Pythagoras
Pythagoras
(Πυθαγόρας)
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Born
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c. 570 BC
Samos |
Died
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c. 495 BC (aged around 75)
Metapontum |
Era
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Region
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Western philosophy
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Main interests
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Notable ideas
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Influenced by[show]
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Influenced[show]
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Pythagoras of Samos (/pɪˈθæɡərəs/;
Ancient Greek: Πυθαγόρας ὁ Σάμιος Pythagóras ho Sámios “Pythagoras
the Samian”, or
simply Πυθαγόρας; Πυθαγόρης in Ionian Greek;
c. 570 BC – c. 495 BC)[1][2]
was an Ionian Greek philosopher,
mathematician, and founder of the religious movement called Pythagoreanism.
Most of the information about Pythagoras was written down centuries after he
lived, so very little reliable information is known about him. He was born on
the island of Samos, and
might have travelled widely in his youth, visiting Egypt and other places seeking knowledge. Around 530 BC, he moved
to Croton, in Magna Graecia,
and there set up a religious sect. His followers pursued the religious rites
and practices developed by Pythagoras, and studied his philosophical theories.
The society took an active role in the politics of Croton, but this eventually
led to their downfall. The Pythagorean meeting-places were burned, and
Pythagoras was forced to flee the city. He is said to have died in Metapontum.
Pythagoras made influential
contributions to philosophy and religious teaching in the late 6th century BC. He is
often revered as a great mathematician,
mystic and scientist,
but he is best known for the Pythagorean theorem which bears his name. However, because legend and
obfuscation cloud his work even more than that of the other pre-Socratic philosophers, one can give only a tentative account of his teachings,
and some have questioned whether he contributed much to mathematics
and natural philosophy. Many of the accomplishments credited to Pythagoras may
actually have been accomplishments of his colleagues and successors. Whether or
not his disciples believed that everything was related to mathematics and that
numbers were the ultimate reality is unknown. It was said that he was the first
man to call himself a philosopher, or lover of wisdom,[3] and
Pythagorean ideas exercised a marked influence on Plato, and through him, all of Western philosophy.
Contents
- 1 Biographical sources
- 2 Life
- 3 Writings
- 4 Mathematics
- 5 Religion and science
- 6 Pythagoreans
- 7 Influence
- 8 See also
- 9 References
- 10 Sources
- 11 External links
Biographical
sources
Accurate facts about the life of
Pythagoras are so few, and most information concerning him is of so late a
date, and so untrustworthy, that it is impossible to provide more than a vague
outline of his life. The lack of information by contemporary writers, together
with the secrecy which surrounded the Pythagorean brotherhood, meant that
invention took the place of facts. The stories which were created were eagerly
sought by the Neoplatonist writers who provide most of the details about Pythagoras,
but who were uncritical concerning anything which related to the gods or which
was considered divine.[4]
Thus many myths were created – such as that Apollo was his
father; that Pythagoras gleamed with a supernatural
brightness; that he had a golden thigh; that Abaris came flying to him on a golden arrow; that he was seen in different places at the same time.[5]
With the exception of a few remarks by Xenophanes,
Heraclitus,
Herodotus,
Plato, Aristotle,
and Isocrates,
we are mainly dependent on Diogenes Laërtius, Porphyry, and Iamblichus
for the biographical details. Aristotle had written a separate work on the
Pythagoreans, which unfortunately has not survived.[6]
His disciples Dicaearchus, Aristoxenus, and Heraclides Ponticus had written on the same subject. These writers, late as
they are, were among the best sources from whom Porphyry and Iamblichus drew,
besides the legendary accounts and their own inventions. Hence historians are
often reduced to considering the statements based on their inherent
probability, but even then, if all the credible stories concerning Pythagoras
were supposed true, his range of activity would be impossibly vast.[7]
Life
Herodotus, Isocrates, and other early writers all agree that Pythagoras was born
on Samos, the
Greek island in the eastern Aegean, and we
also learn that Pythagoras was the son of Mnesarchus.[8]
His father was a gem-engraver or a merchant. 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.[9]
A late source gives his mother's name as Pythais.[10]
As to the date of his birth, Aristoxenus
stated that Pythagoras left Samos in the reign of Polycrates,
at the age of 40, which would give a date of birth around 570 BC.[11]
It was natural for the ancient
biographers to inquire as to the origins of Pythagoras' remarkable system. In
the absence of reliable information, however, a huge range of teachers were
assigned to Pythagoras. Some made his training almost entirely Greek, others
exclusively Egyptian and Oriental. We find mentioned as his instructors Creophylus,[12]
Hermodamas
of Samos,[13]
Bias,[12]
Thales,[12]
Anaximander,[14]
and Pherecydes of Syros.[15]
He is said too, to have been taught by a Delphic priestess
named Themistoclea, who introduced him to the principles of ethics.[16][17]
The Egyptians are said to have taught him geometry, the Phoenicians
arithmetic, the Chaldeans astronomy, the Magians the principles of religion and practical maxims for the
conduct of life.[18]
Of the various claims regarding his Greek teachers, Pherecydes is mentioned
most often.
Diogenes Laërtius reported that
Pythagoras had undertaken extensive travels, having not only visited Egypt but also "journeyed among the Chaldaeans
and Magi",
for the purpose of collecting all available knowledge and especially to learn
information concerning the secret or mystic cults of the gods.[19]
Plutarch asserted
in his book On Isis and Osiris that during his visit to Egypt, Pythagoras received
instruction from the Egyptian priest Oenuphis of Heliopolis.[20]
Other ancient writers asserted his visit to Egypt.[21]
Enough of Egypt was known to attract the curiosity of an inquiring Greek, and
contact between Samos and other parts of Greece with Egypt is mentioned.[22]
It is not easy to say how much
Pythagoras learned from the Egyptian priests, or indeed, whether he learned
anything at all from them. There was nothing in the symbolism which the
Pythagoreans adopted which showed the distinct traces of Egypt. The secret
religious rites of the Pythagoreans exhibited nothing but what might have been
adopted in the spirit of Greek religion, by those who knew nothing of Egyptian
mysteries. The philosophy and the institutions of Pythagoras might easily have
been developed by a Greek mind exposed to the ordinary influences of the age.
Even the ancient authorities note the similarities between the religious and ascetic
peculiarities of Pythagoras with the Orphic or Cretan mysteries,[23]
or the Delphic oracle.[24]
There is little direct evidence as
to the kind and amount of knowledge which Pythagoras acquired, or as to his
definite philosophical views. Everything of the kind mentioned by Plato and Aristotle is attributed not to Pythagoras, but to the Pythagoreans. Heraclitus
stated that he was a man of extensive learning;[25] and
Xenophanes
claimed that he believed in the transmigration of souls.[26]
Xenophanes mentions the story of his interceding on behalf of a dog that was being beaten, professing to recognise in its cries
the voice of a departed friend. Pythagoras is supposed to have claimed that he
had been Euphorbus, the son of Panthus, in the Trojan war,
as well as various other characters, a tradesman, a courtesan, etc.[27]
In his book The Life of Apollonius of Tyana, Philostratus
wrote that Pythagoras knew not only who he was himself, but also who he had
been.[28]
Many mathematical and scientific
discoveries were attributed to Pythagoras, including his famous theorem,[29]
as well as discoveries in the field of music,[30]
astronomy,[31]
and medicine.[32]
But it was the religious element which made the profoundest impression upon his
contemporaries. Thus the people of Croton were supposed to have identified him
with the Hyperborean Apollo,[33]
and he was said to have practised divination
and prophecy.[34]
In the visits to various places in Greece – Delos, Sparta, Phlius, Crete, etc. which are ascribed to him, he usually appears either
in his religious or priestly guise, or else as a lawgiver.[35]
Croton on the southern coast of
Magna Graecia (Southern Italy), to which Pythagoras ventured after feeling
overburdened in Samos.
After his travels, Pythagoras moved
(around 530 BC) to Croton, in Italy (Magna Graecia).
Possibly the tyranny of Polycrates
in Samos made it difficult for him to achieve his schemes there. His later
admirers claimed that Pythagoras was so overburdened with public duties in
Samos, because of the high estimation in which he was held by his
fellow-citizens, that he moved to Croton.[36]
On his arrival in Croton, he quickly attained extensive influence, and many
people began to follow him. Later biographers tell fantastical stories of the
effects of his eloquent speech in leading the people of Croton to abandon their
luxurious and corrupt way of life and devote themselves to the purer system
which he came to introduce.[37]
His followers established a select
brotherhood or club for the purpose of pursuing the religious and ascetic
practices developed by their master. The accounts agree that what was done and
taught among the members was kept a profound secret. The esoteric teachings
may have concerned science and mathematics, or the secret religious doctrines
and usages which were undoubtedly prominent in the Pythagorean system, and may
have been connected with the worship of Apollo.[38]
Temperance of all kinds seems to have been strictly urged. There is
disagreement among the biographers as to whether Pythagoras forbade all animal
food,[39]
or only certain types.[40]
The club was in practice at once "a philosophical school, a religious
brotherhood, and a political association."[41]
Pythagoras, depicted on a
3rd-century coin
Such an aristocratic and exclusive
club could easily have made many people in Croton jealous and hostile, and this
seems to have led to its destruction. The circumstances, however, are
uncertain. Conflict seems to have broken out between the towns of Sybaris and
Croton. The forces of Croton were headed by the Pythagorean Milo, and it
is likely that the members of the brotherhood took a prominent part. After the
decisive victory by Croton, a proposal for establishing a more democratic
constitution, was unsuccessfully resisted by the Pythagoreans. Their enemies,
headed by Cylon and Ninon, the former of whom is said to have been irritated by his
exclusion from the brotherhood, roused the populace against them. An attack was
made upon them while assembled either in the house of Milo, or in some other
meeting-place. The building was set on fire, and many of the assembled members
perished; only the younger and more active escaping.[42]
Similar commotions ensued in the other cities of Magna Graecia in which
Pythagorean clubs had been formed.
As an active and organised
brotherhood the Pythagorean order was everywhere suppressed, and did not again
revive. Still the Pythagoreans continued to exist as a sect, the members of
which kept up among themselves their religious observances and scientific
pursuits, while individuals, as in the case of Archytas, acquired
now and then great political influence. Concerning the fate of Pythagoras
himself, the accounts varied. Some say that he perished in the temple with his
disciples,[43]
others that he fled first to Tarentum, and
that, being driven from there, he escaped to Metapontum,
and there starved himself to death.[44]
His tomb was shown at Metapontum in the time of Cicero.[45]
According to some accounts
Pythagoras married Theano, a lady of Croton. Their children are variously stated to
have included a son, Telauges, and three daughters, Damo, Arignote, and Myia.
Writings
No texts by Pythagoras are known to
have survived, 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.
Mathematics
The Pythagorean theorem: The sum of the areas of the two squares on the legs (a
and b) equals the area of the square on the hypotenuse (c).
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.
There are good reasons to believe
that Pythagoras never dealt with Mathematics at all.[46]
Therefore, although the mathematical ideas exposed below circulated among the
Pythagoreans, they may well not be due to Pythagoras himself.
Pythagorean
theorem
Main article: Pythagorean theorem
A visual proof of the Pythagorean
theorem
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 area of the square on the hypotenuse (the side opposite the right
angle) is equal to the sum of the areas of the squares of the other two
sides—that is, 
.
.
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.[47]
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.[48]
The earliest known mention of Pythagoras's name in connection with the theorem
occurred five centuries after his death, in the writings of Cicero and Plutarch.
Musical
theories and investigations
Medieval woodcut showing Pythagoras
with bells and other instruments in Pythagorean tuning
According to legend, the way
Pythagoras discovered that musical notes could be translated into mathematical
equations was when one day he passed blacksmiths at work, and thought that the
sounds emanating from their anvils being hit were beautiful and harmonious and
decided that whatever scientific law caused this to happen must be mathematical
and could be applied to music. He went to the blacksmiths to learn how this had
happened by looking at their tools, he discovered that it was because the hammers were "simple ratios of each other, one was half the
size of the first, another was 2/3 the size, and so on."
This legend has since proven to be
false by virtue of the fact that these ratios are only relevant to string
length (such as the string of a monochord),
and not to hammer weight.[49][50]
However, it may be that Pythagoras was indeed responsible for discovering these
properties of string length.
Pythagoreans elaborated on a theory
of numbers, the exact meaning of which is still debated among scholars. Another
belief attributed to Pythagoras was that of the "harmony of the spheres". Thus the planets and stars moved according to
mathematical equations, which corresponded to musical notes and thus produced a
symphony.[51]
Tetractys
Pythagoras was also credited with
devising the tetractys, the triangular figure of four rows, which add up to the
perfect number, ten. As a mystical symbol, it was very important to the worship
of the Pythagoreans, who would swear oaths by it:
And the inventions were so
admirable, and so divinised by those who understood them, that the members used
them as forms of oath: "By him who handed to our generation the tetractys,
source of the roots of ever-flowing nature."
—Iamblichus, Vit. Pyth., 29
Religion
and science
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 immortal. His ideas of
reincarnation were influenced by ancient Greek religion. Heraclides Ponticus reports the story that Pythagoras claimed that he had lived
four previous lives that he could remember in detail.[52]
One of his past lives, as reported by Aulus Gellius,
was as a beautiful courtesan.[53]
According to Xenophanes, Pythagoras heard the cry of his dead friend in the bark of
a dog.[54]
Lore
Pythagoras 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 Muslim tradition, Pythagoras was said to have been initiated by Hermes (Egyptian Thoth).[55]
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.[56]
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.[57]
Another legend 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.[58]
Pythagoreans
See also: Pythagoreanism
Both Plato and Isocrates affirm that, above all else, Pythagoras was famous for
leaving behind him a way of life.[59]
Both Iamblichus and Porphyry give detailed accounts of the organisation of the school,
although the primary interest of both writers is not historical accuracy, but
rather to present Pythagoras as a divine figure, sent by the gods to
benefit humankind.[60]
Pythagoras set up an organization
which was in some ways a school, in some ways a brotherhood (and here it should
be noted that sources indicate that as well as men there were many women among
the adherents of Pythagoras),[61]
and in some ways a monastery. It was based upon the religious teachings of
Pythagoras and was very secretive. The adherents were bound by a vow to Pythagoras and each other, for the purpose of pursuing
the religious
and ascetic
observances, and of studying his religious and philosophical
theories. The claim that they put all their property into a common stock is
perhaps only a later inference from certain Pythagorean maxims and practices.[62]
As to the internal arrangements of
the sect, we are informed that what was done and taught among the members was
kept a profound secret towards all. Porphyry stated that this silence was "of
no ordinary kind." Candidates had to pass through a period of probation,
in which their powers of maintaining silence (echemythia) were
especially tested, as well as their general temper, disposition, and mental
capacity.[63]
There were also gradations among the members themselves. It was an old
Pythagorean maxim, that every thing was not to be told to every body.[64]
Thus the Pythagoreans were divided into an inner circle called the mathematikoi
("learners") and an outer circle called the akousmatikoi
("listeners").[65]
Iamblichus describes them in terms of esoterikoi and exoterikoi
(or alternatively Pythagoreioi and Pythagoristai),[66]
according to the degree of intimacy which they enjoyed with Pythagoras.
Porphyry wrote "the mathematikoi learned the more detailed and
exactly elaborated version of this knowledge, the akousmatikoi (were)
those who had heard only the summary headings of his (Pythagoras's) writings,
without the more exact exposition."
There were ascetic practices
(many of which had, perhaps, a symbolic meaning) in the way of life of the
sect.[67]
Some represent Pythagoras as forbidding all animal food, advocating a
plant-based diet, and prohibiting consumption of beans. This may have been due
to the doctrine of metempsychosis.[68]
Other authorities contradict the statement. According to Aristoxenus,[69]
he allowed the use of all kinds of animal food except the flesh of oxen used for ploughing,
and rams.[70]
There is a similar discrepancy as to the prohibition of fish and beans.[71]
But temperance of all kinds seems to have been urged. It is also stated that
they had common meals, resembling the Spartan system,
at which they met in companies of ten.[72]
Considerable importance seems to
have been attached to music and gymnastics
in the daily exercises of the disciples. Their whole discipline is represented
as encouraging a lofty serenity and self-possession, of which, there were
various anecdotes in antiquity.[73]
Iamblichus (apparently on the authority of Aristoxenus)[74]
gives a long description of the daily routine of the members, which suggests
many similarities with Sparta. The members of the sect showed a devoted
attachment to each other, to the exclusion of those who did not belong to their
ranks.[75]
There were even stories of secret symbols, by which members of the sect could
recognise each other, even if they had never met before.[76]
Influence
Influence
on Plato
Pythagoras, depicted as a medieval
scholar in the Nuremberg Chronicle
Pythagoras, or in a broader sense,
the Pythagoreans, allegedly exercised an important influence on the work of Plato. According to R. M. Hare,
this influence consists of three points: (1) 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. (2) 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".
(3) Plato and Pythagoras shared a "mystical approach to the soul and its
place in the material world". It is probable that both were influenced by Orphism.[77]
Aristotle claimed that the
philosophy of Plato closely followed the teachings of the Pythagoreans,[78] and
Cicero repeats this claim: Platonem ferunt didicisse Pythagorea omnia
("They say Plato learned all things Pythagorean").[79]
Bertrand Russell, in his 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 philosophers.
Politics
and science
Pythagoras was the first person
known to have taught the earth was spherical, with antipodes
and that it revolved around the sun. Pythagoras was also said to have spread
the seeds of political liberty to Crotona, Sybaris, Meapontum, Rhegium, Sicily,
Tauromenium, Catana, Agrigentum and Himera.[80]
Influence
on esoteric groups
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 Freemasonry
and Rosicrucianism, both of which were scientific/mystical groups dedicated to
the study of mathematics/geometry and logical reasoning as opposed to religious dogma. Both
Freemasonry and Rosicrucianism have 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 Ages entitled "Pythagorean Mathematics".[81]
See
also
References
1.
Jump up ^
"The dates of his life cannot be fixed exactly, but assuming the
approximate correctness of the statement of Aristoxenus (ap. Porph. V.P.
9) that he left Samos to escape the tyranny of Polycrates at the age of forty,
we may put his birth round about 570 BC, or a few years earlier. The length of
his life was variously estimated in antiquity, but it is agreed that he lived
to a fairly ripe old age, and most probably he died at about seventy-five or
eighty." William
Keith Chambers Guthrie, (1978), A
history of Greek philosophy, Volume 1: The earlier Presocratics and the
Pythagoreans, page 173. Cambridge University Press
3.
Jump up ^
Cicero, Tusculan
Disputations, 5.3.8–9 = Heraclides Ponticus fr. 88 Wehrli, Diogenes Laërtius 1.12, 8.8, Iamblichus
VP 58. Burkert attempted to discredit this ancient tradition, but it has
been defended by C.J. De Vogel, Pythagoras and Early Pythagoreanism
(1966), pp. 97–102, and C. Riedweg, Pythagoras: His Life, Teaching, And
Influence (2005), p. 92.
7.
Jump up ^
This
article incorporates text from a publication now in the public domain: Smith,
William, ed. (1867). "article name needed". Dictionary of Greek and Roman Biography and Mythology.
This
article incorporates text from a publication now in the public domain: Smith,
William, ed. (1867). "article name needed". Dictionary of Greek and Roman Biography and Mythology.
8.
Jump up ^
Herodotus, iv. 95, Isocrates, Busiris, 28–9; Later writers called him a
Tyrrhenian or Phliasian, and gave Marmacus, or Demaratus, as the name of his
father, Diogenes Laërtius, viii. 1; Porphyry, Vit. Pyth. 1, 2; Justin,
xx. 4; Pausanias, ii. 13.
9.
Jump up ^
Riedweg, Christoph (2005). Pythagoras: His Life, Teaching and Influence.
Cornell University. pp. 5–6, 59, 73.
13. Jump up ^
Porphyry, Vit. Pyth. 2, Diogenes Laërtius, viii. 2 C. Riedweg, S. Rendall
ISBN 0-8014-7452-3
Retrieved 2012-02-08
17. Jump up ^
Malone, John C. (30 June 2009). Psychology: Pythagoras to present. MIT Press. p. 22. ISBN 978-0-262-01296-6. Retrieved 25 October 2010.
19. Jump up ^
Diogenes Laërtius, viii. 2; Porphyry, Vit. Pyth. 11, 12; Iamblichus, Vit.
Pyth. 14, etc.
21. Jump up ^
Antiphon. ap. Porphyry, Vit. Pyth. 7; Isocrates, Busiris, 28–9;
Cicero, de Finibus, v. 27; Strabo, xiv.
27. Jump up ^
Porphyry, Vit. Pyth. 26; Pausanias, ii. 17; Diogenes Laërtius, viii. 5;
Horace, Od. i. 28,1. 10
28. Jump up ^
Flavius Philostratus, The Life of Apollonius of Tyana , , trad. F. C.
Conybeare, Vol. 2, London, 1912, Book VI, p. 39.
33. Jump up ^
Porphyry, Vit. Pyth. 20; Iamblichus, Vit. Pyth. 31, 140; Aelian, Varia
Historia, ii. 26; Diogenes Laërtius, viii. 36.
35. Jump up ^
Iamblichus, Vit. Pyth. 25; Porphyry, Vit. Pyth. 17; Diogenes
Laërtius, viii. 3, 13; Cicero, Tusc. Qu. v. 3
38. Jump up ^
Aelian, Varia Historia, ii. 26; Diogenes Laërtius, viii. 13; Iamblichus,
Vit. Pyth. 8, 91, 141
39. Jump up ^
as Empedocles did afterwards, Aristotle, Rhet. i. 14. § 2; Sextus
Empiricus, ix. 127. This was also one of the Orphic precepts, Aristoph. Ran.
1032
40. Jump up ^
Aristo ap. Diogenes Laërtius, viii. 20; comp. Porphyry, Vit. Pyth. 7;
Iamblichus, Vit. Pyth. 85, 108
42. Jump up ^
Iamblichus, Vit. Pyth. 255–259; Porphyry, Vit. Pyth. 54–57;
Diogenes Laërtius, viii. 39; comp. Plutarch, de Gen. Socr. p. 583
44. Jump up ^
Diogenes Laërtius, viii. 39, 40; Porphyry, Vit. Pyth. 56; Iamblichus, Vit.
Pyth. 249; Plutarch, de Stoic. Rep. 37
46. Jump up ^
See Burkert's Lore and Science in Ancient Pythagoreanism, Burnyeat's Other
Lives, or Machiavelo's Pythagoras, Facts and Legends.
47. Jump up ^
There are about 100,000 unpublished cuneiform sources in the British Museum
alone. Babylonian knowledge of proof of the Pythagorean Theorem is discussed by
J. Høyrup, 'The Pythagorean "Rule" and "Theorem" – Mirror
of the Relation between Babylonian and Greek Mathematics,' in: J. Renger
(red.): Babylon. Focus mesopotamischer Geschichte, Wiege früher
Gelehrsamkeit, Mythos in der Moderne (1999).
48. Jump up ^
From Christoph Riedweg , Pythagoras, His Life, Teaching and Influence,
Cornell: Cornell University Press, 2005: "Had Pythagoras and his teachings
not been since the early Academy overwritten with Plato's philosophy, and had
this 'palimpsest' not in the course of the Roman Empire achieved unchallenged
authority among Platonists, it would be scarcely conceivable that scholars from
the Middle Ages and modernity down to the present would have found the
Presocratic charismatic from Samos so fascinating. In fact, as a rule it was
the image of Pythagoras elaborated by Neopythagoreans and Neoplatonists that
determined the idea of what was Pythagorean over the centuries."
49. Jump up ^
Weiss, Piero, and Richard Taruskin, eds. Music in the Western World: A History
in Documents. 2nd ed. N.p.: Thomson Schirmer, 1984. 3. Print.
50. Jump up ^
Christensen, Thomas, ed. The Cambridge history of Western
music theory. Cambridge: Cambridge University Press, 2002. 143. Print.
51. Jump up ^
Christoph Riedweg, Pythagoras: His Life, Teaching and Influence, Cornell:
Cornell University Press, 2005 .
60. Jump up ^
John Dillon and Jackson Hershbell, (1991), Iamblichus, On the Pythagorean
Way of Life, page 14. Scholars Press.; D. J. O'Meara, (1989), Pythagoras
Revived. Mathematics and Philosophy in Late Antiquity, pages 35–40.
Clarendon Press.
71. Jump up ^
Diogenes Laërtius, viii. 19, 34; Aulus Gellius, iv. 11; Porphyry, Vit. Pyth.
34, de Abst. i. 26; Iamblichus, Vit. Pyth. 98
75. Jump up ^
Aristonexus ap. Iamblichus, Vit. Pyth. 94, 101, etc., 229, etc.; comp.
the story of Damon and Phintias; Porphyry, Vit. Pyth. 60; Iamblichus, Vit.
Pyth. 233, etc.
77. Jump up ^
R.M. Hare, Plato in C.C.W. Taylor, R.M. Hare and Jonathan Barnes, Greek
Philosophers, Socrates, Plato, and Aristotle, Oxford: Oxford University Press,
1999 (1982), 103–189, here 117–9.
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