Who is pythagoras and what items did he discover




















Since they felt that fire was more important than earth, the focal point of the universe must be fire. Pythagoras was the first to perceive that Venus at night and Venus toward the beginning of the day were the same planet. Pythagoras started the idea of a numerical system, and therefore the beginning of mathematics. To the Pythagoreans, genuine numbers were the most vital thing, and numbers make up the world.

A soul was believed to exist in both animal and vegetable life, even though there is no proof to indicate that Pythagoras thought the spirit could be contained in a plant. It could, however, be contained in the body of a creature, and Pythagoras professed to have heard the voice of a dead companion in the wail of a dog being beaten. Pythagoras is most famous for his ideas in geometry. He was the first to propose that the square of the hypotenuse the side of the triangle opposite to the right angle is equivalent to the sum of the squares of the opposite two sides.

Even though this hypothesis was first put forward by the Babylonians, Pythagoras was first to demonstrate it. It is additionally thought that he invented the tetractys, a triangular figure consisting of 10 points arranged in four rows, with one, two, three, and four points in each row. Pythagoras believed that 10 was the ideal number. Plato may also have obtained the idea that mathematical and dynamic ideas are behind logic, science, and morality from Pythagoras.

Plato and Pythagoras shared a magical way to deal with the spirit and its place in the material world, and it is likely that both were impacted by Orphism, a set of religious beliefs and practices originating in the ancient Greek world. Pythagoras established the mysterious society of the Pythagoreans in southern Italy.

Porphyry VP 48—53 explicitly cites Moderatus of Gades as one of his sources. In the Pythagorean Memoirs , Pythagoras is said to have adopted the Monad and the Indefinite Dyad as incorporeal principles, from which arise first the numbers, then plane and solid figures and finally the bodies of the sensible world Diogenes Laertius VIII.

Diels, Doxographi Graeci I. The testimony of Aristotle makes completely clear, however, that this was the philosophical system of Plato in his later years and not that of Pythagoras or even the later Pythagoreans. This view of Pythagoreanism finds its way into the doxography of Aetius either because Theophrastus followed the early Academy rather than his teacher Aristotle Burkert a, 66 or because the Theophrastan doxography on Pythagoras was rewritten in the first century BCE under the influence of Neopythagoreanism Diels , ; Zhmud a, The evidence for the early Academy is, however, very limited and some reject the thesis that its members assigned late Platonic metaphysics to Pythagoras Zhmud , — Proclus quotes a passage in which Speusippus assigns to the ancients, who in this context are the Pythagoreans, the One and the Indefinite Dyad.

Some scholars argue that this is not a genuine fragment of Speusippus but rather a later fabrication see Zhmud a, — and for a response Dillon , If the Academy did not assign the One and the Dyad to Pythagoras, however, it becomes less clear how these principles came to be assigned to him. If we step back for a minute and compare the sources for Pythagoras with those available for other early Greek philosophers, the extent of the difficulties inherent in the Pythagorean Question becomes clear.

Since Pythagoras wrote no books, this most fundamental of all sources is denied us. In dealing with Heraclitus, the modern scholar turns with reluctance next to the doxographical tradition, the tradition represented by Aetius in the first century CE, which preserves in handbook form a systematic account of the beliefs of the Greek philosophers on a series of topics having to do with the physical world and its first principles.

Here again the case of Pythagoras is exceptional. Thus, the second standard source for evidence for early Greek philosophy is, in the case of Pythagoras, corrupted. Whatever views Pythagoras might have had are replaced by late Platonic metaphysics in the doxographical tradition.

A third source of evidence for early Greek philosophy is regarded with great skepticism by most scholars and, in the case of most early Greek philosophers, used only with great caution. This is the biographical tradition represented by the Lives of the Philosophers written by Diogenes Laertius. Unfortunately, these two additional lives are written by authors Iamblichus and Porphyry whose goal is explicitly non-historical, and all three of the lives rely heavily on authors in the Neopythagorean tradition, whose goal was to show that all later Greek philosophy, insofar as it was true, had been stolen from Pythagoras.

The historian Timaeus of Tauromenium ca. In some cases, the fragments of these early works are clearly identified in the later lives, but in other cases we may suspect that they are the source of a given passage without being able to be certain. Large problems remain even in the case of these sources. They were all written — years after the death of Pythagoras; given the lack of written evidence for Pythagoras, they are based largely on oral traditions.

Even among fourth-century authors that had at least some pretensions to historical accuracy and who had access to the best information available, there are widely divergent presentations, simply because such contradictions were endemic to the evidence available in the fourth century.

What we can hope to obtain from the evidence presented by Aristotle, Aristoxenus, Dicaearchus, and Timaeus is thus not a picture of Pythagoras that is consistent in all respects but rather a picture that at least defines the main areas of his achievement. This testimony is extremely limited, about twenty brief references, but this dearth of evidence is not unique to Pythagoras. The pre-Aristotelian testimony for Pythagoras is more extensive than for most other early Greek philosophers and is thus testimony to his fame.

In the case of Pythagoras, what is striking is the essential agreement of Plato and Aristotle in their presentation of his significance.

Aristotle frequently discusses the philosophy of Pythagoreans, whom he dates to the middle and second half of the fifth century and who posited limiters and unlimiteds as first principles. Aristotle strikingly may never refer to Pythagoras himself in his extant writings Metaph. In the fragments of his now lost two-book treatise on the Pythagoreans, Aristotle does discuss Pythagoras himself, but the references are all to Pythagoras as a founder of a way of life, who forbade the eating of beans Fr.

Zhmud a, — argues that in one place Aristotle also describes Pythagoras as a mathematician Fr. For Aristotle Pythagoras did not belong to the succession of thinkers starting with Thales, who were attempting to explain the basic principles of the natural world, and hence he could not see what sense it made to call a fifth-century thinker like Philolaus, who joined that succession by positing limiters and unlimiteds as first principles, a Pythagorean.

Plato is often thought to be heavily indebted to the Pythagoreans, but he is almost as parsimonious in his references to Pythagoras as Aristotle and mentions him only once in his writings. In the Philebus , Plato does describe the philosophy of limiters and unlimiteds, which Aristotle assigns to the so-called Pythagoreans of the fifth century and which is found in the fragments of Philolaus, but like Aristotle he does not ascribe this philosophy to Pythagoras himself.

Scholars, both ancient and modern, under the influence of the later glorification of Pythagoras, have supposed that the Prometheus, whom Plato describes as hurling the system down to men, was Pythagoras e. The fragments of Philolaus show that he was the primary figure of this group. For both Plato and Aristotle, then, Pythagoras is not a part of the cosmological and metaphysical tradition of Presocratic philosophy nor is he closely connected to the metaphysical system presented by fifth-century Pythagoreans like Philolaus; he is instead the founder of a way of life.

References to Pythagoras by Xenophanes ca. For the details of his life we have to rely on fourth-century sources such as Aristoxenus, Dicaearchus and Timaeus of Tauromenium. There is a great deal of controversy about his origin and early life, but there is agreement that he grew up on the island of Samos, near the birthplace of Greek philosophy, Miletus, on the coast of Asia Minor.

There are a number of reports that he traveled widely in the Near East while living on Samos, e. To some extent reports of these trips are an attempt to claim the ancient wisdom of the east for Pythagoras and some scholars totally reject them Zhmud , 83—91 , but relatively early sources such as Herodotus II.

Aristoxenus says that he left Samos at the age of forty, when the tyranny of Polycrates, who came to power ca. This chronology would suggest that he was born ca. He then emigrated to the Greek city of Croton in southern Italy ca. There are a variety of stories about his death, but the most reliable evidence Aristoxenus and Dicaearchus suggests that violence directed against Pythagoras and his followers in Croton ca.

There is little else about his life of which we can be confident. The evidence suggests that Pythagoras did not write any books.

No source contemporaneous with Pythagoras or in the first two hundred years after his death, including Plato, Aristotle and their immediate successors in the Academy and Lyceum, quotes from a work by Pythagoras or gives any indication that any works written by him were in existence. Several later sources explicitly assert that Pythagoras wrote nothing e.

This fragment shows only that Pythagoras read the writings of others, however, and says nothing about him writing something of his own. The second of these is a Sacred Discourse , which some have wanted to trace back to Pythagoras himself. The idea that Pythagoras wrote such a Sacred Discourse seems to arise from a misreading of the early evidence. Herodotus says that the Pythagoreans agreed with the Egyptians in not allowing the dead to be buried in wool and then asserts that there is a sacred discourse about this II.

For an interesting but ultimately unconvincing attempt to argue that the historical Pythagoras did write books, see Riedweg , 42—43 and the response by Huffman a, — One of the manifestations of the attempt to glorify Pythagoras in the later tradition is the report that he, in fact, invented the word philosophy. This story goes back to the early Academy, since it is first found in Heraclides of Pontus Cicero, Tusc. Moreover, the story depends on a conception of a philosopher as having no knowledge but being situated between ignorance and knowledge and striving for knowledge.

Such a conception is thoroughly Platonic, however see, e. For a recent attempt to defend at least the partial accuracy of the story, see Riedweg 90—97 and the response by Huffman a—; see also Zhmud a, — Even if he did not invent the word, what can we say about the philosophy of Pythagoras? For the reasons given in 1. The Pythagorean Question and 2. There is general agreement as to what the pre-Aristotelian evidence is, although there are differences in interpretation of it. It is crucial to decide this question before developing a picture of the philosophy of Pythagoras since chapter 19, if it is by Dicaearchus, is our earliest summary of Pythagorean philosophy.

Porphyry is very reliable about quoting his sources. He explicitly cites Dicaearchus at the beginning of Chapter 18 and names Nicomachus as his source at the beginning of chapter The material in chapter 19 follows seamlessly on chapter the description of the speeches that Pythagoras gave upon his arrival in Croton in chapter 18 is followed in chapter 19 by an account of the disciples that he gained as the result of those speeches and a discussion of what he taught these disciples.

Thus, the onus is on anyone who would claim that Porphyry changes sources before the explicit change at the beginning of chapter Wehrli gives no reason for not including chapter 19 and the great majority of scholars accept it as being based on Dicaearchus see the references in Burkert a, , n. Zhmud a, following Philip , argues that the passage cannot derive from Dicaearchus, since it presents immortality of the soul with approval, whereas Dicaearchus did not accept its immortality.

However, the passage merely reports that Pythagoras introduced the notion of the immortality of the soul without expressing approval or disapproval. Zhmud lists other features of the chapter that he regards as unparalleled in fourth-century sources a, but, since the evidence is so fragmentary, such arguments from silence can carry little weight. In the face of the Pythagorean question and the problems that arise even regarding the early sources, it is reasonable to wonder if we can say anything about Pythagoras.

A minimalist might argue that the early evidence only allows us to conclude that Pythagoras was a historical figure who achieved fame for his wisdom but that it is impossible to determine in what that wisdom consisted. We might say that he was interested in the fate of the soul and taught a way of life, but we can say nothing precise about the nature of that life or what he taught about the soul Lloyd There is some reason to believe, however, that something more than this can be said.

The earliest evidence makes clear that above all Pythagoras was known as an expert on the fate of our soul after death. Herodotus tells the story of the Thracian Zalmoxis, who taught his countrymen that they would never die but instead go to a place where they would eternally possess all good things IV.

Among the Greeks the tradition arose that this Zalmoxis was the slave of Pythagoras. Ion of Chios 5 th c. Although Xenophanes clearly finds the idea ridiculous, the fragment shows that Pythagoras believed in metempsychosis or reincarnation, according to which human souls were reborn into other animals after death. According to Herodotus, the Egyptians believed that the soul was reborn as every sort of animal before returning to human form after 3, years.

Without naming names, he reports that some Greeks both earlier and later adopted this doctrine; this seems very likely to be a reference to Pythagoras earlier and perhaps Empedocles later. Many doubt that Herodotus is right to assign metempsychosis to the Egyptians, since none of the other evidence we have for Egyptian beliefs supports his claim, but it is nonetheless clear that we cannot assume that Pythagoras accepted the details of the view Herodotus ascribes to them. Similarly both Empedocles see Inwood , 55—68 and Plato e.

Did he think that we ever escape the cycle of reincarnations? We simply do not know. The fragment of Ion quoted above may suggest that the soul could have a pleasant existence after death between reincarnations or even escape the cycle of reincarnation altogether, but the evidence is too weak to be confident in such a conclusion. In the fourth century several authors report that Pythagoras remembered his previous human incarnations, but the accounts do not agree on the details.

Dicaearchus Aulus Gellius IV. Dicaearchus continues the tradition of savage satire begun by Xenophanes, when he suggests that Pythagoras was the beautiful prostitute, Alco, in another incarnation Huffman b, — It is not clear how Pythagoras conceived of the nature of the transmigrating soul but a few tentative conjectures can be made Huffman Transmigration does not require that the soul be immortal; it could go through several incarnations before perishing.

It has often been assumed that the transmigrating soul is immaterial, but Philolaus seems to have a materialistic conception of soul and he may be following Pythagoras. Similarly, it is doubtful that Pythagoras thought of the transmigrating soul as a comprehensive soul that includes all psychic faculties. His ability to recognize something distinctive of his friend in the puppy if this is not pushing the evidence of a joke too far and to remember his own previous incarnations show that personal identity was preserved through incarnations.

The manuscript was prepared in and published in Loomis received literally hundreds of new proofs from after his book was released up until his death, but he could not keep up with his compendium.

As for the exact number of proofs, no one is sure how many there are. Surprisingly, geometricians often find it quite difficult to determine whether certain proofs are in fact distinct proofs.

He died on 11 December , and the obituary was published as he had written it, except for the date of his death and the addresses of some of his survivors.

According to his autobiography, a preteen Albert Einstein Figure 8. Many known proofs use similarity arguments, but this one is notable for its elegance, simplicity and the sense that it reveals the connection between length and area that is at the heart of the theorem.

At the age of 12, I experienced a second wonder of a totally different nature: in a little book dealing with Euclidean plane geometry, which came into my hands at the beginning of a school year.

Here were assertions, as for example the intersection of the three altitudes of a triangle in one point, which — though by no means evident — could nevertheless be proved with such certainty that any doubt appeared to be out of the question. This lucidity and certainty made an indescribable impression upon me. For example I remember that an uncle told me the Pythagorean Theorem before the holy geometry booklet had come into my hands.

Einstein Figure 9 used the Pythagorean Theorem in the Special Theory of Relativity in a four-dimensional form , and in a vastly expanded form in the General Theory of Relatively.

The following excerpts are worthy of inclusion. Special relativity is still based directly on an empirical law, that of the constancy of the velocity of light. The fact that such a metric is called Euclidean is connected with the following. The postulation of such a metric in a three-dimensional continuum is fully equivalent to the postulation of the axioms of Euclidean Geometry.

The defining equation of the metric is then nothing but the Pythagorean Theorem applied to the differentials of the co-ordinates. Such transformations are called Lorentz transformations.

From the latest results of the theory of relativity, it is probable that our three-dimensional space is also approximately spherical , that is, that the laws of disposition of rigid bodies in it are not given by Euclidean geometry, but approximately by spherical geometry. According to the general theory of relativity , the geometrical properties of space are not independent, but they are determined by matter. I wished to show that space time is not necessarily something to which one can ascribe to a separate existence, independently of the actual objects of physical reality.

Physical objects are not in space, but these objects are spatially extended. The above excerpts — from the genius himself — precede any other person's narrative of the Theory of Relativity and the Pythagorean Theorem. Accordingly, I now provide a less demanding excerpt, albeit one that addresses the effects of the Special and General theories of relativity.

The system of units in which the speed of light c is the unit of velocity allows to cast all formulas in a very simple form. The Pythagorean Theorem graphically relates energy, momentum and mass. Euclid of Alexandria was a Greek mathematician Figure 10 , and is often referred to as the Father of Geometry. The date and place of Euclid's birth, and the date and circumstances of his death, are unknown, but it is thought that he lived circa BCE.

His work Elements , which includes books and propositions, is the most successful textbook in the history of mathematics. In it, the principles of what is now called Euclidean Geometry were deduced from a small set of axioms.

When Euclid wrote his Elements around BCE , he gave two proofs of the Pythagorean Theorem: The first, Proposition 47 of Book I, relies entirely on the area relations and is quite sophisticated; the second, Proposition 31 of Book VI, is based on the concept of proportion and is much simpler. He may have used Book VI Proposition 31, but, if so, his proof was deficient, because the complete theory of Proportions was only developed by Eudoxus, who lived almost two centuries after Pythagoras.

Euclid's Elements furnishes the first and, later, the standard reference in geometry. It is a mathematical and geometric treatise consisting of 13 books. It comprises a collection of definitions, postulates axioms , propositions theorems and constructions and mathematical proofs of the propositions.

Euclid provided two very different proofs, stated below, of the Pythagorean Theorem. This is probably the most famous of all the proofs of the Pythagorean proposition. In right-angled triangles the figure on the side opposite the right angle equals the sum of the similar and similarly described figures on the sides containing the right angle.

In right-angled triangles the square on the side opposite the right angle equals the sum of the squares on the sides containing the right angle. Euclid I 47 is often called the Pythagorean Theorem , called so by Proclus — a Greek philosopher who became head of Plato's Academy and is important mathematically for his commentaries on the work of other mathematicians — and others centuries after Pythagoras and even centuries after Euclid.

Although many of the results in Elements originated with earlier mathematicians, one of Euclid's accomplishments was to present them in a single, logically coherent framework, making them easy to use and easy to reference, including a system of rigorous mathematical proofs that remains the basis of mathematics twenty-three centuries later.

Although best known for its geometric results, Elements also includes number theory. It considers the connection between perfect numbers and Mersenne primes, the infinitude of prime numbers and the Euclidean algorithm for finding the greatest common divisor of two numbers. The geometrical system described in the Elements was long known simply as geometry , and was considered to be the only geometry possible.

Today, however, this system is often referred to as Euclidean Geometry to distinguish it from other so-called Non-Euclidean geometries that mathematicians discovered in the nineteenth century. At this point in my plotting of the year-old story of Pythagoras, I feel it is fitting to present one proof of the famous theorem. For me, the simplest proof among the dozens of proofs that I read in preparing this article is that shown in Figure Start with four copies of the same triangle.

See upper part of Figure See lower part of Figure In the seventeenth century, Pierre de Fermat — Figure 14 investigated the following problem: for which values of n are there integer solutions to the equation. Fermat conjectured that there were no non-zero integer solutions for x and y and z when n was greater than 2.

He did not leave a proof, though. Instead, in the margin of a textbook, he wrote that he knew that this relationship was not possible, but he did not have enough room on the page to write it down.

Thales 's pupil, Anaximander, lectured on Miletus and Pythagoras attended these lectures. Anaximander certainly was interested in geometry and cosmology and many of his ideas would influence Pythagoras's own views. In about BC Pythagoras went to Egypt. This happened a few years after the tyrant Polycrates seized control of the city of Samos.

There is some evidence to suggest that Pythagoras and Polycrates were friendly at first and it is claimed [ 5 ] that Pythagoras went to Egypt with a letter of introduction written by Polycrates. In fact Polycrates had an alliance with Egypt and there were therefore strong links between Samos and Egypt at this time.

The accounts of Pythagoras's time in Egypt suggest that he visited many of the temples and took part in many discussions with the priests. According to Porphyry [ 12 ] and [ 13 ] Pythagoras was refused admission to all the temples except the one at Diospolis where he was accepted into the priesthood after completing the rites necessary for admission.

It is not difficult to relate many of Pythagoras's beliefs, ones he would later impose on the society that he set up in Italy, to the customs that he came across in Egypt. For example the secrecy of the Egyptian priests, their refusal to eat beans, their refusal to wear even cloths made from animal skins, and their striving for purity were all customs that Pythagoras would later adopt.

Porphyry in [ 12 ] and [ 13 ] says that Pythagoras learnt geometry from the Egyptians but it is likely that he was already acquainted with geometry, certainly after teachings from Thales and Anaximander. Polycrates abandoned his alliance with Egypt and sent 40 ships to join the Persian fleet against the Egyptians. Pythagoras was taken prisoner and taken to Babylon.

Iamblichus writes that Pythagoras see [ 8 ] Whilst he was there he gladly associated with the Magoi He also reached the acme of perfection in arithmetic and music and the other mathematical sciences taught by the Babylonians Polycrates had been killed in about BC and Cambyses died in the summer of BC, either by committing suicide or as the result of an accident.

The deaths of these rulers may have been a factor in Pythagoras's return to Samos but it is nowhere explained how Pythagoras obtained his freedom. Darius of Persia had taken control of Samos after Polycrates' death and he would have controlled the island on Pythagoras's return.

This conflicts with the accounts of Porphyry and Diogenes Laertius who state that Polycrates was still in control of Samos when Pythagoras returned there. Pythagoras made a journey to Crete shortly after his return to Samos to study the system of laws there.

Back in Samos he founded a school which was called the semicircle. Iamblichus [ 8 ] writes in the third century AD that They do this because they think one should discuss questions about goodness, justice and expediency in this place which was founded by the man who made all these subjects his business.

Outside the city he made a cave the private site of his own philosophical teaching, spending most of the night and daytime there and doing research into the uses of mathematics Pythagoras left Samos and went to southern Italy in about BC some say much earlier. Iamblichus [ 8 ] gives some reasons for him leaving.



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