the Spheres�. What is the best-known contribution of the Greek scientist Pythagoras? A. an oath about how doctors should practice medicine B. the idea that all matter is composed of small moving particles C. a method of teaching that uses questioning rather than lecturing D. a theorem on the relationship between the three sides of a right triangle You have new items in your feed. Click to view.

Question and answer What is the best-known contribution of the Greek scientist Pythagoras? A. an oath about how doctors should practice medicine B. the idea that all matter is composed of small moving particles C. a method of teaching that uses questioning rather than lecturing D. a theorem on the relationship between the three sides of a right triangle D. a theorem on the relationship between the three sides of a right triangle is the best-known contribution of the Greek scientist Pythagoras. Expert answered| OxTornado007 |Points 5908| Pythagoras Pythagoras Monday, 19 September 2011 12:48 Pythagoras of Samos was an Ionian Greek philosopher and founder of the religious movement called Pythagoreanism. Most of our information about Pythagoras was written down centuries after he lived, thus very little reliable information is known about him. He was born on the island of Samos, and may have travelled widely in his youth, visiting Egypt and other places seeking knowledge. Around 530 BC, he moved to Croton, a Greek colony in southern Italy, and there set up a religious sect.

His followers pursued the religious rites and practices developed by Pythagoras, and studied his philosophical theories. The society took an active role in the politics of Croton, but this eventually led to their downfall. The Pythagorean meeting-places were burned, and Pythagoras was forced to flee the city. He is said by some to have ended his days in Metapontum. Pythagoras made influential contributions to philosophy and religious teaching in the late 6th century BC. He is often revered as a great mathematician, mystic and scientist, and he is best known for the Pythagorean theorem which bears his name. However, because legend and obfuscation cloud his work even more than with the other pre-Socratic philosophers, one can say little with confidence about his teachings, and some have questioned whether he contributed much to mathematics and natural philosophy. Many of the accomplishments credited to Pythagoras may actually have been accomplishments of his colleagues and successors. We do know that his disciples believed that everything was related to mathematics and that numbers were the ultimate reality. It was said that he was the first man to call himself a philosopher, or lover of wisdom and Pythagorean ideas exercised a marked influence on Plato, and through him, all of western philosophy.

Since the fourth century AD, Pythagoras has commonly been given credit for discovering the Pythagorean theorem, a theorem in geometry that states that in a right-angled triangle the square of the hypotenuse (the side opposite the right angle), c, is equal to the sum of the squares of the other two sides, b and a—that is a2 + b2 = c2. While the theorem that now bears his name was known and previously utilized by the Babylonians and Indians, he, or his students, are often said to have constructed the first proof. It must, however, be stressed that the way in which the Babylonians handled Pythagorean numbers, implies that they knew that the principle was generally applicable, and knew some kind of proof, which has not yet been found in the (still largely unpublished) cuneiform sources. Because of the secretive nature of his school and the custom of its students to attribute everything to their teacher, there is no evidence that Pythagoras himself worked on or proved this theorem. For that matter, there is no evidence that he worked on any mathematical or meta-mathematical problems.

Some attribute it as a carefully constructed myth by followers of Plato over two centuries after the death of Pythagoras, mainly to bolster the case for Platonic meta-physics, which resonate well with the ideas they attributed to Pythagoras. This attribution has stuck, down the centuries up to modern times. The earliest known mention of Pythagoras's name in connection with the theorem occurred five centuries after his death, in the writings of Cicero and Plutarch. History Menu Pythagoras (-570 - -495) - Genealogy Pythagoras Isle of Samos, Ancient Greece Death: in Metapontum, Gulf of Tarentum, Italy Immediate Family: mother About Pythagoras of Samos Pythagoras of Samos (Ancient Greek: Ὁ Πυθαγόρας ὁ Σάμιος Ho Pythagóras ho Sámios "Pythagoras the Samian", or simply Ὁ Πυθαγόρας; c. 570–c. 495 BCE) was an Ionian Greek philosopher, mathematician, and founder of the religious movement called Pythagoreanism.

Most of the information about Pythagoras was written down centuries after he lived, so very little reliable information is known about him. He was born on the island of Samos, and might have travelled widely in his youth, visiting Egypt and other places seeking knowledge. He had a teacher named Themistoclea, who introduced him to the principles of ethics. Around 530 BCE, he moved to Croton, a Greek colony in southern Italy, and there set up a religious sect. His followers pursued the religious rites and practices developed by Pythagoras, and studied his philosophical theories. The society took an active role in the politics of Croton, but this eventually led to their downfall. The Pythagorean meeting-places were burned, and Pythagoras was forced to flee the city. He is said to have ended his days in Metapontum. Pythagoras made influential contributions to philosophy and religious teaching in the late 6th century BCE. He is often revered as a great mathematician, mystic and scientist, but he is best known for the Pythagorean theorem which bears his name.

However, because legend and obfuscation cloud his work even more than with the other pre-Socratic philosophers, one can give account of his teachings to a little extent, and some have questioned whether he contributed much to mathematics and natural philosophy. Many of the accomplishments credited to Pythagoras may actually have been accomplishments of his colleagues and successors. Whether or not his disciples believed that everything was related to mathematics and that numbers were the ultimate reality is unknown. It was said that he was the first man to call himself a philosopher, or lover of wisdom, and Pythagorean ideas exercised a marked influence on Plato, and through him, all of Western philosophy. Mathematics The Pythagorean theorem: The sum of the areas of the two squares on the legs (a and b) equals the area of the square on the hypotenuse . The so-called Pythagoreans, who were the first to take up mathematics, not only advanced this subject, but saturated with it, they fancied that the principles of mathematics were the principles of all things. —Aristotle, Metaphysics 1–5 , cc.

350 BCE Pythagorean theorem Since the fourth century CE, Pythagoras has commonly been given credit for discovering the Pythagorean theorem, a theorem in geometry that states that in a right-angled triangle the area of the square on the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares of the other two sides—that is, a*2 + b*2 = c*2. While the theorem that now bears his name was known and previously utilized by the Babylonians and Indians, he, or his students, are often said to have constructed the first proof. It must, however, be stressed that the way in which the Babylonians handled Pythagorean numbers implies that they knew that the principle was generally applicable, and knew some kind of proof, which has not yet been found in the (still largely unpublished) cuneiform sources. Because of the secretive nature of his school and the custom of its students to attribute everything to their teacher, there is no evidence that Pythagoras himself worked on or proved this theorem. For that matter, there is no evidence that he worked on any mathematical or meta-mathematical problems.

Some attribute it as a carefully constructed myth by followers of Plato over two centuries after the death of Pythagoras, mainly to bolster the case for Platonic meta-physics, which resonate well with the ideas they attributed to Pythagoras. This attribution has stuck down the centuries up to modern times. The earliest known mention of Pythagoras's name in connection with the theorem occurred five centuries after his death, in the writings of Cicero and Plutarch. Musical theories and investigations According to legend, the way Pythagoras discovered that musical notes could be translated into mathematical equations was when one day he passed blacksmiths at work, and thought that the sounds emanating from their anvils being hit were beautiful and harmonious and decided that whatever scientific law caused this to happen must be mathematical and could be applied to music.

He went to the blacksmiths to learn how this had happened by looking at their tools, he discovered that it was because the hammers were "simple ratios of each other, one was half the size of the first, another was 2/3 the size, and so on." This legend has since proven to be false by virtue of the fact that these ratios are only relevant to string length (such as the string of a monochord), and not to hammer weight. However, it may be that Pythagoras was indeed responsible for discovering these properties of string length. Pythagoreans elaborated on a theory of numbers, the exact meaning of which is still debated among scholars. Another belief attributed to Pythagoras was that of the "harmony of the spheres". Thus the planets and stars moved according to mathematical equations, which corresponded to musical notes and thus produced a symphony. Tetractys Pythagoras was also credited with devising the tetractys, the triangular figure of four rows, which add up to the perfect number, ten.

As a mystical symbol, it was very important to the worship of the Pythagoreans, who would swear oaths by it: And the inventions were so admirable, and so divinised by those who understood them, that the members used them as forms of oath: "By him who handed to our generation the tetractys, source of the roots of ever-flowing nature." —Iamblichus, Vit. Pyth., 29 Pythagoreanism Both Plato and Isocrates affirm that, above all else, Pythagoras was famous for leaving behind him a way of life.[55] Both Iamblichus and Porphyry give detailed accounts of the organisation of the school, although the primary interest of both writers is not historical accuracy, but rather to present Pythagoras as a divine figure, sent by the gods to benefit humankind. Pythagoras set up an organization which was in some ways a school, in some ways a brotherhood (and here it should be noted that sources indicate that as well as men there were many women among the adherents of Pythagoras), and in some ways a monastery. It was based upon the religious teachings of Pythagoras and was very secretive.

The adherents were bound by a vow to Pythagoras and each other, for the purpose of pursuing the religious and ascetic observances, and of studying his religious and philosophical theories. The claim that they put all their property into a common stock is perhaps only a later inference from certain Pythagorean maxims and practices. As to the internal arrangements of the sect, we are informed that what was done and taught among the members was kept a profound secret towards all. Porphyry stated that this silence was "of no ordinary kind." Candidates had to pass through a period of probation, in which their powers of maintaining silence (echemythia) were especially tested, as well as their general temper, disposition, and mental capacity. There were also gradations among the members themselves. It was an old Pythagorean maxim, that every thing was not to be told to every body. Thus the Pythagoreans were divided into an inner circle called the mathematikoi ("learners") and an outer circle called the akousmatikoi ("listeners").

Iamblichus describes them in terms of esoterikoi and exoterikoi (or alternatively Pythagoreioi and Pythagoristai), according to the degree of intimacy which they enjoyed with Pythagoras. Porphyry wrote "the mathematikoi learned the more detailed and exactly elaborated version of this knowledge, the akousmatikoi (were) those who had heard only the summary headings of his (Pythagoras's) writings, without the more exact exposition." There were ascetic practices (many of which had, perhaps, a symbolic meaning) in the way of life of the sect. Some represent Pythagoras as forbidding all animal food. This may have been due to the doctrine of metempsychosis. Other authorities contradict the statement. According to Aristoxenus, he allowed the use of all kinds of animal food except the flesh of oxen used for ploughing, and rams. There is a similar discrepancy as to the prohibition of fish and beans. But temperance of all kinds seems to have been urged. It is also stated that they had common meals, resembling the Spartan system, at which they met in companies of ten.

Considerable importance seems to have been attached to music and gymnastics in the daily exercises of the disciples. Their whole discipline is represented as encouraging a lofty serenity and self-possession, of which, there were various anecdotes in antiquity. Iamblichus (apparently on the authority of Aristoxenus) gives a long description of the daily routine of the members, which suggests many similarities with Sparta. The members of the sect showed a devoted attachment to each other, to the exclusion of those who did not belong to their ranks. There were even stories of secret symbols, by which members of the sect could recognise each other, even if they had never met before. Wikimedia Commons has media related to Pythagoras Pythagoras facts, information, pictures | Encyclopedia.com articles about 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 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. Cite this article Pick a style below, and copy the text for your bibliography. MLA 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. Cite this article PYTHAGORAS PYTHAGORAS. Pythagoras (c. 580–c. 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 non–meat 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 Pythagoras - Ascension Glossary Pythagoras Jump to: navigation , search Pythagoras of Samos (c. 570 – c. 495 BC)[was an Ionian Greek philosopher, mathematician, and founder of the religious movement called Pythagoreanism. Most of the information about Pythagoras was written down centuries after he lived, so very little reliable information is known about him. He was born on the island of Samos, and might have travelled widely in his youth, visiting Egypt and other places seeking knowledge. Around 530 BC, he moved to Croton, in Magna Graecia, and there set up a religious sect. His followers pursued the religious rites and practices developed by Pythagoras and studied his philosophical theories. The society took an active role in the politics of Croton but this eventually led to their downfall.

Pythagorean meeting-places were burned and Pythagoras was forced to flee the city. He is said to have died in Metapontum. Pythagoras made influential contributions to philosophy and religion in the late 6th century BC. He is often revered as a great mathematician, mystic, and scientist and is best known for the Pythagorean theorem which bears his name. [1] Contents 7 See Also: Sacred Decad These nine divisions of the Magnum Opus of Alchemical Law meet in Ophiuchus and are three (3) times three (3), plus the Universal Prima Materia (Azoth) life force which equal ten (10), the sacred decad of Pythagoras. The Hebrew Tetragrammaton (YHWH-4) and ten spheres of emanation in the Tree of Life are based on The Sacred Decad or Tetractys. This is a triangular figure consisting of ten points arranged in four rows. As a mystical symbol, this was very important to the secret worship of the Pythagoreans which became the basis of many mystical teachings. Kabbalah , Freemasonry and Rosicrucianism have claimed to have evolved out of the Pythagorean Brotherhood.

Ten is the all-inclusive structure of the four dimensional Universe, which also represents maintaining the domination and order contained within the structure. Thus, ten also represents the “Matrix” of that construct which hypnotizes and keeps one locked in the illusion of the third dimensional world. Tetractys Tetractys The Sacred Decad (10) or Tetractys symbolize the Four Living Creatures and the elements that made them through the Prima Materia; fire, air, water, and earth. This also intersects into the four quadrants of cardinal directions and represents the divine calculus of the Universe. The Sacred Decad geometry is held within the Ophiuchus constellation transmission (13=4) which represent the organization of dimension and space in the Cosmos: • the first row represents zero-dimensions (a point) • the second row represents one-dimension (a line of two points) • the third row represents two-dimensions (a plane defined by a triangle of three points) • the fourth row represents three-dimensions (a tetrahedron defined by four points) Four as the Universal Prototype According to the Pythagorean symbolism in the Tetrad, Four is considered the root of all things and the source of all nature.

It is the number of numbers because The Sacred Decad or 10, includes 1, 2, 3 and 4, because it is a symbol of the four of the first four numbers. The tetrad connects them throughout all numbers, elements, and temporary structures which manifest as recurrent cycles in time. A Master Builder or Mason is quantified by the number 22, which when broken down aligns back to the structure of Four. The Master Builder sees the archetype, and brings it down to earth in some material form. The Universal Prototype in the Law of Structure related to the Four is revealing its truth to us now. That which has been created with falsified archetypes or with deception is being destroyed. While that which holds the sacred and perfect number of the Universal Law will remain and reveal its truth.

To consider the importance of the Four as the Universal Prototype in the Law of Structure , there are many aspects in creation [2] : The Four "Corners" of Creation East South West North The Fixed Signs of the Zodiac Aquarius Leo Scorpio Taurus The Four Living Creatures (Cherubim) Man Lion Eagle Bull The Four Seasons Spring Winter Autumn Summer Planetary Grid Networks GEG GWL SBC FFM The Ages of Hu-Man Childhood Youth Maturity Age The Stages of Existence Birth Growth Maturity Decay The Parts of Man's Constitution Spirit Soul Mind Body The Four Elements Air Fire Water Earth The Four Worlds Emanation Creation Formation Action The Royal Stars Aldebaran Fomalhaut Antares Regulus Sumerian Artifact There are compositions known as The Sacred Decad , which have been confirmed in several literary catalogues of tablets dating back to ancient Sumeria-Egypt. Sumerian artifacts are the basis of all written history believed to have been established and is the earliest known record of developed human civilization. The Decad contains the narrative of the Sumerian Epic of Gilgamesh, which has influenced modern literature, religions and culture.

All themes from the Sumerian Epic of Gilgamesh are found in later biblical text stemming from the time of the Annunaki invasion. [3] The Prophet Ezekiel (Born c. 622 BCE Jerusalem Died c. 570 BCE Babylon (Iraq) is viewed as the author of the Book of Ezekiel that reveals a sequence of prophecies around the restoration of Israel (God’s Kingdom) and the Third Temple. Ezekiel goes into great detail about his encounter with God and the Four Living Creatures or Cherubim with four wheels. Recent intelligence about Ezekiel is being shared to give deeper clarity on the impact of the changes transpiring with the Four Living Creatures in our Planetary Grid Network . Ezekiel was the Hierophant of Pythagoras and the Father of Pythagorean Mystery Schools. [4] References ↑ The Golden Gate, December 2014 ↑ The Golden Gate, December 2014

Question: The Greek mathematician, mystic and scientist Pythagoras is best known for the Pythagorean theorem, which concerns what?
Answer: Triangles