Ten years later, when Persia invaded Egypt, Pythagoras was taken prisoner and sent to Babylon in what is now Iraq , where he met the Magoi, priests who taught him sacred rites. Iamblichus AD , a Syrian philosopher, wrote about Pythagoras, "He also reached the acme of perfection in arithmetic and music and the other mathematical sciences taught by the Babylonians His methods of teaching were not popular with the leaders of Samos, and their desire for him to become involved in politics did not appeal to him, so he left.
Pythagoras settled in Crotona, a Greek colony in southern Italy, about BC, and founded a philosophical and religious school where his many followers lived and worked. The Pythagoreans lived by rules of behavior, including when they spoke, what they wore and what they ate. Pythagoras was the Master of the society, and the followers, both men and women, who also lived there, were known as mathematikoi.
They had no personal possessions and were vegetarians. Another group of followers who lived apart from the school were allowed to have personal possessions and were not expected to be vegetarians. He suggests that the Pythagorean way of life differed little from standard aristocratic morality Zhmud a, If, however, the Pythagorean way of life was little out of the ordinary, why do Plato and Isocrates specifically comment on how distinctive those who followed it were?
The silence of fifth-century sources about people practicing acusmata is not terribly surprising given the very meager sources for the Greek cities in southern Italy in the period. We would then have lots of people who followed the acusmata of the name in the catalogue appear nowhere else.
Moreover, other scholars argue that archaic Greek society in southern Italy was pervaded by religion and the presence of similar precepts in authors such as Hesiod show that adherence to taboos such as are found in the acusmata would not have caused a scandal and adherence to many of them would have gone unobserved by outsiders Gemelli Marciano , — Once again a problem of source criticism raises its head.
Zhmud argues that the split between acusmatici who blindly followed the acusmata and the mathematici who learned the reasons for them see the fifth paragraph of section 5 below is a creation of the later tradition, appearing first in Clement of Alexandria and disappearing after Iamblichus Zhmud a, — He also notes that the term acusmata appears first in Iamblichus On the Pythagorean Life 82—86 and suggests that it is also a creation of the later tradition.
The Pythagorean maxims did exist earlier, as the testimony of Aristotle shows, but they were known as symbola , were originally very few in number and were mainly a literary phenomena rather than being tied to people who actually practiced them Zhmud a, — Indeed, the description of the split in what is likely to be the original version Iamblichus, On General Mathematical Science So the question of whether Pythagoras taught a way of life tightly governed by the acusmata turns again on whether key passages in Iamblichus On the Pythagorean Life 81—87, On General Mathematical Science If they do, we have very good reason to believe that Pythagoras taught such a life, if they do not the issue is less clear.
The testimony of fourth-century authors such as Aristoxenus and Dicaearchus indicates that the Pythagoreans also had an important impact on the politics and society of the Greek cities in southern Italy.
Dicaearchus reports that, upon his arrival in Croton, Pythagoras gave a speech to the elders and that the leaders of the city then asked him to speak to the young men of the town, the boys and the women Porphyry, VP The acusmata teach men to honor their wives and to beget children in order to insure worship for the gods Iamblichus, VP 84—6. Dicaearchus reports that the teaching of Pythagoras was largely unknown, so that Dicaearchus cannot have known of the content of the speech to the women or of any of the other speeches; the speeches presented in Iamblichus VP 37—57 are thus likely to be later forgeries Burkert a, , but there is early evidence that he gave different speeches to different groups Antisthenes V A On the other hand, it is noteworthy that Plato explicitly presents Pythagoras as a private rather than a public figure R.
It seems most likely that the Pythagorean societies were in essence private associations but that they also could function as political clubs see Zhmud a, — , while not being a political party in the modern sense; their political impact should perhaps be better compared to modern fraternal organizations such as the Masons. Thus, the Pythagoreans did not rule as a group but had political impact through individual members who gained positions of authority in the Greek city-states in southern Italy.
See further Burkert a, ff. It should be clear from the discussion above that, while the early evidence shows that Pythagoras was indeed one of the most famous early Greek thinkers, there is no indication in that evidence that his fame was primarily based on mathematics or cosmology.
Neither Plato nor Aristotle treats Pythagoras as having contributed to the development of Presocratic cosmology, although Aristotle in particular discusses the topic in some detail in the first book of the Metaphysics and elsewhere.
Thus, for Dicaearchus too, it is not as a mathematician or Presocratic writer on nature that Pythagoras is famous. At first sight, it appears that Eudemus did assign Pythagoras a significant place in the history of geometry. Eudemus is reported as beginning with Thales and an obscure figure named Mamercus, but the third person mentioned by Proclus in this report is Pythagoras, immediately before Anaxagoras.
There is no mention of the Pythagorean theorem, but Pythagoras is said to have transformed the philosophy of geometry into a form of liberal education, to have investigated its theorems in an immaterial and intellectual way and specifically to have discovered the study of irrational magnitudes and the construction of the five regular solids. Proclus elsewhere quotes long passages from Iamblichus and is doing the same here.
Even those who want to assign Pythagoras a larger role in early Greek mathematics recognize that most of what Proclus says here cannot go back to Eudemus Zhmud a, — Thus, not only is Pythagoras not commonly known as a geometer in the time of Plato and Aristotle, but also the most authoritative history of early Greek geometry assigns him no role in the history of geometry in the overview preserved in Proclus. Eudemus does not assign the discoveries to any specific Pythagorean, and they are hard to date.
The crucial point to note is that Eudemus does not assign these discoveries to Pythagoras himself. The first Pythagorean whom we can confidently identify as an accomplished mathematician is Archytas in the late fifth and the first half of the fourth century. Are we to conclude, then, that Pythagoras had nothing to do with mathematics or cosmology?
The evidence is not quite that simple. Several things need to be noted about this tradition, however, in order to understand its true significance. Although a number of modern scholars have speculated on what sort of proof Pythagoras might have used e. All that this tradition ascribes to Pythagoras, then, is discovery of the truth contained in the theorem. The truth may not have been in general form but rather focused on the simplest such triangle with sides 3, 4 and 5 , pointing out that such a triangle and all others like it will have a right angle.
Robson It is possible, then, that Pythagoras just passed on to the Greeks a truth that he learned from the East. If the story is to have any force and if it dates to the fourth century, it shows that Pythagoras was famous for a connection to a certain piece of geometrical knowledge, but it also shows that he was famous for his enthusiastic response to that knowledge, as evidenced in his sacrifice of oxen, not for any geometric proof.
What emerges from this evidence, then, is not Pythagoras as the master geometer, who provides rigorous proofs, but rather Pythagoras as someone who recognizes and celebrates certain geometrical relationships as of high importance. It is striking that a very similar picture of Pythagoras emerges from the evidence for his cosmology. A famous discovery is attributed to Pythagoras in the later tradition, i. The only early source to associate Pythagoras with the whole number ratios that govern the concords is Xenocrates Fr.
It may be once again that Pythagoras knew of the relationship without either having discovered it or having demonstrated it scientifically. The relationship was probably first discovered by instrument makers, and specifically makers of wind instruments rather than stringed instruments Barker , Here in the acusmata , these four numbers are identified with one of the primary sources of wisdom in the Greek world, the Delphic oracle. This acusma thus seems to be based on the knowledge of the relationship between the concords and the whole number ratios.
The picture of Pythagoras that emerges from the evidence is thus not of a mathematician, who offered rigorous proofs, or of a scientist, who carried out experiments to discover the nature of the natural world, but rather of someone who sees special significance in and assigns special prominence to mathematical relationships that were in general circulation.
Some might suppose that this is a reference to a rigorous treatment of arithmetic, such as that hypothesized by Becker , who argued that Euclid IX.
There is, moreover, no talk of mathematical proof or a deductive system in the passage from Aristoxenus just quoted. Pythagoras is known for the honor he gives to number and for removing it from the practical realm of trade and instead pointing to correspondences between the behavior of number and the behavior of things. The doxographical tradition reports that Pythagoras discovered the sphericity of the earth, the five celestial zones and the identity of the evening and morning star Diogenes Laertius VIII.
In each case, however, Burkert has shown that these reports seem to be false and the result of the glorification of Pythagoras in the later tradition, since the earliest and most reliable evidence assigns these same discoveries to someone else a, ff. Thus, Theophrastus, who is the primary basis of the doxographical tradition, says that it was Parmenides who discovered the sphericity of the earth Diogenes Laertius VIII.
Parmenides is also identified as the discoverer of the identity of the morning and evening star Diogenes Laertius IX. The identification of the five celestial zones depends on the discovery of the obliquity of the ecliptic, and some of the doxography duly assigns this discovery to Pythagoras as well and claims that Oenopides stole it from Pythagoras Aetius II.
As was shown above, Pythagoras saw the cosmos as structured according to number insofar as the tetraktys is the source of all wisdom. His cosmos was also imbued with a moral significance, which is in accordance with his beliefs about reincarnation and the fate of the soul West , —; Huffman , 60— Zhmud calls these cosmological acusmata into question a, — , noting that some only appear in Porphyry, but Porphyry explicitly identifies Aristotle as his source and we have no reason to doubt him VP The best analogy for the type of account of the cosmos which Pythagoras gave might be some of the myths which appear at the end of Platonic dialogues such as the Phaedo , Gorgias or Republic , where cosmology has a primarily moral purpose.
Should the doctrine of the harmony of the spheres be assigned to Pythagoras? Certainly the acusma which talks of the sirens singing in the harmony represented by the tetraktys suggests that there might have been a cosmic music and that Pythagoras may well have thought that the heavenly bodies, which we see move across the sky at night, made music by their motions.
The first such cosmic model in the Pythagorean tradition is that of Philolaus in the second half of the fifth century, a model which still shows traces of the connection to the moral cosmos of Pythagoras in its account of the counter-earth and the central fire see Philolaus. If Pythagoras was primarily a figure of religious and ethical significance, who left behind an influential way of life and for whom number and cosmology primarily had significance in this religious and moral context, how are we to explain the prominence of rigorous mathematics and mathematical cosmology in later Pythagoreans such as Philolaus and Archytas?
It is important to note that this is not just a question asked by modern scholars but was already a central question in the fourth century BCE. What is the connection between Pythagoras and fifth-century Pythagoreans? The tradition of a split between two groups of Pythagoreans in the fifth century, the mathematici and the acusmatici , points to the same puzzlement.
The evidence for this split is quite confused in the later tradition, but Burkert a, ff. The acusmatici , who are clearly connected by their name to the acusmata , are recognized by the other group, the mathematici , as genuine Pythagoreans, but the acusmatici do not regard the philosophy of the mathematici as deriving from Pythagoras but rather from Hippasus. The mathematici appear to have argued that, while the acusmatici were indeed Pythagoreans, it was the mathematici who were the true Pythagoreans; Pythagoras gave the acusmata to those who did not have the time to study the mathematical sciences, so that they would at least have moral guidance, while to those who had the time to fully devote themselves to Pythagoreanism he gave training in the mathematical sciences, which explained the reasons for this guidance.
This tradition thus shows that all agreed that the acusmata represented the teaching of Pythagoras, but that some regarded the mathematical work associated with the mathematici as not deriving from Pythagoras himself, but rather from Hippasus on the controversy about the evidence for this split into two groups of Pythagoreans see the fifth paragraph of section 4. For fourth-century Greeks as for modern scholars, the question is whether the mathematical and scientific side of later Pythagoreanism derived from Pythagoras or not.
The picture of Pythagoras presented above is inevitably based on crucial decisions about sources and has been recently challenged in a searching critique Zhmud a. In many cases, he argues, the evidence suggests that early Pythagoreanism was more scientific and that religious and mythic elements only gained in importance in the later tradition. One of the central pieces of evidence for this view is that the tetraktys does not first appear until late in the tradition, in Aetius in the first century CE DK 1.
Zhmud himself agrees that sections 82—86 of On the Pythagorean Life as a whole go back to Aristotle but suggests that the acusma about the tetraktys was a post-Aristotelian addition a, — Once again source criticism is crucial.
If the acusma in question goes back to Aristotle then there is good evidence for the tetraktys in early Pythagoreanism. If we regard it as a later insertion into Aristotelian material, the early Pythagorean credentials of the tetraktys are less clear. Although there is no explicit evidence, Pythagoras is the most likely candidate to fill these gaps.
Thus between Thales, whom Eudemus identifies as the first geometer, and Hippocrates of Chios, who produced the first Elements , someone turned geometry into a deductive science Zhmud a, In each case Zhmud suggests that Pythagoras is that someone. Such speculations have some plausibility but they highlight even more the puzzle as to why, if Pythagoras played this central role in early Greek mathematics, no early source explicitly ascribes it to him.
Of course, some scholars argue that the majority have overlooked key passages that do assign mathematical achievements to Pythagoras. In order to gain a rounded view of the Pythagorean question it is thus appropriate to look at the most controversial of these passages. Some scholars who regard Pythagoras as a mathematician and rational cosmologist, such as Guthrie, admit that the earliest evidence does not support this view Lloyd , 25 , but maintain that the prominence of Pythagoras the mathematician in the late tradition must be based on something early.
Others maintain that there is evidence in the sixth- and fifth-century BCE for Pythagoras as a mathematician and cosmologist. Thus the description of Pythagoras as a wise man who practiced inquiry is simply too general to aid in deciding what sort of figure Herodotus and Heraclitus saw him as being.
There is more controversy about the fourth-century evidence. Zhmud argues that Isocrates regards Pythagoras as a philosopher and mathematician a, However, it is hard to see how the passage in question Busiris 28—29 supports this view.
Nowhere in it does Isocrates ascribe mathematical work or a rational cosmology to Pythagoras. What Isocrates emphasizes about Pythagoras is what the rest of the early tradition emphasizes, his interest in religious rites. The same situation arises with Fr. If the words in question were by Aristotle they would be his sole statement that Pythagoras was a natural philosopher. Pythagoras studied properties of numbers which would be familiar to mathematicians today, such as even and odd numbers, triangular numbers , perfect numbers etc.
However to Pythagoras numbers had personalities which we hardly recognise as mathematics today [ 3 ] :- Each number had its own personality - masculine or feminine, perfect or incomplete, beautiful or ugly.
This feeling modern mathematics has deliberately eliminated, but we still find overtones of it in fiction and poetry. Of course today we particularly remember Pythagoras for his famous geometry theorem. Although the theorem, now known as Pythagoras's theorem, was known to the Babylonians years earlier he may have been the first to prove it.
Proclus , the last major Greek philosopher, who lived around AD wrote see [ 7 ] :- After [ Thales , etc. Again Proclus , writing of geometry, said:- I emulate the Pythagoreans who even had a conventional phrase to express what I mean "a figure and a platform, not a figure and a sixpence", by which they implied that the geometry which is deserving of study is that which, at each new theorem, sets up a platform to ascend by, and lifts the soul on high instead of allowing it to go down among the sensible objects and so become subservient to the common needs of this mortal life.
Heath [ 7 ] gives a list of theorems attributed to Pythagoras, or rather more generally to the Pythagoreans. We should note here that to Pythagoras the square on the hypotenuse would certainly not be thought of as a number multiplied by itself, but rather as a geometrical square constructed on the side. To say that the sum of two squares is equal to a third square meant that the two squares could be cut up and reassembled to form a square identical to the third square.
This is certainly attributed to the Pythagoreans but it does seem unlikely to have been due to Pythagoras himself. This went against Pythagoras's philosophy the all things are numbers, since by a number he meant the ratio of two whole numbers. However, because of his belief that all things are numbers it would be a natural task to try to prove that the hypotenuse of an isosceles right angled triangle had a length corresponding to a number.
It is thought that Pythagoras himself knew how to construct the first three but it is unlikely that he would have known how to construct the other two. He also recognised that the orbit of the Moon was inclined to the equator of the Earth and he was one of the first to realise that Venus as an evening star was the same planet as Venus as a morning star. References show. Biography in Encyclopaedia Britannica. M Cerchez, Pythagoras Romanian Bucharest, Diogenes Laertius, Lives of eminent philosophers New York, P Gorman, Pythagoras, a life C Byrne, The left-handed Pythagoras, Math.
Intelligencer 12 3 , 52 - Canada 7 2 , - Ense nanza Univ. G Tarr, Pythagoras and his theorem, Nepali Math. Annalen - 49 , - , - Additional Resources show. Is seems that Pythagoras was the first person to define the consonant acoustic relationships between strings of proportional lengths. Specifically, strings of equal tension of proportional lengths create tones of proportional frequencies when plucked. For example, a string that is 2 feet long will vibrate x times per second that is, hertz, a unit of frequency equal to one cycle per second , while a string that is 1 foot long will vibrate twice as fast: 2 x.
Furthermore, those two frequencies create a perfect octave. The Pythagoreans were so troubled over the finding of irrational numbers that they swore each other to secrecy about its existence. It is known that one Pythagorean did tell someone outside the school, and he was never to be found thereafter, that is, he was murdered, as Pythagoras himself was murdered by oppressors of the Semicircle of Pythagoras.
Mesopotamia arrow 1 in Figure 2 was in the Near East in roughly the same geographical position as modern Iraq. Mesopotamia was one of the great civilizations of antiquity, rising to prominence years ago. Thousands of clay tablets, found over the past two centuries, confirm a people who kept accurate records of astronomical events, and who excelled in the arts and literature.
Only a small fraction of this vast archeological treasure trove has been studied by scholars. The great majority of tablets lie in the basements of museums around the world, awaiting their turn to be deciphered and to provide a glimpse into the daily life of ancient Babylon. The marks are in wedge-shaped characters, carved with a stylus into a piece of soft clay that was then dried in the sun or baked in an oven.
They turn out to be numbers, written in the Babylonian numeration system that used the base In this sexagestimal system, numbers up to 59 were written in essentially the modern base numeration system, but without a zero. Units were written as vertical Y-shaped notches, while tens were marked with similar notches written horizontally. What is the breadth?
Its size is not known. And 5 times 5 is You take 16 from 25 and there remains 9. What times what shall I take in order to get 9? The number along the upper left side is easily recognized as The conclusion is inescapable. This was probably the first number known to be irrational. Two factors with regard to this tablet are particularly significant. First, it proves that the Babylonians knew how to compute the square root of a number with remarkable accuracy.
The unknown scribe who carved these numbers into a clay tablet nearly years ago showed a simple method of computing: multiply the side of the square by the square root of 2. But there remains one unanswered question: Why did the scribe choose a side of 30 for his example? Probably, 30 was used for convenience, as it was part of the Babylonian system of sexagesimal, a base numeral system. To Pythagoras it was a geometric statement about areas.
It was with the rise of modern algebra, circa CE , that the theorem assumed its familiar algebraic form. In any right triangle, the area of the square whose side is the hypotenuse the side opposite the right angle is equal to the sum of the areas of the squares whose sides are the two legs the two sides that meet at a right angle.
An area interpretation of this statement is shown in Figure 5. The square of the hypotenuse of a right triangle is equal to the sum of the squares on the other two sides.
Ancient Egyptians arrow 4, in Figure 2 , concentrated along the middle to lower reaches of the Nile River arrow 5, in Figure 2 , were a people in Northeastern Africa. The ancient civilization of the Egyptians thrived miles to the southwest of Mesopotamia. The two nations coexisted in relative peace for over years, from circa BCE to the time of the Greeks.
As to the claim that the Egyptians knew and used the Pythagorean Theorem in building the great pyramids, there is no evidence to support this claim. Egypt has over pyramids, most built as tombs for their country's Pharaohs. Egypt arrow 4, in Figure 2 and its pyramids are as immortally linked to King Tut as are Pythagoras and his famous theorem.
King Tut ruled from the age of 8 for 9 years, — BC. He was born in BC and died some believe he was murdered in BC at the age of Elisha Scott Loomis — Figure 7 , an eccentric mathematics teacher from Ohio, spent a lifetime collecting all known proofs of the Pythagorean Theorem and writing them up in The Pythagorean Proposition , a compendium of proofs. 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.
0コメント