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Archytas was born in -428.

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Brachypeza archytas was created in 1891.

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Archippus died in 0##.

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Archytas of Tarentum

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Archytas was an ancient Greek philosopher, mathematician, and statesman from the city of Tarentum in Southern Italy. He is known for his contributions to mathematics, particularly in geometry and music theory, as well as for his friendship with Plato and his involvement in politics. Archytas is considered one of the prominent figures in the early history of Greek philosophy.

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Archytas of Tarentum, he was a philosopher and inventor much like Archimedes.

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Archytas of Tarentum invented a birdlike self-propelled device which flew for around 200 meters. His came up with the harmonic mean, which is useful for projective geometry. This type of geometry is used for rocket making and launching calculations.

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Archytas was an ancient Greek mathematician, philosopher, astronomer, statesman, and strategist. He was a prominent figure in the Pythagorean school of thought and made significant contributions to mathematics, particularly in the field of geometry. Archytas is also known for his contributions to music theory and his work in aerodynamics.

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The very first robot was invented by Ancient Greek mathematician Archytas of Tarentum. It was a flying wooden dove that flapped its wings.

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Maude F. Christian-Meier has written:

'Interactions of resistant corn cultivars, Spodoptera frugiperda (J.E. Smith) and Archytas marmoratus (Townsend)' -- subject(s): Accessible book

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The history of kites is such a long one that nobody knows about its invention. According to some people a Greek named Archytas invented the kite in the 4th century B.C., but it has been proved that kites were flown in oriental countries long before that time.

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We do not know exactly. The screw may already have existed around 700 BC in Mesopotamia during the reign of King Sennacherib. Many credit the Greek Archytas of Tarentum with its invention around 400 BC; it was then mainly used for wine and olive presses. Archimedes was the first to adapt and use the screw principle for "industrial" purposes.

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  • According to Aulus gellius, Archytas, the Ancient Greekphilosopher, mathematician, astronomer, statesman, and strategist, was reputed to have designed and built the first artificial, self-propelled flying device, a bird-shaped model propelled by a jet of what was probably steam, said to have actually flown some 200 meters. This machine, which its inventor called The Pigeon.

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The lever was likely invented by Archimedes in ancient Greece, while the screw was invented by either Archytas of Tarentum or Archimedes. Both inventions revolutionized the fields of science and engineering, and are still fundamental components in modern machinery and technology.

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The first rockets were invented in China sometime between the 9th and 13th centuries (A.D). These were black powder rockets that were essentially what we would call fireworks today, but were technically what are known as solid fuel rockets.

The Russian Konstantine Tsiolkovsky was the first to seriously propose a liquid fueled rocket in 1903.

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The idea of robotics dates back to 350 BC when Greek mathematician Archytas built a mechanical bird dubbed "the pigeon". It moves through using steam and serves as one of histories earliest studies of flight. In 200 BC Greek inventor and physicist Ctesibus of Alexandria designs water clocks that have movable figures on them. Da Vinci in 1495 designed a mechanical device that looked like a armored knight. It moved as if there was a real person inside. Jacques de Vaucanson began building autoamata in 1738 in Grenoble France. It was a flute player that could play 12 songs. The list goes on through history of various people who did robotics for various reasons. It is still going on.

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It is believed that the first kite was built by the ancient Chinese around 2,000 years ago. The exact creator is unknown, but kites were initially used for military purposes before evolving into a popular recreational activity.

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Aeroplane is not something which was discovered in a day or so. Rather it had an ancient history. Around 400 BC in Greece, Archytas, was reputed to have designed and built the first artificial, self-propelled flying device, a bird-shaped model propelled by a jet of what was probably steam, said to have flown some 200 m.1 there hundreds of other such reports of such flights. But the first documented and recorded flight was that of Wright brothers on 17 Dec, 1903. They made several flight in their home-made aircraft. Afterwards, the first commercial aeroplane was the de Havilland Comet, was introduced in 1952.2

[1]&[2] source: Wikipedia

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In antiquity, the screw was first used as part of the screw pump of Sennacherib, King of Assyria, for the water systems at the Hanging Gardens of Babylon and Nineveh in the 7th century BC.

The screw was later described by the Greek mathematician Archytas of Tarentum (428 - 350 BC). By the 1st century BC, wooden screws were commonly used throughout the Mediterranean world in devices such as oil and wine presses. Metal screws used as fasteners did not appear in Europe until the 1400s.

In 1770, English instrument maker, Jesse Ramsden (1735-1800) invented the first satisfactory screw-cutting lathe. The British engineer Henry Maudslay (1771-1831) patented a screw-cutting lathe in 1797; a similar device was patented by David Wilkinson in the United States in 1798.

In 1908, square-drive screws were invented by Canadian P. L. Robertson, becoming a North American standard. In the early 1930s, the Phillips head screw was invented by Henry F. Phillips.

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Alchemical research in China between the 9th and 13th centuries yielded most early discoveries in gunpowder and rocketry. Reliable documentation of rocket use first emerges in the 13th century, from Chinese accounts and subsequent European accounts of Mongol use.

short answer - China

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The first rocket which could fly high enough to claim the title "first space rocket" was invented by German scientists Werner Von BraunA and Walter Riedel.B

The rocket was called V2 but was part of a program from Nazi Germany to attack Britain during World War 2. The first V2 to reach space was launched by Von Braun's team on the 3rd of October 1942.C

Essential to the success of the rocket were three key technologies. These were the large liquid fuel rocket engines, supersonic aerodynamics, and guidance and control. This rocket could reach maximum speeds of 5760 Km/h or mach 4+.

American Scientist Robert Goddard created and launched the first liquid fueled rocket on March 16, 1926.D From 1930 to 1935 he launched rockets that attained speeds of up to 885 km/h (550 mph). Though his work in the field was revolutionary, he was sometimes ridiculed for his theories. He received little support during his lifetime, but would eventually be recognized as one of the fathers of modern rocketry.

Other important predecessors:

A Greek named Archytas (a few centuries BC) made small wooden devices that flew, propelled by a jet of steam. Some people point to this as the first rocket.

In the centuries CE (around 8th century) the Chinese invented fireworks that were rocket like. Later (12 to 14 century) they made more advanced designs. These devices were actually used for warfare.

Sources:

A Wernher von Braun: Wikipedia Entry

B Walter Riedel: Wikipedia Entry

C V-2: Wikipedia Entry

D Robert H. Goddard: Wikipedia Entry

See related links.

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Geometry History

Geometry was thoroughly organized in about 300 BC, when the Greek mathematician Euclid gathered what was known at the time, added original work of his own, and arranged 465 propositions into 13 books, called 'Elements'. The books covered not only plane and solid geometry but also much of what is now known as algebra, trigonometry, and advanced arithmetic.

Through the ages, the propositions have been rearranged, and many of the proofs are different, but the basic idea presented in the 'Elements' has not changed. In the work facts are not just cataloged but are developed in a fashionable way.

Even in 300 BC, geometry was recognized to be not just for mathematicians. Anyone can benefit from the basic learning of geometry, which is how to follow lines of reasoning, how to say precisely what is intended, and especially how to prove basic concepts by following these lines of reasoning. Taking a course in geometry is beneficial for all students, who will find that learning to reason and prove convincingly is necessary for every profession. It is true that not everyone must prove things, but everyone is exposed to proof. Politicians, advertisers, and many other people try to offer convincing arguments. Anyone who cannot tell a good proof from a bad one may easily be persuaded in the wrong direction. Geometry provides a simplified universe, where points and lines obey believable rules and where conclusions are easily verified. By first studying how to reason in this simplified universe, people can eventually, through practice and experience, learn how to reason in a complicated world.

Geometry in ancient times was recognized as part of everyone's education. Early Greek philosophers asked that no one come to their schools that had not learned the Elements' of Euclid. There were, and still are, many who resisted this kind of education.

Ancient knowledge of the sciences was often wrong and wholly unsatisfactory by modern standards. However not all of the knowledge of the more learned peoples of the past was false. In fact without people like Euclid or Plato we may not have been as advanced in this age as we are. Mathematics is an adventure in ideas. Within the history of mathematics, one finds the ideas and lives of some of the most brilliant people in the history of mankind's' populace upon Earth. First man created a number system of base 10. Certainly, it is not just coincidence that man just so happens to have ten fingers or ten toes, for when our primitive ancestors first discovered the need to count they definitely would have used their fingers to help them along just like a child today. When primitive man learned to count up to ten he somehow differentiated him from other animals. As an object of a higher thinking, man invented ten number-sounds. The needs and possessions of primitive man were not many. When the need to count over ten aroused, he simply combined the number-sounds related with his fingers. So, if he wished to define one more than ten, he simply said one-ten. Thus our word eleven is simply a modern form of the Teutonic ein-lifon. Since those first sounds were created, man has only added five new basic number-sounds to the ten primary ones. They are "hundred," "thousand," "million," "billion" (a thousand millions in America, a million millions in England), "trillion" (a million millions in America, a million-million millions in England). Because primitive man invented the same number of number-sounds as he had fingers, our number system is a decimal one, or a scale based on ten, consisting of limitless repetitions of the first ten number sounds. Undoubtedly, if nature had given man thirteen fingers instead of ten, our number system would be much changed. For instance, with a base thirteen number system we would call fifteen, two-thirteen's. While some intelligent and well-schooled scholars might argue whether or not base ten is the most adequate number system, base ten is the irreversible favorite among all the nations. Of course, primitive man most certainly did not realize the concept of the number system he had just created. Man simply used the number-sounds loosely as adjectives. So an amount of ten fish was ten fish, whereas ten is an adjective describing the noun fish. Soon the need to keep tally on one's counting raised. The simple solution was to make a vertical mark. Thus, on many caves we see a number of marks that the resident used to keep track of his possessions such a fish or knives. This way of record keeping is still taught today in our schools under the name of tally marks.

The earliest continuous record of mathematical activity is from the second millennium BC when one of the few wonders of the world was created mathematics was necessary. Even the earliest Egyptian pyramid proved that the makers had a fundamental knowledge of geometry and surveying skills. The approximate time period was 2900 BC The first proof of mathematical activity in written form came about one thousand years later. The best-known sources of ancient Egyptian mathematics in written format are the Rhind Papyrus and the Moscow Papyrus. The sources provide undeniable proof that the later Egyptians had intermediate knowledge of the following mathematical problems, applications to surveying, salary distribution, calculation of area of simple geometric figures' surfaces and volumes, simple solutions for first and second degree equations. Egyptians used a base ten number system most likely because of biologic reasons (ten fingers as explained above). They used the Natural Numbers (1,2,3,4,5,6, etc.) also known as the counting numbers. The word digit, which is Latin for finger, is also another name for numbers that explains the influence of fingers upon numbers once again. The Egyptians produced a more complex system then the tally system for recording amounts. Hieroglyphs stood for groups of tens, hundreds, and thousands. The higher powers of ten made it much easier for the Egyptians to calculate into numbers as large as one million. Our number system which is both decimal and positional (52 is not the same value as 25) differed from the Egyptian, which was additive, but not positional. The Egyptians also knew more of pi then its mere existence. They found pi to equal C/D or 4(8/9)ª whereas a equals 2. The method for ancient peoples arriving at this numerical equation was fairly easy. They simply counted how many times a string that fit the circumference of the circle fitted into the diameter, thus the rough approximation of 3. The biblical value of pi can be found in the Old Testament (I Kings vii.23 and 2 Chronicles iv.2)in the following verse "Also, he made a molten sea of ten cubits from brim to brim, round in compass, and five cubits the height thereof; and a line of thirty cubits did compass it round about." The molten sea, as we are told is round, and measures thirty cubits round about (in circumference) and ten cubits from brim to brim (in diameter). Thus the biblical value for pi is 30/10 = 3.

Now we travel to ancient Mesopotamia, home of the early Babylonians. Unlike the Egyptians, the Babylonians developed a flexible technique for dealing with fractions. The Babylonians also succeeded in developing more sophisticated base ten arithmetic that were positional and they also stored mathematical records on clay tablets. Despite all this, the greatest and most remarkable feature of Babylonian Mathematics was their complex usage of a sexagesimal place-valued system in addition a decimal system much like our own modern one. The Babylonians counted in both groups of ten and sixty. Because of the flexibility of a sexagismal system with fractions, the Babylonians were strong in both algebra and number theory. Remaining clay tablets from the Babylonian records show solutions to first, second, and third degree equations. Also the calculations of compound interest, squares and square roots were apparent in the tablets. The sexagismal system of the Babylonians is still commonly in usage today. Our system for telling time revolves around a sexagesimal system. The same system for telling time that is used today was also used by the Babylonians. Also, we use base sixty with circles (360 degrees to a circle). Usage of the sexagesimal system was principally for economic reasons. Being, the main units of weight and money were mina,(60 shekels) and talent (60 mina). This sexagesimal arithmetic was used in commerce and in astronomy. The Babylonians used many of the more common cases of the Pythagorean Theorem for right triangles. They also used accurate formulas for solving the areas, volumes and other measurements of the easier geometric shapes as well as trapezoids. The Babylonian value for pi was a very rounded off three. Because of this crude approximation of pi, the Babylonians achieved only rough estimates of the areas of circles and other spherical, geometric objects.

The real birth of modern math was in the era of Greece and Rome. Not only did the philosophers ask the question "how" of previous cultures, but they also asked the modern question of "why." The goal of this new thinking was to discover and understand the reason for mans' existence in the universe and also to find his place. The philosophers of Greece used mathematical formulas to prove propositions of mathematical properties. Some of who, like Aristotle, engaged in the theoretical study of logic and the analysis of correct reasoning. Up until this point in time, no previous culture had dealt with the negated abstract side of mathematics, of with the concept of the mathematical proof. The Greeks were interested not only in the application of mathematics but also in its philosophical significance, which was especially appreciated by Plato (429-348 BC). Plato was of the richer class of gentlemen of leisure. He, like others of his class, looked down upon the work of slaves and crafts worker. He sought relief, for the tiresome worries of life, in the study of philosophy and personal ethics. Within the walls of Plato's academy at least three great mathematicians were taught, Theaetetus, known for the theory of irrational, Eodoxus, the theory of proportions, and also Archytas (I couldn't find what made him great, but three books mentioned him so I will too). Indeed the motto of Plato's academy "Let no one ignorant of geometry enter within these walls" was fitting for the scene of the great minds who gathered here. Another great mathematician of the Greeks was Pythagoras who provided one of the first mathematical proofs and discovered incommensurable magnitudes, or irrational numbers. The Pythagorean theorem relates the sides of a right triangle with their corresponding squares. The discovery of irrational magnitudes had another consequence for the Greeks since the length of diagonals of squares could not be expressed by rational numbers in the form of A over B, the Greek number system was inadequate for describing them. As, you might have realized, without the great minds of the past our mathematical experiences would be quite different from the way they are today.

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Pythagoras of Samos (Greek: Ὁ Πυθαγόρας ὁ Σάμιος, O Pūthagoras o Samios, "Pythagoras the Samian", or simply Ὁ Πυθαγόρας; born between 580 and 572 BC, died between 500 and 490 BC) was an Ionian Greek mathematician and founder of the religious movement called Pythagoreanism. He is often revered as a great mathematician, mystic and scientist; however some have questioned the scope of his contributions to mathematics and natural philosophy. Herodotus referred to him as "the most able philosopher among the Greeks". His name led him to be associated with Pythian Apollo; Aristippus explained his name by saying, "He spoke (agor-) the truth no less than did the Pythian (Pyth-)," and Iamblichus tells the story that the Pythia prophesied that his pregnant mother would give birth to a man supremely beautiful, wise, and beneficial to humankind.[1] He is best known for the Pythagorean theorem, which bears his name. Known as "the father of numbers", Pythagoras made influential contributions to philosophy and religious teaching in the late 6th century BC. Because legend and obfuscation cloud his work even more than with the other pre-Socratics, one can say little with confidence about his life and teachings. We do know that Pythagoras and his students believed that everything was related to mathematics and that numbers were the ultimate reality and, through mathematics, everything could be predicted and measured in rhythmic patterns or cycles. According to Iamblichus of Chalcis, Pythagoras once said that "number is the ruler of forms and ideas and the cause of gods and daemons." He was the first man to call himself a philosopher, or lover of wisdom,[2] and Pythagorean ideas exercised a marked influence on Plato. Unfortunately, very little is known about Pythagoras because none of his writings have survived. Many of the accomplishments credited to Pythagoras may actually have been accomplishments of his colleagues and successors. Pythagoras was born on Samos, a Greek island in the eastern Aegean, off the coast of Asia Minor. He was born to Pythais (his mother, a native of Samos) and Mnesarchus (his father, a Phoenician merchant from Tyre). As a young man, he left his native city for Croton, Calabria, in Southern Italy, to escape the tyrannical government of Polycrates. According to Iamblichus, Thales, impressed with his abilities, advised Pythagoras to head to Memphis in Egypt and study with the priests there who were renowned for their wisdom. He was also discipled in the temples of Tyre and Byblos in Phoenicia. It may have been in Egypt where he learned some geometric principles which eventually inspired his formulation of the theorem that is now called by his name. This possible inspiration is presented as an extraordinaire problem in the Berlin Papyrus. Upon his migration from Samos to Croton, Calabria, Italy, Pythagoras established a secret religious society very similar to (and possibly influenced by) the earlier Orphic cult. Pythagoras undertook a reform of the cultural life of Croton, urging the citizens to follow virtue and form an elite circle of followers around himself called Pythagoreans. Very strict rules of conduct governed this cultural center. He opened his school to both male and female students uniformly. Those who joined the inner circle of Pythagoras's society called themselves the Mathematikoi. They lived at the school, owned no personal possessions and were required to assume a mainly vegetarian diet (meat that could be sacrificed was allowed to be eaten). Other students who lived in neighboring areas were also permitted to attend Pythagoras's school. Known as Akousmatikoi, these students were permitted to eat meat and own personal belongings. Richard Blackmore, in his book The Lay Monastery (1714), saw in the religious observances of the Pythagoreans, "the first instance recorded in history of a monastic life." According to Iamblichus, the Pythagoreans followed a structured life of religious teaching, common meals, exercise, reading and philosophical study. Music featured as an essential organizing factor of this life: the disciples would sing hymns to Apollo together regularly; they used the lyre to cure illness of the soul or body; poetry recitations occurred before and after sleep to aid the memory. Flavius Josephus, in his polemical Against Apion, in defence of Judaism against Greek philosophy, mentions that according to Hermippus of Smyrna, Pythagoras was familiar with Jewish beliefs, incorporating some of them in his own philosophy. Towards the end of his life he fled to Metapontum because of a plot against him and his followers by a noble of Croton named Cylon. He died in Metapontum around 90 years old from unknown causes. Bertrand Russell, in A History of Western Philosophy, contended that the influence of Pythagoras on Plato and others was so great that he should be considered the most influential of all western philosophy.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 BC The organization was in some ways a school, in some ways a brotherhood, and in some ways a monastery. It was based upon the religious teachings of Pythagoras and was very secretive. At first, the school was highly concerned with the morality of society. Members were required to live ethically, love one another, share political beliefs, practice pacifism, and devote themselves to the mathematics of nature. Pythagoras's followers were commonly called "Pythagoreans". They are generally accepted as philosophical mathematicians who had an influence on the beginning of axiomatic geometry, which after two hundred years of development was written down by Euclid in The Elements. The Pythagoreans observed a rule of silence called echemythia, the breaking of which was punishable by death. This was because the Pythagoreans believed that a man's words were usually careless and misrepresented him and that when someone was "in doubt as to what he should say, he should always remain silent". Another rule that they had was to help a man "in raising a burden, but do not assist him in laying it down, for it is a great sin to encourage indolence", and they said "departing from your house, turn not back, for the furies will be your attendants"; this axiom reminded them that it was better to learn none of the truth about mathematics, God, and the universe at all than to learn a little without learning all. (The Secret Teachings of All Ages by Manly P. Hall). In his biography of Pythagoras (written seven centuries after Pythagoras's time), Porphyry stated that this silence was "of no ordinary kind." The Pythagoreans were divided into an inner circle called the mathematikoi("mathematicians") and an outer circle called the akousmatikoi ("listeners"). 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." According to Iamblichus, the akousmatikoi were the exoteric disciples who listened to lectures that Pythagoras gave out loud from behind a veil. The akousmatikoi were not allowed to see Pythagoras and they were not taught the inner secrets of the cult. Instead they were taught laws of behavior and morality in the form of cryptic, brief sayings that had hidden meanings. The akousmatikoi recognized the mathematikoi as real Pythagoreans, but not vice versa. After the murder of a number of the mathematikoi by the cohorts of Cylon, a resentful disciple, the two groups split from each other entirely, with Pythagoras's wife Theano and their two daughters leading the mathematikoi. Theano, daughter of the Orphic initiate Brontinus, was a mathematician in her own right. She is credited with having written treatises on mathematics, physics, medicine, and child psychology, although nothing of her writing survives. Her most important work is said to have been a treatise on the philosophical principle of the golden mean. In a time when women were usually considered property and relegated to the role of housekeeper or spouse, Pythagoras allowed women to function on equal terms in his society.[3] The Pythagorean society is associated with prohibitions such as not to step over a crossbar, and not to eat beans. These rules seem like primitive superstition, similar to "walking under a ladder brings bad luck". The abusive epithet mystikos logos ("mystical speech") was hurled at Pythagoras even in ancient times to discredit him. The prohibition on beans could be linked to favism, which is relatively widespread around the Mediterranean. The key here is that akousmatameans "rules", so that the superstitious taboos primarily applied to the akousmatikoi, and many of the rules were probably invented after Pythagoras's death and independent from the mathematikoi (arguably the real preservers of the Pythagorean tradition). The mathematikoi placed greater emphasis on inner understanding than did the akousmatikoi, even to the extent of dispensing with certain rules and ritual practices. For the mathematikoi, being a Pythagorean was a question of innate quality and inner understanding. There was also another way of dealing with the akousmata - by allegorizing them. We have a few examples of this, one being Aristotle's explanations of them: "'step not over a balance', i.e. be not covetous; 'poke not the fire with a sword', i.e. do not vex with sharp words a man swollen with anger, 'eat not heart', i.e. do not vex yourself with grief," etc. We have evidence for Pythagoreans allegorizing in this way at least as far back as the early fifth century BC. This suggests that the strange sayings were riddles for the initiated. The Pythagoreans are known for their theory of the transmigration of souls, and also for their theory that numbers constitute the true nature of things. They performed purification rites and followed and developed various rules of living which they believed would enable their soul to achieve a higher rank among the gods. Much of their mysticism concerning the soul seem inseparable from the Orphic tradition. The Orphics advocated various purificatory rites and practices as well as incubatory rites of descent into the underworld. Pythagoras is also closely linked with Pherecydes of Syros, the man ancient commentators tend to credit as the first Greek to teach a transmigration of souls. Ancient commentators agree that Pherekydes was Pythagoras's most intimate teacher. Pherekydes expounded his teaching on the soul in terms of a pentemychos ("five-nooks", or "five hidden cavities") - the most likely origin of the Pythagorean use of the pentagram, used by them as a symbol of recognition among members and as a symbol of inner health (ugieia). 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 (c).

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, band 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.[5] 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.[6] 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. Today, Pythagoras is revered as a prophet by the Ahl al-Tawhid or Druze faith along with his fellow Greek, Plato. But Pythagoras also had his critics, such as Heraclitus who said that "much learning does not teach wisdom; otherwise it would have taught Hesiod and Pythagoras, and again Xenophanes and Hecataeus".[7] Pythagoras' religious and scientific views were, in his opinion, inseparably interconnected. Religiously, Pythagoras was a believer of metempsychosis. He believed in transmigration, or the reincarnation of the soul again and again into the bodies of humans, animals, or vegetables until it became moral. His ideas of reincarnation were influenced by ancient Greek religion. He was one of the first to propose that the thought processes and the soul were located in the brain and not the heart. He himself claimed to have lived four lives that he could remember in detail, and heard the cry of his dead friend in the bark of a dog. One of Pythagoras' beliefs was that the essence of being is number. Thus, being relies on stability of all things that create the universe. Things like health relied on a stable proportion of elements; too much or too little of one thing causes an imbalance that makes a being unhealthy. Pythagoras viewed thinking as the calculating with the idea numbers. When combined with the Folk theories, the philosophy evolves into a belief that Knowledge of the essence of being can be found in the form of numbers. If this is taken a step further, one can say that because mathematics is an unseen essence, the essence of being is an unseen characteristic that can be encountered by the study of mathematics. No texts by Pythagoras survive, although forgeries under his name - a few of which remain extant - did circulate in antiquity. Critical ancient sources like Aristotle and Aristoxenus cast doubt on these writings. Ancient Pythagoreans usually quoted their master's doctrines with the phrase autos ephe ("he himself said") - emphasizing the essentially oral nature of his teaching. Pythagoras appears as a character in the last book of Ovid's Metamorphoses, where Ovid has him expound upon his philosophical viewpoints. Pythagoras has been quoted as saying, "No man is free who cannot command himself." There is another side to Pythagoras, as he became the subject of elaborate legends surrounding his historic persona. Aristotle described Pythagoras as a wonder-worker and somewhat of a supernatural figure, attributing to him such aspects as a golden thigh, which was a sign of divinity. According to Aristotle and others' accounts, some ancients believed that he had the ability to travel through space and time, and to communicate with animals and plants.[8] An extract from Brewer's Dictionary of Phrase and Fable's entry entitled "Golden Thigh": Pythagoras is said to have had a golden thigh, which he showed to Abaris, the Hyperborean priest, and exhibited in the Olympic games.[9] Another legend, also taken from Brewer's Dictionary, describes his writing on the moon: Pythagoras asserted he could write on the moon. His plan of operation was to write on a looking-glass in blood, and place it opposite the moon, when the inscription would appear photographed or reflected on the moon's disc.[10]

One of Pythagoras's major accomplishments was the discovery that music was based on proportional intervals of the numbers one through four. He believed that the number system, and therefore the universe system, was based on the sum of these numbers: ten. Pythagoreans swore by the Tetrachtys of the Decad, or ten, rather than by the gods. Odd numbers were masculine and even were feminine. He discovered the theory of mathematical proportions, constructed from three to five geometrical solids. One member of his order, Hippasos, also discovered Irrational Numbers, but the idea was unthinkable to Pythagoras, and according to legend, Hippasos was executed. Pythagoras (or the Pythagoreans) also discovered square numbers. They found that if one took, for example, four small stones and arranged them into a square, each side of the square was not only equivalent to the other, but that when the two sides were multiplied together, they equaled the sum total of stones in the square arrangement, hence the name "Square Root"[11]. He was one of the first to think that the earth was round, that all planets have an axis, and that all the planets travel around one central point. He originally identified that point as Earth, but later renounced it for the idea that the planets revolve around a central "fire" that he never identified as the sun. He also believed that the moon was another planet that he called a "counter-Earth" - furthering his belief in the Limited-Unlimited. Pythagoras or in a broader sense, the Pythagoreans, allegedly exercised an important influence on the work of Plato. According to R. M. Hare, his influence consists of three points: a) the platonic Republic might be related to the idea of "a tightly organized community of like-minded thinkers", like the one established by Pythagoras in Croton. b) there is evidence that Plato possibly took from Pythagoras the idea that mathematics and, generally speaking, abstract thinking is a secure basis for philosophical thinking as well as "for substantial theses in science and morals". c) Plato and Pythagoras shared a "mystical approach to the soul and its place in the material world". It is probable that both have been influenced by Orphism.[12] Plato's harmonics were clearly influenced by the work of Archytas, a genuine Pythagorean of the third generation, who made important contributions to geometry, reflected in Book VIII of Euclid's Elements. In the legends of ancient Rome, Numa Pompilius, the second King of Rome, is said to have studied under Pythagoras. This is unlikely, since the commonly accepted dates for the two lives do not overlap. Pythagoras started a secret society called the Pythagorean brotherhood devoted to the study of mathematics. This had a great effect on future esoteric traditions, such as Rosicrucianism and Freemasonry, both of which were occult groups dedicated to the study of mathematics and both of which claimed to have evolved out of the Pythagorean brotherhood. The mystical and occult qualities of Pythagorean mathematics are discussed in a chapter of Manly P. Hall's The Secret Teachings of All Agesentitled "Pythagorean Mathematics". Pythagorean theory was tremendously influential on later numerology, which was extremely popular throughout the Middle East in the ancient world. The 8th-century Muslim alchemist Jabir ibn Hayyan grounded his work in an elaborate numerology greatly influenced by Pythagorean theory

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