Array ( [0] => {{Short description|Greek philosopher (c. 570 – c. 495 BC)}} [1] => {{redirect|Pythagoras of Samos|the Samian statuary|Pythagoras (sculptor)}} [2] => {{other uses}} [3] => {{good article}} [4] => {{pp-semi-indef}} [5] => {{pp-move-indef}} [6] => {{Use American English|date=February 2024}} [7] => [8] => {{Infobox philosopher [9] => | name = Pythagoras [10] => | image = Pythagoras in the Roman Forum, Colosseum.jpg [11] => | alt = Marble bust of a man with a long, pointed beard, wearing a tainia, a kind of ancient Greek headcovering in this case resembling a turban. The face is somewhat gaunt and has prominent, but thin, eyebrows, which seem halfway fixed into a scowl. The ends of his mustache are long a trail halfway down the length of his beard to about where the bottom of his chin would be if we could see it. None of the hair on his head is visible, since it is completely covered by the tainia. [12] => | caption = Bust of Pythagoras of Samos in the
[[Capitoline Museums]], [[Rome]]{{sfn|Joost-Gaugier|2006|page=143}} [13] => | birth_date = {{circa|570 BC}} [14] => | birth_place = [[Samos]] [15] => | death_date = {{circa|495 BC|lk=no}} (aged around 75) [16] => | death_place = either [[Crotone|Croton]] or [[Metapontum]] [17] => | era = [[Pre-Socratic philosophy]] [18] => | region = [[Western philosophy]] [19] => | school_tradition = [[Pythagoreanism]] [20] => | main_interests = {{hlist |[[Ethics]] |[[Mathematics]] |[[Metaphysics]] |[[Music theory]] |[[Mysticism]] |[[Politics]] |[[Religion]]}} [21] => | notable_ideas = {{ublist |[[Intentional community|Communalism]] |[[Metempsychosis]]| ''[[Musica universalis]]''}}
Attributed ideas: {{ublist |[[Geographical zone|Five climatic zones]] |[[Platonic solids|Five regular solids]] |[[Proportionality (mathematics)|Proportions]] |[[Pythagorean theorem]] |[[Pythagorean tuning]] |[[Spherical Earth|Sphericity of the Earth]] |[[Vegetarianism]]}} [22] => }} [23] => [24] => '''Pythagoras of Samos'''{{efn|{{IPAc-en|p|aɪ|ˈ|θ|æ|ɡ|ər|ə|s}} {{respell|py|THAG|ər|əs}},{{citation |url=http://www.collinsdictionary.com/dictionary/english/pythagoras?showCookiePolicy=true |title=British: Pythagoras |access-date=25 September 2014 |publisher=Collins Dictionary |date=n.d.}} {{IPAc-en|also|US|p|ɪ|ˈ|θ|æ|ɡ|ər|ə|s}} {{respell|pih|-}}{{citation |url=http://www.collinsdictionary.com/dictionary/american/pythagoras?showCookiePolicy=true |title=American: Pythagoras |access-date=25 September 2014 |publisher=Collins Dictionary |date=n.d.}}}} ({{lang-grc|Πυθαγόρας ὁ Σάμιος|Pythagóras ho Sámios|Pythagoras the [[Samos|Samian]]}}, or simply {{lang|grc|Πυθαγόρας}}; {{lang|grc|Πυθαγόρης}} in [[Ionian Greek]]; {{circa|570|495 BC|lk=no}}){{efn|"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'', p. 173. Cambridge University Press}} was an ancient [[Ionians|Ionian]] [[Ancient Greek philosophy|Greek philosopher]], [[polymath]] and the eponymous founder of [[Pythagoreanism]]. His political and religious teachings were well known in [[Magna Graecia]] and influenced the philosophies of [[Plato]], [[Aristotle]], and, through them, the [[Western philosophy|West]] in general. Knowledge of his life is clouded by legend; modern scholars disagree regarding Pythagoras's education and influences, but they do agree that, around 530 BC, he travelled to [[Crotone|Croton]] in southern Italy, where he founded a school in which initiates were sworn to secrecy and lived a communal, [[asceticism|ascetic]] lifestyle. This lifestyle entailed a number of dietary prohibitions, traditionally said to have included aspects of [[vegetarianism]].{{Cite book |url=https://www.cambridge.org/core/books/history-of-pythagoreanism/A8D4A4D8F103C30A769EC3FBE6DB0DBE |title=A History of Pythagoreanism |date=2014 |publisher=Cambridge University Press |isbn=978-1-107-01439-8 |editor-last=Huffman |editor-first=Carl A. |location=Cambridge|doi=10.1017/CBO9781139028172 }} [25] => [26] => The teaching most securely identified with Pythagoras is ''[[metempsychosis]]'', or the "transmigration of souls", which holds that every [[soul]] is [[immortality|immortal]] and, upon death, [[reincarnation|enters into a new body]]. He may have also devised the doctrine of ''[[musica universalis]]'', which holds that the [[Classical planet|planets]] move according to [[mathematics|mathematical]] [[equation]]s and thus resonate to produce an inaudible symphony of music. Scholars debate whether Pythagoras developed the [[Numerology|numerological]] and musical teachings attributed to him, or if those teachings were developed by his later followers, particularly [[Philolaus|Philolaus of Croton]]. Following Croton's decisive victory over [[Sybaris]] in around 510 BC, Pythagoras's followers came into conflict with supporters of [[Greek democracy|democracy]], and Pythagorean meeting houses were burned. Pythagoras may have been killed during this persecution, or he may have escaped to [[Metapontum]] and died there. [27] => [28] => In antiquity, Pythagoras was credited with many mathematical and scientific discoveries, including the [[Pythagorean theorem]], [[Pythagorean tuning]], the [[Platonic solids|five regular solids]], the [[Proportionality (mathematics)|Theory of Proportions]], the [[Spherical Earth|sphericity of the Earth]], and the identity of the [[Phosphorus (morning star)|morning]] and [[Hesperus|evening stars]] as the planet [[Venus]]. It was said that he was the first man to call himself a philosopher ("lover of wisdom"){{efn|[[Cicero]], ''[[Tusculan Disputations]]'', 5.3.8–9 (citing [[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 [[Cornelia Johanna de Vogel|C. J. De Vogel]], ''Pythagoras and Early Pythagoreanism'' (1966), pp. 97–102, and C. Riedweg, ''Pythagoras: His Life, Teaching, And Influence'' (2005), p. 92.}} and that he was the first to divide the globe into [[Geographical zone|five climatic zones]]. Classical historians debate whether Pythagoras made these discoveries, and many of the accomplishments credited to him likely originated earlier or were made by his colleagues or successors. Some accounts mention that the philosophy associated with Pythagoras was related to mathematics and that numbers were important, but it is debated to what extent, if at all, he actually contributed to mathematics or [[natural philosophy]]. [29] => [30] => Pythagoras influenced Plato, whose [[Socratic dialogue|dialogues]], especially his ''[[Timaeus (dialogue)|Timaeus]]'', exhibit Pythagorean teachings. Pythagorean ideas on mathematical perfection also impacted [[ancient Greek art]]. His teachings underwent a major revival in the first century BC among [[Middle Platonism|Middle Platonists]], coinciding with the rise of [[Neopythagoreanism]]. Pythagoras continued to be regarded as a great philosopher throughout the [[Middle Ages]] and his philosophy had a major impact on scientists such as [[Nicolaus Copernicus]], [[Johannes Kepler]], and [[Isaac Newton]]. Pythagorean symbolism was used throughout early modern [[Western esotericism|European esotericism]], and his teachings as portrayed in [[Ovid]]'s ''[[Metamorphoses]]'' influenced the modern vegetarian movement. [31] => [32] => == Biographical sources == [33] => No authentic writings of Pythagoras have survived,{{sfn|Joost-Gaugier|2006|page=11}}{{sfn|Celenza|2010|page=796}}{{sfn|Ferguson|2008|page=4}} and almost nothing is known for certain about his life.{{sfn|Ferguson|2008|pages=3–5}}{{sfn|Gregory|2015|pages=21–23}}{{sfn|Copleston|2003|page=29}} The earliest sources on Pythagoras's life are brief, ambiguous, and often [[satire|satirical]].{{sfn|Ferguson|2008|page=4}}{{sfn|Kahn|2001|page=2}}{{sfn|Burkert|1985|page=299}} The earliest source on Pythagoras's teachings is a satirical poem probably written after his death by [[Xenophanes of Colophon]], who had been one of his contemporaries.{{sfn|Joost-Gaugier|2006|page=12}}{{sfn|Riedweg|2005|page=62}} In the poem, Xenophanes describes Pythagoras interceding on behalf of a [[dog]] that is being beaten, professing to recognize in its cries the voice of a departed friend.{{sfn|Burkert|1985|page=299}}{{sfn|Joost-Gaugier|2006|page=12}}Diogenes Laërtius, viii. 36{{sfn|Copleston|2003|page=31}} [[Alcmaeon of Croton]], a doctor who lived in Croton at around the same time Pythagoras lived there,{{sfn|Joost-Gaugier|2006|page=12}} incorporates many Pythagorean teachings into his writings{{sfn|Joost-Gaugier|2006|pages=12–13}} and alludes to having possibly known Pythagoras personally.{{sfn|Joost-Gaugier|2006|pages=12–13}} The poet [[Heraclitus of Ephesus]], who was born across a few miles of sea away from Samos and may have lived within Pythagoras's lifetime,{{sfn|Joost-Gaugier|2006|page=13}} mocked Pythagoras as a clever charlatan,{{sfn|Kahn|2001|page=2}}{{sfn|Joost-Gaugier|2006|page=13}} remarking that "Pythagoras, son of Mnesarchus, practiced inquiry more than any other man, and selecting from these writings he manufactured a wisdom for himself—much learning, artful knavery."{{sfn|Kahn|2001|page=2}}{{sfn|Joost-Gaugier|2006|page=13}} [34] => [[File:Pythagoras in Thomas Stanley History of Philosophy.jpg|thumb|Fictionalized portrait of Pythagoras from a 17th-century engraving]] [35] => The Greek poets [[Ion of Chios]] ({{circa|480|421 BC|lk=on}}) and [[Empedocles of Acragas]] ({{circa|493|432 BC}}) both express admiration for Pythagoras in their poems.{{sfn|Joost-Gaugier|2006|pages=14–15}} The first concise description of Pythagoras comes from the historian [[Herodotus of Halicarnassus]] ({{circa|484|420 BC|lk=no}}),{{sfn|Joost-Gaugier|2006|page=16}} who describes him as "not the most insignificant" of Greek sages4. 95. and states that Pythagoras taught his followers how to attain [[immortality]].{{sfn|Joost-Gaugier|2006|page=16}} The accuracy of the works of Herodotus is controversial.{{harvp|Marincola|2001|p=59}}{{harvp|Roberts|2011|p=2}}{{harvp|Sparks|1998|p=58}}{{harvp|Asheri|Lloyd|Corcella|2007}}{{harvp|Cameron|2004|p=156}} The writings attributed to the Pythagorean philosopher [[Philolaus|Philolaus of Croton]], who lived in the late fifth century BC, are the earliest texts to describe the numerological and musical theories that were later ascribed to Pythagoras.{{sfn|Joost-Gaugier|2006|page=88}} The [[Ancient Athens|Athenian]] [[rhetoric]]ian [[Isocrates]] (436–338 BC) was the first to describe Pythagoras as having visited Egypt.{{sfn|Joost-Gaugier|2006|page=16}} [[Aristotle]] wrote a treatise ''On the Pythagoreans'', which no longer exists.He alludes to it himself, ''Met.'' i. 5. p. 986. 12, ed. Bekker. Some of it may be preserved in the ''[[Protrepticus (Aristotle)|Protrepticus]]''. Aristotle's disciples [[Dicaearchus]], [[Aristoxenus]], and [[Heraclides Ponticus]] also wrote on the same subject.{{sfn|Burkert|1972|page=109}} [36] => [37] => Most of the major sources on Pythagoras's life are from the [[Ancient Greece#Roman Greece|Roman period]],{{sfn|Kahn|2001|page=5}} by which point, according to the German classicist [[Walter Burkert]], "the history of Pythagoreanism was already{{nbsp}}... the laborious reconstruction of something lost and gone."{{sfn|Burkert|1972|page=109}} Three [[Historical biography|ancient biographies]] of Pythagoras have survived from late antiquity,{{sfn|Copleston|2003|page=29}}{{sfn|Kahn|2001|page=5}} all of which are filled primarily with myths and legends.{{sfn|Copleston|2003|page=29}}{{sfn|Kahn|2001|page=5}}{{sfn|Zhmud|2012|page=9}} The earliest and most respectable of these is the one from [[Diogenes Laërtius]]'s ''[[Lives and Opinions of Eminent Philosophers]]''.{{sfn|Kahn|2001|page=5}}{{sfn|Zhmud|2012|page=9}} The two later biographies were written by the [[Neoplatonism|Neoplatonist]] philosophers [[Porphyry (philosopher)|Porphyry]] and [[Iamblichus]]{{sfn|Kahn|2001|page=5}}{{sfn|Zhmud|2012|page=9}} and were partially intended as polemics against the [[rise of Christianity]].{{sfn|Zhmud|2012|page=9}} The later sources are much lengthier than the earlier ones,{{sfn|Kahn|2001|page=5}} and even more fantastic in their descriptions of Pythagoras's achievements.{{sfn|Kahn|2001|page=5}}{{sfn|Zhmud|2012|page=9}} Porphyry and Iamblichus used material from the lost writings of Aristotle's disciples{{sfn|Burkert|1972|page=109}} and material taken from these sources is generally considered to be the most reliable.{{sfn|Burkert|1972|page=109}} [38] => [39] => == Life == [40] => === Early life === [41] => {{blockquote|There is not a single detail in the life of Pythagoras that stands uncontradicted. But it is possible, from a more or less critical selection of the data, to construct a plausible account.|[[Walter Burkert]], 1972{{sfn|Burkert|1972|page=106}}}} [42] => [43] => [[Herodotus]],[[Herodotus]], [[Histories (Herodotus)|''Histories'']], 4.95 (in [https://el.wikisource.org/wiki/%CE%99%CF%83%CF%84%CE%BF%CF%81%CE%AF%CE%B1%CE%B9_(%CE%97%CF%81%CE%BF%CE%B4%CF%8C%CF%84%CE%BF%CF%85)/%CE%9C%CE%B5%CE%BB%CF%80%CE%BF%CE%BC%CE%AD%CE%BD%CE%B7#95 Greek original text] and English translations: [https://en.wikisource.org/wiki/The_History_of_Herodotus_(Macaulay)/Book_IV] [https://en.wikisource.org/wiki/The_History_of_Herodotus_(Rawlinson)/Book_4]) [[Isocrates]], and other early writers agree that Pythagoras was the son of Mnesarchus,{{sfn|Joost-Gaugier|2006|page=16}}{{sfn|Kahn|2001|page=6}} and that he was born on the Greek island of [[Samos]] in the eastern [[Aegean Sea|Aegean]].{{sfn|Joost-Gaugier|2006|page=11}}{{sfn|Kahn|2001|page=6}}{{sfn|Ferguson|2008|page=12}}{{sfn|Kenny|2004|page=9}} According to these biographers, Pythagoras' father was not born on the island, although he got naturalized there,{{sfn|Ferguson|2008|page=12}} but according to [[Iamblichus]] he was a native of the island.{{sfn|Ferguson|2008|page=11}} He is said to have been a gem-engraver or a wealthy merchant,[[Porphyry (philosopher)|Porphyry]], ''Vita Pythagorae'', Leipzig, 1886; [[Porphyry (philosopher)|Porphyry]], ''Life of Pythogoras in M. Hadas and M. Smith, Heroes and Gods'', London, 1965.Clemens von Alexandria: ''Stromata'' I 62, 2–3, cit. {{citation|url=https://books.google.com/books?id=teoyAQAAQBAJ&pg=PA15|title=Iamblichus and the Foundations of Late Platonism|publisher=Brill|year=2012|editor1=Eugene V. Afonasin|editor2=John M. Dillon|editor3=John Finamore|place=Leiden and Boston|page=15|isbn=978-90-04-23011-8}}{{sfn|Joost-Gaugier|2006|page=21}} but his ancestry is disputed and unclear.{{efn|Some writers call him a native [[Samos|Samian]], a [[Tyrrhenians|Tyrrhenian]] from [[Lemnos]], or a [[Phlius|Phliasian]] from Peloponnesus, and give Marmacus or Demaratus as his name (see Diogenes Laërtius, viii. 1; Porphyry, ''Vit. Pyth.'' 1, 2; Justin, xx. 4; Pausanias, ii. 13; Iamblichus, ii. 4). Porphyry also presents a conflicting report by Neanthes, that Mnesarchus was born in Tyre (in Syria), or that he was a Tyrrhenian from Lemnos; however, the confusion was possibly due to the similarity of the names "Tyre" and "Tyrrhenian", while it has also been suggested that Porphyry's own background from Tyre, could explain why his work was the only one, out of the three biographies of Pythagoras that have survived from late antiquity, to have linked Pythagoras' father with it.{{sfn|Ferguson|2008|pp=11–12, 198}} Due to this obscurity, some modern scholars deem it safer to accept "that Pythagoras and his father were pure-blooded Greeks, of undiluted Samian stock".{{sfn|Jacoby|Bollansée|1999|pp=256–257|loc=n. 73}}}} His mother was a native of [[Samos]], descending from a ''[[Geomori (Samos)|geomoroi]]'' family.{{sfn|Ferguson|2008|page=15}} [[Apollonius of Tyana]], gives her name as Pythaïs.{{sfn|Taub|2017|page=[https://books.google.com/books?id=odm7DgAAQBAJ&pg=PA122 122]}}[[Apollonius of Tyana]] ap. Porphyry, ''Vit. Pyth.'' 2. [[Iamblichus]] tells the story that the Pythia prophesied to her while she was pregnant with him that she would give birth to a man supremely beautiful, wise, and beneficial to humankind.{{sfn|Riedweg|2005|page=59}} 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.Porphyry, ''Vit. Pyth.'' 9 Pythagoras's name led him to be associated with [[Pythia]]n [[Apollo]] ({{transliteration|grc|Pūthíā}}); [[Aristippus of Cyrene]] in the 4th century BC explained his name by saying, "He spoke [{{lang|grc|[[wikt:ἀγορεύω|ἀγορεύω]]}}, {{transliteration|grc|agoreúō}}] the truth no less than did the Pythian [{{lang|grc|[[wikt:Πυθία|πυθικός]]}} {{transliteration|grc|puthikós}}]".{{sfn|Riedweg|2005|page=59}} [44] => [45] => During Pythagoras's formative years, Samos was a thriving cultural hub known for its feats of advanced architectural engineering, including the building of the [[Tunnel of Eupalinos]], and for its riotous festival culture.{{sfn|Riedweg|2005|pages=45–47}} It was a major center of trade in the Aegean where traders brought goods from the [[Near East]].{{sfn|Joost-Gaugier|2006|page=11}} According to Christiane L. Joost-Gaugier, these traders almost certainly brought with them Near Eastern ideas and traditions.{{sfn|Joost-Gaugier|2006|page=11}} Pythagoras's early life also coincided with the flowering of early Ionian [[natural philosophy]].{{sfn|Kahn|2001|page=6}}{{sfn|Riedweg|2005|pages=44–45}} He was a contemporary of the philosophers [[Anaximander]], [[Anaximenes of Miletus|Anaximenes]], and the historian [[Hecataeus of Miletus|Hecataeus]], all of whom lived in [[Miletus]], across the sea from Samos.{{sfn|Riedweg|2005|pages=44–45}} [46] => [47] => === Reputed travels === [48] => {{anchor|Alleged travels}} [49] => Pythagoras is traditionally thought to have received most of his education in the Near East.{{sfn|Riedweg|2005|page=7}} Modern scholarship has shown that the culture of [[Archaic Greece]] was heavily influenced by those of [[Levant]]ine and [[Mesopotamia]]n cultures.{{sfn|Riedweg|2005|page=7}} Like many other important Greek thinkers, Pythagoras was said to have studied in [[Ancient Egypt|Egypt]].{{sfn|Joost-Gaugier|2006|page=16}}{{sfn|Riedweg|2005|pages=7–8}}{{sfn|Gregory|2015|pages=22–23}} By the time of Isocrates in the fourth century BC, Pythagoras's reputed studies in Egypt were already taken as fact.{{sfn|Joost-Gaugier|2006|page=16}}{{sfn|Riedweg|2005|page=59}} The writer [[Antiphon (writer)|Antiphon]], who may have lived during the Hellenistic Era, claimed in his lost work ''On Men of Outstanding Merit'', used as a source by Porphyry, that Pythagoras learned to speak [[Egyptian language|Egyptian]] from the [[Pharaoh]] [[Amasis II]] himself, that he studied with the Egyptian priests at [[Thebes, Egypt|Diospolis]] (Thebes), and that he was the only foreigner ever to be granted the privilege of taking part in their worship.{{sfn|Riedweg|2005|page=7}}Porphyry, ''Vit. Pyth.'' 6. The [[Middle Platonist]] biographer [[Plutarch]] ({{circa|46|120 AD|lk=on}}) writes in his treatise ''[[Moralia|On Isis and Osiris]]'' that, during his visit to Egypt, Pythagoras received instruction from the Egyptian priest [[Oenuphis of Heliopolis]] (meanwhile [[Solon]] received lectures from a [[Sonchis of Sais]]).Plutarch, ''[https://penelope.uchicago.edu/Thayer/E/Roman/Texts/Plutarch/Moralia/Isis_and_Osiris*/A.html On Isis And Osiris]'', ch. 10. According to the [[Christian theologian]] [[Clement of Alexandria]] ({{circa|150|215 AD|lk=no}}), "Pythagoras was a disciple of Sonchis, an Egyptian archprophet, as well as [[Plato]] of [[Sechnuphis]]."{{sfn|Press|2003|page=83}} Some ancient writers claimed that Pythagoras learned geometry and the doctrine of [[metempsychosis]] from the Egyptians.{{sfn|Riedweg|2005|pages=7–8}}cf. Antiphon. ap. Porphyry, ''Vit. Pyth.'' 7; Isocrates, ''Busiris'', 28–9; Cicero, ''de Finibus'', v. 29; Strabo, 14.1.16. [50] => [51] => Other ancient writers, however, claimed that Pythagoras had learned these teachings from the [[Magi]] in [[Persia]] or even from [[Zoroaster]] himself.Diogenes Laërtius, viii. 1, 3.{{sfn|Riedweg|2005|page=8}} Diogenes Laërtius asserts that Pythagoras later visited [[Crete]], where he went to the [[Mount Ida|Cave of Ida]] with [[Epimenides]]. The [[Phoenicia]]ns are reputed to have taught Pythagoras [[arithmetic]] and the [[Chaldea]]ns to have taught him astronomy.{{sfn|Riedweg|2005|page=8}} By the third century BC, Pythagoras was already reported to have studied under the [[Jews]] as well.{{sfn|Riedweg|2005|page=8}} Contradicting all these reports, the novelist [[Antonius Diogenes]], writing in the second century BC, reports that Pythagoras discovered all his doctrines himself by [[Dream interpretation|interpreting dreams]].{{sfn|Riedweg|2005|page=8}} The third-century AD [[Sophist]] [[Philostratus]] claims that, in addition to the Egyptians, Pythagoras also studied under sages or [[gymnosophist]]s in [[India]].{{sfn|Riedweg|2005|page=8}} Iamblichus expands this list even further by claiming that Pythagoras also studied with the [[Celts]] and [[Iberians]].{{sfn|Riedweg|2005|page=8}} [52] => [53] => === Alleged Greek teachers === [54] => [[File:Pythagoras Bust Vatican Museum (cropped).jpg|thumb|upright|alt=Bust of a somewhat elderly and rather tired-looking man with a short, curly beard, vaguely similar to Greek busts of Homer|Bust of Pythagoras in the [[Vatican Museums]], [[Vatican City]], showing him as a "tired-looking older man"{{sfn|Joost-Gaugier|2006|page=143}}]] [55] => [[File:Archytas of Tarentum MAN Napoli Inv5607.jpg|thumb|upright|alt=Bronze bust of a man with a short, curly beard wearing a tainia, which resembles a turban. Short curls hang out from underneath the tainia. The face is much broader than the other busts and the neck much fatter. The brow ridges are very prominent.|Bronze bust of a philosopher wearing a ''[[Tainia (costume)|tainia]]'' from [[Villa of the Papyri]], [[Herculaneum]], possibly a fictional bust of Pythagoras{{sfn|Dillon|2005|page=163}}{{sfn|Joost-Gaugier|2006|page=143}}]] [56] => [57] => Ancient sources also record Pythagoras having studied under a variety of native Greek thinkers.{{sfn|Riedweg|2005|page=8}} Some identify Hermodamas of Samos as a possible tutor.{{sfn|Riedweg|2005|page=8}}Porphyry, ''Vit. Pyth.'' 2, Diogenes Laërtius, viii. 2. Hermodamas represented the indigenous Samian [[Rhapsode|rhapsodic]] tradition and his father Creophylos was said to have been the host of his rival poet [[Homer]].{{sfn|Riedweg|2005|page=8}} Others credit [[Bias of Priene]], Thales,Iamblichus, ''Vit. Pyth.'' 9. or [[Anaximander]] (a pupil of Thales).{{sfn|Riedweg|2005|page=8}}Porphyry, ''Vit. Pyth.'' 2. Other traditions claim the mythic bard [[Orpheus]] as Pythagoras's teacher, thus representing the [[Orphism (religion)|Orphic Mysteries]].{{sfn|Riedweg|2005|page=8}} The Neoplatonists wrote of a "sacred discourse" Pythagoras had written on the gods in the [[Doric Greek]] dialect, which they believed had been dictated to Pythagoras by the Orphic priest Aglaophamus upon his initiation to the orphic Mysteries at [[Leibethra]].{{sfn|Riedweg|2005|page=8}} Iamblichus credited Orpheus with having been the model for Pythagoras's manner of speech, his spiritual attitude, and his manner of worship.{{sfn|Riedweg|2005|pages=8–9}} Iamblichus describes Pythagoreanism as a synthesis of everything Pythagoras had learned from Orpheus, from the Egyptian priests, from the [[Eleusinian Mysteries]], and from other religious and philosophical traditions.{{sfn|Riedweg|2005|pages=8–9}} Riedweg states that, although these stories are fanciful, Pythagoras's teachings were definitely influenced by Orphism to a noteworthy extent.{{sfn|Riedweg|2005|page=9}} [58] => [59] => Of the various Greek sages claimed to have taught Pythagoras, [[Pherecydes of Syros]] is mentioned most often.{{sfn|Riedweg|2005|page=9}}Aristoxenus and others in Diogenes Laërtius, i. 118, 119; Cicero, ''de Div.'' i. 49 Similar miracle stories were told about both Pythagoras and Pherecydes, including one in which the hero predicts a shipwreck, one in which he predicts the conquest of [[Messina]], and one in which he drinks from a well and predicts an earthquake.{{sfn|Riedweg|2005|page=9}} [[Apollonius Paradoxographus]], a paradoxographer who may have lived in the second century BC, identified Pythagoras's [[Thaumaturgy|thaumaturgic]] ideas as a result of Pherecydes's influence.{{sfn|Riedweg|2005|page=9}} Another story, which may be traced to the Neopythagorean philosopher [[Nicomachus]], tells that, when Pherecydes was old and dying on the island of [[Delos]], Pythagoras returned to care for him and pay his respects.{{sfn|Riedweg|2005|page=9}} [[Duris of Samos|Duris]], the historian and [[tyrant]] of Samos, is reported to have patriotically boasted of an epitaph supposedly penned by Pherecydes which declared that Pythagoras's wisdom exceeded his own.{{sfn|Riedweg|2005|page=9}} On the grounds of all these references connecting Pythagoras with Pherecydes, Riedweg concludes that there may well be some historical foundation to the tradition that Pherecydes was Pythagoras's teacher.{{sfn|Riedweg|2005|page=9}} Pythagoras and Pherecydes also appear to have shared similar views on the soul and the teaching of metempsychosis.{{sfn|Riedweg|2005|page=9}} [60] => [61] => Before 520 BC, on one of his visits to Egypt or Greece, Pythagoras might have met [[Thales of Miletus]], who would have been around fifty-four years older than him. Thales was a philosopher, scientist, mathematician, and engineer,{{citation |last=Boyer |first=Carl B. |url=https://archive.org/details/AHistoryOfMathematics |title=A History of Mathematics |year=1968}} also known for a [[Thales' Theorem|special case]] of the [[inscribed angle theorem]]. Pythagoras's birthplace, the island of [[Samos]], is situated in the Northeast [[Aegean Sea]] not far from [[Miletus]].{{sfn|Zhmud|2012|pages=2, 16}} Diogenes Laërtius cites a statement from [[Aristoxenus]] (fourth century BC) stating that Pythagoras learned most of his [[moral]] doctrines from the [[Pythia|Delphic priestess]] [[Themistoclea]].Diogenes Laërtius, [https://www.perseus.tufts.edu/hopper/text?doc=Perseus:text:1999.01.0258:book=8:chapter=1&highlight=themistoclea ''Lives of Eminent Philosophers''], viii. 1, 8.{{citation|url=https://books.google.com/books?id=x7ngECDpxmMC|title=Ancient Women Philosophers: 600 B.C.-500 A. D.|first=M. E.|last=Waithe|date=April 30, 1987|publisher=Springer Science & Business Media|via=Google Books|isbn=9789024733682}}{{citation |last=Malone |first=John C. |title=Psychology: Pythagoras to present |url=https://books.google.com/books?id=e6Qa6cMQj8AC&pg=PA22 |access-date=25 October 2010 |date=30 June 2009 |publisher=MIT Press |isbn=978-0-262-01296-6 |page=22}} Porphyry agrees with this assertion,Porphyry, [http://www.tertullian.org/fathers/porphyry_life_of_pythagoras_02_text.htm ''Life of Pythagoras''], 41. but calls the priestess Aristoclea (''Aristokleia'').[[Gilles Ménage]]: ''The history of women philosophers''. Translated from the Latin with an introduction by Beatrice H. Zedler. University Press of America, Lanham 1984, p. 47. "The person who is referred to as Themistoclea in Laërtius and Theoclea in Suidas, Porphyry calls Aristoclea." Ancient authorities furthermore note the similarities between the religious and [[ascetic]] peculiarities of Pythagoras with the [[Orphism (religion)|Orphic]] or [[Greco-Roman mysteries|Cretan mysteries]],Iamblichus, ''Vit. Pyth.'' 25; Porphyry, ''Vit. Pyth.'' 17; Diogenes Laërtius, viii. 3. or the [[Pythia|Delphic oracle]].Ariston. ap. Diogenes Laërtius, viii. 8, 21; Porphyry, ''Vit. Pyth.'' 41. [62] => [63] => === In Croton === [64] => {{ [65] => Location map+|Italy|width=250|float=left|caption=Map of Italy showing locations associated with Pythagoras|places= [66] => {{Location map~|Italy|lat=39.0808|N|long=17.1271|E|region:IT_type:city(60157)|position=right|label_size=75 |label=[[Crotone|Croton]]}} [67] => {{Location map~|Italy|lat=40.3706|N|long=16.8137|E|region:IT_type:city(60157)|position=below|label_size=75 |label=[[Metapontum]]}} [68] => {{Location map~|Italy|lat=39.7500|N|long=16.4667|E|region:IT_type:city(60157)|position=below|label_size=75 |label=[[Sybaris]]}} [69] => {{Location map~|Italy|lat=40.4644|N|long=17.2470|E|region:IT_type:city(60157)|position=right|label_size=75 |label=[[Taranto|Tarentum]]}} [70] => {{Location map~|Italy|lat=37.3094|N|long=13.5860|E|region:IT_type:city(60157)|position=right|label_size=75 |label=[[Agrigento|Acragas]]}} [71] => }} [72] => [73] => Porphyry repeats an account from [[Antiphon (writer)|Antiphon]], who reported that, while he was still on Samos, Pythagoras founded a school known as the "semicircle".{{sfn|Riedweg|2005|page=10}}{{sfn|Cornelli|McKirahan|2013|page=64}} Here, Samians debated matters of public concern.{{sfn|Riedweg|2005|page=10}}{{sfn|Cornelli|McKirahan|2013|page=64}} Supposedly, the school became so renowned that the brightest minds in all of Greece came to Samos to hear Pythagoras teach.{{sfn|Riedweg|2005|page=10}} Pythagoras himself dwelled in a secret cave, where he studied in private and occasionally held discourses with a few of his close friends.{{sfn|Riedweg|2005|page=10}}{{sfn|Cornelli|McKirahan|2013|page=64}} Christoph Riedweg, a German scholar of early Pythagoreanism, states that it is entirely possible Pythagoras may have taught on Samos,{{sfn|Riedweg|2005|page=10}} but cautions that Antiphon's account, which makes reference to a specific building that was still in use during his own time, appears to be motivated by Samian patriotic interest.{{sfn|Riedweg|2005|page=10}} [74] => [75] => Around 530 BC, when Pythagoras was about forty years old, he left Samos.{{sfn|Joost-Gaugier|2006|page=11}}{{sfn|Kahn|2001|page=6}}{{sfn|Riedweg|2005|page=11}}{{sfn|Ferguson|2008|page=5}}{{sfn|Gregory|2015|page=22}} His later admirers claimed that he left because he disagreed with the [[tyrant|tyranny]] of [[Polycrates]] in Samos,{{sfn|Riedweg|2005|page=11}} Riedweg notes that this explanation closely aligns with Nicomachus's emphasis on Pythagoras's purported love of freedom, but that Pythagoras's enemies portrayed him as having a proclivity towards tyranny.{{sfn|Riedweg|2005|page=11}} Other accounts claim that Pythagoras left Samos because he was so overburdened with public duties in Samos, because of the high estimation in which he was held by his fellow-citizens.Iamblichus, ''Vit. Pyth.'' 28; Porphyry, ''Vit. Pyth.'' 9 He arrived in the Greek colony of Croton (today's [[Crotone]], in [[Calabria]]) in what was then [[Magna Graecia]].{{sfn|Kahn|2001|page=6}}{{sfn|Gregory|2015|page=22}}[[Cornelia J. de Vogel]]: ''Pythagoras and Early Pythagoreanism.'' Assen 1966, pp. 21ff. Cfr. Cicero, ''De re publica'' 2, 28–30.{{sfn|Riedweg|2005|pages=11–12}} All sources agree that Pythagoras was charismatic and quickly acquired great political influence in his new environment.{{sfn|Kahn|2001|page=6}}[[Cornelia J. de Vogel]]: Pythagoras and Early Pythagoreanism, Assen 1966, S. 148–150.{{sfn|Riedweg|2005|pages=12–13}} He served as an advisor to the elites in Croton and gave them frequent advice.{{sfn|Riedweg|2005|pages=12–18}} Later biographers tell fantastical stories of the effects of his eloquent speeches 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.Porphyry, ''Vit. Pyth.'' 18; Iamblichus, ''Vit. Pyth.'' 37, etc.{{sfn|Riedweg|2005|pages=13–18}} [76] => [77] => === Family and friends === [78] => [[File:The story of the greatest nations; a comprehensive history, extending from the earliest times to the present, founded on the most modern authorities, and including chronological summaries and (14783288925).jpg|thumb|upright=1.4|alt=Illustration showing Pythagoras teaching a class of women|Illustration from 1913 showing Pythagoras teaching a class of women. Many prominent members of his school were women{{sfn|Kahn|2001|page=8}}{{sfn|Pomeroy|2013|page=1}} and some modern scholars think that he may have believed that women should be taught philosophy as well as men.{{sfn|Pomeroy|2013|page=xvi}}]] [79] => [80] => Diogenes Laërtius states that Pythagoras "did not indulge in the pleasures of love"{{sfn|Ferguson|2008|page=58}} and that he cautioned others to only have sex "whenever you are willing to be weaker than yourself".{{sfn|Ferguson|2008|page=59}} According to Porphyry, Pythagoras married [[Theano (philosopher)|Theano]], a lady of [[Crete]] and the daughter of Pythenax{{sfn|Ferguson|2008|page=59}} and had several children with her.{{sfn|Ferguson|2008|page=59}} Porphyry writes that Pythagoras had two sons named [[Telauges]] and [[Arignote]],{{sfn|Ferguson|2008|page=59}} and a daughter named Myia,{{sfn|Ferguson|2008|page=59}} who "took precedence among the maidens in Croton and, when a wife, among married women."{{sfn|Ferguson|2008|page=59}} Iamblichus mentions none of these children{{sfn|Ferguson|2008|page=59}} and instead only mentions a son named Mnesarchus after his grandfather.{{sfn|Ferguson|2008|page=59}} This son was raised by Pythagoras's appointed successor Aristaeus and eventually took over the school when Aristaeus was too old to continue running it.{{sfn|Ferguson|2008|page=59}} [[Suda]] writes that Pythagoras had 4 children (Telauges, Mnesarchus, Myia and Arignote).{{citation| url = https://www.cs.uky.edu/~raphael/sol/sol-entries/theta/84| title = Suda Encyclopedia, th.84}} [81] => [82] => The wrestler [[Milo of Croton]] was said to have been a close associate of Pythagoras{{sfn|Riedweg|2005|pages=5–6, 59, 73}} and was credited with having saved the philosopher's life when a roof was about to collapse.{{sfn|Riedweg|2005|pages=5–6, 59, 73}} This association may have been the result of confusion with a different man named Pythagoras, who was an athletics trainer.{{sfn|Riedweg|2005|page=10}} Diogenes Laërtius records Milo's wife's name as Myia.{{sfn|Ferguson|2008|page=59}} Iamblichus mentions Theano as the wife of Brontinus of Croton.{{sfn|Ferguson|2008|page=59}} Diogenes Laërtius states that the same Theano was Pythagoras's pupil{{sfn|Ferguson|2008|page=59}} and that Pythagoras's wife Theano was her daughter.{{sfn|Ferguson|2008|page=59}} Diogenes Laërtius also records that works supposedly written by Theano were still extant during his own lifetime{{sfn|Ferguson|2008|page=59}} and quotes several opinions attributed to her.{{sfn|Ferguson|2008|page=59}} These writings are now known to be [[pseudepigraphica]]l.{{sfn|Ferguson|2008|page=59}} [83] => [84] => === Death === [85] => Pythagoras's emphasis on dedication and asceticism are credited with aiding in Croton's decisive victory over the neighboring colony of [[Sybaris]] in 510 BC.{{sfn|Kahn|2001|pages=6–7}} After the victory, some prominent citizens of Croton proposed a [[Greek democracy|democratic constitution]], which the Pythagoreans rejected.{{sfn|Kahn|2001|pages=6–7}} The supporters of democracy, headed by [[Cylon of Croton|Cylon]] and Ninon, the former of whom is said to have been irritated by his exclusion from Pythagoras's brotherhood, roused the populace against them.{{sfn|Riedweg|2005|page=19}} Followers of Cylon and Ninon attacked the Pythagoreans during one of their meetings, either in the house of Milo or in some other meeting-place.{{sfn|Kahn|2001|page=7}}{{sfn|Riedweg|2005|pages=19–20}} Accounts of the attack are often contradictory and many probably confused it with the later anti-Pythagorean rebellions, such as the one in Metapontum in 454 BC.{{sfn|Riedweg|2005|page=19}}Plutarch'', On the Sign of Socrates'' 583a The building was apparently set on fire,{{sfn|Kahn|2001|page=7}} and many of the assembled members perished;{{sfn|Kahn|2001|page=7}} only the younger and more active members managed to escape.Iamblichus, ''Vit. Pyth.'' 255–259; Porphyry, ''Vit. Pyth.'' 54–57; Diogenes Laërtius, viii. 39; comp. Plutarch, ''de Gen. Socr.'' p. 583 [86] => [87] => Sources disagree regarding whether Pythagoras was present when the attack occurred and, if he was, whether or not he managed to escape.{{sfn|Burkert|1972|page=106}}{{sfn|Riedweg|2005|pages=19–20}} In some accounts, Pythagoras was not at the meeting when the Pythagoreans were attacked because he was on Delos tending to the dying Pherecydes.{{sfn|Riedweg|2005|pages=19–20}} According to another account from Dicaearchus, Pythagoras was at the meeting and managed to escape,{{sfn|Riedweg|2005|page=20}} leading a small group of followers to the nearby city of [[Locris]], where they pleaded for sanctuary, but were denied.{{sfn|Riedweg|2005|page=20}} They reached the city of [[Metapontum]], where they took shelter in the temple of the [[Muses]] and died there of starvation after forty days without food.{{sfn|Burkert|1972|page=106}}{{sfn|Kahn|2001|page=7}}{{sfn|Riedweg|2005|page=20}}{{sfn|Grant|1989|page=278}} Another tale recorded by Porphyry claims that, as Pythagoras's enemies were burning the house, his devoted students laid down on the ground to make a path for him to escape by walking over their bodies across the flames like a bridge.{{sfn|Riedweg|2005|page=20}} Pythagoras managed to escape, but was so despondent at the deaths of his beloved students that he committed suicide.{{sfn|Riedweg|2005|page=20}} A different legend reported by both Diogenes Laërtius and Iamblichus states that Pythagoras almost managed to escape, but that he came to a fava bean field and refused to run through it, since doing so would violate his teachings, so he stopped instead and was killed.{{sfn|Riedweg|2005|page=20}}{{sfn|Simoons|1998|pages=225–228}} This story seems to have originated from the writer Neanthes, who told it about later Pythagoreans, not about Pythagoras himself.{{sfn|Riedweg|2005|page=20}} [88] => [89] => == Teachings == [90] => === Metempsychosis === [91] => [[File:Cropped image of Pythagoras from Raphael's School of Athens.jpg|thumb|upright=1.3|alt=Scene from Raphael's School of Athens showing Pythagoras as a balding, bearded man writing in a book with a quill. He is dressed in a long-sleeved tunic with a cloak spread across his legs as he kneels to write, using his left thigh to support the book. In front of him, a boy with long hair presents him with a chalk board showing a diagrammatic representation of a lyre above the symbol of the tetractys. Averroes, shown as a stereotypical Middle Easterner with a mustache and wearing a turban, peers over his left shoulder while another bearded, balding philosopher in classical garb, probably Anaxagoras, peers over his right shoulder, taking notes into a much smaller notepad. A very feminine looking figure with long hair stands behind the boy, dressed in a white cloak.|In [[Raphael]]'s fresco ''[[The School of Athens]]'', Pythagoras is shown writing in a book as a young man presents him with a tablet showing a [[diagram]]matic representation of a [[lyre]] above a drawing of the sacred ''[[tetractys]]''.{{sfn|Bruhn|2005|page=66}}]] [92] => [93] => Although the exact details of Pythagoras's teachings are uncertain,{{sfn|Burkert|1972|pages=106–109}}{{sfn|Kahn|2001|pages=5–6}} it is possible to reconstruct a general outline of his main ideas.{{sfn|Burkert|1972|pages=106–109}}{{sfn|Kahn|2001|pages=9–11}} Aristotle writes at length about the teachings of the Pythagoreans,{{sfn|Copleston|2003|page=31}}{{sfn|Burkert|1972|pages=29–30}} but without mentioning Pythagoras directly.{{sfn|Copleston|2003|page=31}}{{sfn|Burkert|1972|pages=29–30}} One of Pythagoras's main doctrines appears to have been ''[[metempsychosis]]'',{{sfn|Ferguson|2008|page=5}}{{sfn|Kahn|2001|page=11}}{{sfn|Zhmud|2012|page=232}}{{sfn|Burkert|1985|pages=300–301}}{{sfn|Gregory|2015|pages=24–25}}{{sfn|Copleston|2003|pages=30–31}} the belief that all [[soul]]s are immortal and that, after death, a soul is transferred into a new body.{{sfn|Kahn|2001|page=11}}{{sfn|Gregory|2015|pages=24–25}} This teaching is referenced by Xenophanes, Ion of Chios, and Herodotus.{{sfn|Kahn|2001|page=11}}Diogenes Laërtius, viii. 36, comp. Aristotle, ''de Anima'', i. 3; Herodotus, ii. 123. Nothing whatsoever, however, is known about the nature or mechanism by which Pythagoras believed metempsychosis to occur.{{sfn|Gregory|2015|page=25}} [94] => [95] => [[Empedocles]] alludes in one of his poems that Pythagoras may have claimed to possess the ability to recall his former incarnations.{{sfn|Kahn|2001|page=12}} Diogenes Laërtius reports an account from [[Heraclides Ponticus]] that Pythagoras told people that he had lived four previous lives that he could remember in detail.Diogenes Laërtius, viii. 3–4{{sfn|Cornelli|McKirahan|2013|pages=164–167}}Porphyry, ''Vit. Pyth.'' 26; Pausanias, ii. 17; Diogenes Laërtius, viii. 5; Horace, ''Od.'' i. 28,1. 10 The first of these lives was as [[Aethalides]] the son of [[Hermes]], who granted him the ability to remember all his past incarnations.{{sfn|Cornelli|McKirahan|2013|pages=164–165}} Next, he was incarnated as [[Euphorbus]], a minor hero from the [[Trojan War]] briefly mentioned in the ''[[Iliad]]''.{{sfn|Cornelli|McKirahan|2013|pages=165–166}} He then became the philosopher [[Hermotimus of Clazomenae|Hermotimus]],{{sfn|Cornelli|McKirahan|2013|page=167}} who recognized the shield of Euphorbus in the temple of Apollo.{{sfn|Cornelli|McKirahan|2013|page=167}} His final incarnation was as Pyrrhus, a fisherman from [[Delos]].{{sfn|Cornelli|McKirahan|2013|page=167}} One of his past lives, as reported by [[Dicaearchus]], was as a beautiful courtesan.{{sfn|Zhmud|2012|page=232}}Aulus Gellius, iv. 11 [96] => [97] => === Mysticism === [98] => Another belief attributed to Pythagoras was that of the "[[Musica universalis|harmony of the spheres]]",{{sfn|Riedweg|2005|pages=29–30}}{{sfn|Gregory|2015|pages=38–39}} which maintained that the planets and stars move according to mathematical equations, which correspond to musical notes and thus produce an inaudible symphony.{{sfn|Riedweg|2005|pages=29–30}}{{sfn|Gregory|2015|pages=38–39}} According to Porphyry, Pythagoras taught that the seven [[Muse]]s were actually the [[Classical planet|seven planets]] singing together.{{sfn|Riedweg|2005|page=30}} In his philosophical dialogue [[Protrepticus (Aristotle)|''Protrepticus'']], [[Aristotle]] has his literary double say:{{citation|author1=D. S. Hutchinson|author2=Monte Ransome Johnson|url=http://blog.protrepticus.info/2015/01/new-reconstruction-includes-greek-text.html|title=New Reconstruction, includes Greek text|date=25 January 2015|page=48}} [99] => [100] => {{blockquote|When Pythagoras was asked [why humans exist], he said, "to observe the heavens", and he used to claim that he himself was an observer of nature, and it was for the sake of this that he had passed over into life.}} [101] => [102] => Pythagoras was said to have practiced [[divination]] and [[prophecy]].Cicero, ''de Divin.'' i. 3, 46; Porphyry, ''Vit. Pyth.'' 29. The earliest mentions of divination by [[isopsephy]] in Greek literature associate it with Pythagoras; he was viewed as the founder of this practice.{{harvnb|Barry|1999|pp=24, 30–31, 83}}; {{harvnb|Acevedo|2020|pp=42–44, 50}} According to his biographer, Iamblichus, he taught his method of divination to a Scythian priest of Apollo by the name of [[Abaris the Hyperborean]]:{{sfn|Barry|1999|pp=30–31}} [103] => [104] => {{Blockquote [105] => |text =Abaris stayed with Pythagoras, and was compendiously taught physiology and theology; and instead of divining by the entrails of beasts, he revealed to him the art of prognosticating by numbers, conceiving this to be a method purer, more divine, and more kindred to the celestial numbers of the Gods. [106] => |author =Iamblichus [107] => |title =The Life of Pythagoras [108] => |source =19.93 [109] => }} [110] => [111] => This shouldn't be confused with a simplified version known today as "[[Numerology#Pythagorean method|Pythagorean numerology]]", involving a variant of an isopsephic technique known – among other names – as {{lang|grc|pythmenes}} {{gloss|roots}}{{harvnb|Barry|1999|pp=120, 194–196}}; {{harvnb|Gregory|2015|pp=32–34}}; {{harvnb|Kalvesmaki|2013|pp=82–83}} or {{gloss|base numbers}},{{harvnb|Zhmud|2012|p=277}} by means of which the base values of letters in a word were mathematically reduced by addition or division, in order to obtain a single value from one to nine for the whole name or word; these 'roots' or 'base numbers' could then be interpreted with other techniques, such as traditional Pythagorean attributions.{{harvnb|Barry|1999|pp=120, 124, 194–196}}; {{harvnb|Kalvesmaki|2013|pp=82–83}} This latter form of numerology flourished during the [[Byzantine Empire|Byzantine era]], and was first attested among the [[Gnosticism|Gnostics]] of the second century AD. By that time, isopsephy had developed into several different techniques that were used for a variety of purposes; including divination, doctrinal allegory, and medical prognosis and treatment. [112] => [113] => 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.Iamblichus, ''Vit. Pyth.'' 25; Porphyry, ''Vit. Pyth.'' 17; Diogenes Laërtius, viii. 3, 13; Cicero, ''Tusc. Qu.'' v. 3. [114] => [115] => === Numerology === [116] => {{blockquote|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 (Aristotle)|Metaphysics]]'' 1–5, c. 350 BC}} [117] => [[File:Tetractys.svg|thumb|alt=Diagram showing the tetractys, an equilateral triangle made up of ten dots, with one dot in the top row, two in the second, three in the third, and four in the bottom.|Pythagoras is credited with having devised the [[tetractys]],{{sfn|Bruhn|2005|pages=65–66}}{{sfn|Gregory|2015|pages=28–29}} an important [[Religious symbol|sacred symbol]] in later Pythagoreanism.{{sfn|Riedweg|2005|page=29}}{{sfn|Kahn|2001|pages=1–2}}]] [118] => [119] => According to Aristotle, the Pythagoreans used mathematics for solely mystical reasons, devoid of practical application.{{sfn|Burkert|1972|pages=467–468}} They believed that all things were made of numbers.{{sfn|Burkert|1972|page=265}}{{sfn|Kahn|2001|page=27}} The number one (the [[Monad (philosophy)|monad]]) represented the origin of all things{{sfn|Riedweg|2005|page=23}} and the number two (the [[Dyad (Greek philosophy)|dyad]]) represented matter.{{sfn|Riedweg|2005|page=23}} The number three was an "ideal number" because it had a beginning, middle, and end{{sfn|Joost-Gaugier|2006|pages=170–172}} and was the smallest number of points that could be used to define a plane triangle, which they revered as a symbol of the god [[Apollo]].{{sfn|Joost-Gaugier|2006|pages=170–172}} The number four signified the [[Seasons|four seasons]] and the [[Classical element|four elements]].{{sfn|Joost-Gaugier|2006|page=172}} The number seven was also sacred because it was the number of planets and the number of strings on a lyre,{{sfn|Joost-Gaugier|2006|page=172}} and because Apollo's birthday was celebrated on the seventh day of each month.{{sfn|Joost-Gaugier|2006|page=172}} They believed that [[Parity (mathematics)|odd numbers]] were [[masculine]],{{sfn|Burkert|1972|page=433}} that even numbers were [[feminine]],{{sfn|Burkert|1972|page=433}} and that the number five represented marriage, because it was the sum of two and three.{{sfn|Burkert|1972|page=467}}{{sfn|Joost-Gaugier|2006|page=170}} [120] => [121] => Ten was regarded as the "perfect number"{{sfn|Burkert|1972|pages=467–468}} and the Pythagoreans honored it by never gathering in groups larger than ten.{{sfn|Joost-Gaugier|2006|page=161}} Pythagoras was credited with devising the [[tetractys]], the triangular figure of four rows which add up to the perfect number, ten.{{sfn|Bruhn|2005|pages=65–66}}{{sfn|Gregory|2015|pages=28–29}} The Pythagoreans regarded the tetractys as a symbol of utmost mystical importance.{{sfn|Bruhn|2005|pages=65–66}}{{sfn|Gregory|2015|pages=28–29}}{{sfn|Riedweg|2005|page=29}} Iamblichus, in his ''Life of Pythagoras'', states that the tetractys was "so admirable, and so divinised by those who understood [it]," that Pythagoras's students would swear oaths by it.{{sfn|Bruhn|2005|page=66}}{{sfn|Gregory|2015|pages=28–29}}{{sfn|Riedweg|2005|page=29}}Iamblichus, ''Vit. Pyth.'', 29 Andrew Gregory concludes that the tradition linking Pythagoras to the tetractys is probably genuine.{{sfn|Gregory|2015|page=28}} [122] => [123] => Modern scholars debate whether these numerological teachings were developed by Pythagoras himself or by the later Pythagorean philosopher [[Philolaus of Croton]].{{sfn|Joost-Gaugier|2006|pages=87–88}} In his landmark study ''Lore and Science in Ancient Pythagoreanism'', Walter Burkert argues that Pythagoras was a charismatic political and religious teacher,{{sfn|Kahn|2001|pages=2–3}} but that the number philosophy attributed to him was really an innovation by Philolaus.{{sfn|Kahn|2001|page=3}} According to Burkert, Pythagoras never dealt with numbers at all, let alone made any noteworthy contribution to mathematics.{{sfn|Kahn|2001|pages=2–3}} Burkert argues that the only mathematics the Pythagoreans ever actually engaged in was simple, [[Mathematical proof|proofless]] [[arithmetic]],{{sfn|Burkert|1972|pages=428–433}} but that these arithmetic discoveries did contribute significantly to the beginnings of mathematics.{{sfn|Burkert|1972|page=465}} [124] => [125] => == Pythagoreanism == [126] => === Communal lifestyle === [127] => [[File:Bronnikov gimnpifagoreizev.jpg|thumb|right|upright=1.5|alt=Painting showing a group of people dressed in white classical garb standing at the edge of a cliff overlooking the sea watching as the sun rises. The central figure, probably Pythagoras himself, is turned away from our view towards the sunrise. He has long braids and his long beard is partially visible from the side. Both of his arms are raised into the air. The three men closest to him, two on his left and one on his right, are kneeling and making frantic gestures, possibly weeping. Behind them, an older man plays a harp and two women play lyres. A young man without a beard and an middle-aged man with a beard play lyres as well, while another young man plays the aulos. A man in the foreground at the back of the group kneels prayerfully towards the sunrise. In the background, at the far left side of the painting, a woman, a girl, a boy, and a young, naked child watch the Pythagoreans. The woman and the girl are carrying pots, indicating they have been fetching water.|''Pythagoreans Celebrate the Sunrise'' (1869) by [[Fyodor Bronnikov]]]] [128] => {{Main|Pythagoreanism}} [129] => Both [[Plato]] and [[Isocrates]] state that, above all else, Pythagoras was known as the founder of a new way of life.Plato, ''Republic'', 600a, Isocrates, ''Busiris'', 28{{sfn|Cornelli|McKirahan|2013|page=168}}{{sfn|Grant|1989|page=277}} The organization Pythagoras founded at Croton was called a "school",Porphyry, ''Vit. Pyth.'' 19Thirlwall, ''Hist. of Greece'', vol. ii. p. 148 but, in many ways, resembled a [[monastery]].{{sfn|Riedweg|2005|page=31}} 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.comp. Cicero, ''de Leg.'' i. 12, ''de Off.'' i. 7; Diogenes Laërtius, viii. 10 The members of the sect [[community ownership|shared all their possessions in common]]{{sfn|Cornelli|McKirahan|2013|page=65}} and were devoted to each other to the exclusion of outsiders.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.{{sfn|Cornelli|McKirahan|2013|pages=68–69}} Ancient sources record that the Pythagoreans ate meals in common after the manner of the [[Sparta]]ns.Iamblichus, ''Vit. Pyth.'' 98; Strabo, vi.{{sfn|Kenny|2004|page=10}} One Pythagorean [[maxim (literature)|maxim]] was "''koinà tà phílōn''" ("All things in common among friends").{{sfn|Cornelli|McKirahan|2013|page=65}} Both Iamblichus and Porphyry provide detailed accounts of the organization 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 [[Greek gods|gods]] to benefit humankind.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. Iamblichus, in particular, presents the "Pythagorean Way of Life" as a pagan alternative to the Christian monastic communities of his own time.{{sfn|Riedweg|2005|page=31}} For Pythagoreans, the highest reward a human could attain was for their soul to join in the life of the gods and thus escape the cycle of reincarnation.Charles H. Kahn (2001). ''Pythagoras and the Pythagoreans''. Hackett Publishing. p. 52. Two groups existed within early Pythagoreanism: the ''mathematikoi'' ("learners") and the ''akousmatikoi'' ("listeners").{{sfn|Zhmud|2012|pages=2, 16}}{{sfn|Gregory|2015|page=31}} The ''akousmatikoi'' are traditionally identified by scholars as "old believers" in mysticism, numerology, and religious teachings;{{sfn|Gregory|2015|page=31}} whereas the ''mathematikoi'' are traditionally identified as a more intellectual, modernist faction who were more rationalist and scientific.{{sfn|Gregory|2015|page=31}} Gregory cautions that there was probably not a sharp distinction between them and that many Pythagoreans probably believed the two approaches were compatible.{{sfn|Gregory|2015|page=31}} The study of mathematics and music may have been connected to the worship of Apollo.Aelian, ''Varia Historia'', ii. 26; Diogenes Laërtius, viii. 13; Iamblichus, ''Vit. Pyth.'' 8, 91, 141 The Pythagoreans believed that music was a purification for the soul, just as medicine was a purification for the body.{{sfn|Riedweg|2005|page=30}} One anecdote of Pythagoras reports that when he encountered some drunken youths trying to break into the home of a virtuous woman, he sang a solemn tune with long [[spondee]]s and the boys' "raging willfulness" was quelled.{{sfn|Riedweg|2005|page=30}} The Pythagoreans also placed particular emphasis on the importance of [[physical exercise]];{{sfn|Riedweg|2005|page=31}} therapeutic [[dance|dancing]], daily morning walks along [[scenic route]]s, and [[Sport of athletics|athletics]] were major components of the Pythagorean lifestyle.{{sfn|Riedweg|2005|page=31}} Moments of contemplation at the beginning and end of each day were also advised.{{sfn|Riedweg|2005|pages=33–34}} [130] => [131] => === Prohibitions and regulations === [132] => [[File:Do Not Eat Beans.jpg|thumb|alt=Old manuscript illustration showing a cloaked and hooded man labelled "Pythagoras" raising his arms and turning his face away from a fava bean plant, labelled "Fabe."|French manuscript from 1512/1514, showing Pythagoras turning his face away from [[Vicia faba|fava beans]] in revulsion]] [133] => [134] => Pythagorean teachings were known as "symbols" (''symbola''){{sfn|Kahn|2001|page=8}} and members took a vow of silence that they would not reveal these symbols to non-members.{{sfn|Kahn|2001|page=8}}{{sfn|Cornelli|McKirahan|2013|page=168}}Scholion ad Aristophanes, ''Nub.'' 611; Iamblichus, ''Vit. Pyth.'' 237, 238 Those who did not obey the laws of the community were expelled{{sfn|Cornelli|McKirahan|2013|page=69}} and the remaining members would erect [[tombstone]]s for them as though they had died.{{sfn|Cornelli|McKirahan|2013|page=69}} A number of "oral sayings" (''akoúsmata'') attributed to Pythagoras have survived,{{sfn|Copleston|2003|page=31}}{{sfn|Riedweg|2005|pages=64–67}} dealing with how members of the Pythagorean community should perform sacrifices, how they should honor the gods, how they should "move from here", and how they should be buried.{{sfn|Riedweg|2005|page=64}} Many of these sayings emphasize the importance of ritual purity and avoiding defilement.{{sfn|Riedweg|2005|page=65}}{{sfn|Copleston|2003|pages=30–31}} For instance, a saying which Leonid Zhmud concludes can probably be genuinely traced back to Pythagoras himself forbids his followers from wearing woolen garments.{{sfn|Zhmud|2012|page=200}} Other extant oral sayings forbid Pythagoreans from breaking bread, poking fires with swords, or picking up crumbs{{sfn|Kenny|2004|page=10}} and teach that a person should always put the right sandal on before the left.{{sfn|Kenny|2004|page=10}} The exact meanings of these sayings, however, are frequently obscure.{{sfn|Riedweg|2005|pages=65–67}} Iamblichus preserves Aristotle's descriptions of the original, ritualistic intentions behind a few of these sayings,{{sfn|Riedweg|2005|pages=65–66}} but these apparently later fell out of fashion, because Porphyry provides markedly different ethical-philosophical interpretations of them:{{sfn|Riedweg|2005|pages=66–67}} [135] => [136] => {| class="wikitable" [137] => |- [138] => ! Pythagorean saying [139] => ! Original ritual purpose according to Aristotle/Iamblichus [140] => ! Porphyry's philosophical interpretation [141] => |- [142] => | "Do not take roads traveled by the public."{{sfn|Riedweg|2005|page=66}}{{sfn|Copleston|2003|page=31}} [143] => | "Fear of being defiled by the impure"{{sfn|Riedweg|2005|page=66}} [144] => | "with this he forbade following the opinions of the masses, yet to follow the ones of the few and the educated".{{sfn|Riedweg|2005|page=66}} [145] => |- [146] => | "and [do] not wear images of the gods on rings"{{sfn|Riedweg|2005|page=66}} [147] => | "Fear of defiling them by wearing them."{{sfn|Riedweg|2005|page=66}} [148] => | "One should not have the teaching and knowledge of the gods quickly at hand and visible [for everyone], nor communicate them to the masses."{{sfn|Riedweg|2005|page=66}} [149] => |- [150] => | "and pour libations for the gods from a drinking cup's handle [the 'ear']"{{sfn|Riedweg|2005|page=66}} [151] => | "Efforts to keep the divine and the human strictly separate"{{sfn|Riedweg|2005|page=66}} [152] => | "thereby he enigmatically hints that the gods should be honored and praised with music; for it goes through the ears".{{sfn|Riedweg|2005|page=66}} [153] => |} [154] => [155] => New initiates were allegedly not permitted to meet Pythagoras until after they had completed a five-year initiation period,{{sfn|Cornelli|McKirahan|2013|page=64}} during which they were required to remain silent.{{sfn|Cornelli|McKirahan|2013|page=64}} Sources indicate that Pythagoras himself was unusually progressive in his attitudes towards women{{sfn|Pomeroy|2013|page=xvi}} and female members of Pythagoras's school appear to have played an active role in its operations.{{sfn|Kahn|2001|page=8}}{{sfn|Pomeroy|2013|page=xvi}} Iamblichus provides a list of 235 famous Pythagoreans,{{sfn|Pomeroy|2013|page=1}} seventeen of whom are women.{{sfn|Pomeroy|2013|page=1}} In later times, many prominent female philosophers contributed to the development of [[Neopythagoreanism]].{{sfn|Pomeroy|2013|pages=xvi–xvii}} [156] => [157] => Pythagoreanism also entailed a number of dietary prohibitions.{{sfn|Copleston|2003|pages=30–31}}{{sfn|Kenny|2004|page=10}}comp. Porphyry, ''Vit. Pyth.'' 32; Iamblichus, ''Vit. Pyth.'' 96, etc. It is more or less agreed that Pythagoras issued a prohibition against the consumption of [[Vicia faba|fava beans]]{{sfn|Zhmud|2012|pages=137, 200}}{{sfn|Kenny|2004|page=10}} and the meat of non-sacrificial animals such as fish and poultry.{{sfn|Zhmud|2012|page=200}}{{sfn|Kenny|2004|page=10}} Both of these assumptions, however, have been contradicted.{{sfn|Copleston|2003|page=30}}Diogenes Laërtius, viii. 19, 34; Aulus Gellius, iv. 11; Porphyry, ''Vit. Pyth.'' 34, ''de Abst.'' i. 26; Iamblichus, ''Vit. Pyth.'' 98 Pythagorean dietary restrictions may have been motivated by belief in the doctrine of [[metempsychosis]].{{sfn|Cornelli|McKirahan|2013|page=168}}Plutarch, ''de Esu Carn.'' pp. 993, 996, 997{{sfn|Kahn|2001|page=9}}{{sfn|Kenny|2004|pages=10–11}} Some ancient writers present Pythagoras as enforcing a strictly [[Vegetarianism|vegetarian]] diet.{{efn|as Empedocles did afterwards, Aristotle, ''Rhet.'' i. 14. § 2; Sextus Empiricus, ix. 127. This was also one of the Orphic precepts, Aristoph. ''Ran.'' 1032}}{{sfn|Cornelli|McKirahan|2013|page=168}}{{sfn|Kahn|2001|page=9}} [[Eudoxus of Cnidus]], a student of Archytas, writes, "Pythagoras was distinguished by such purity and so avoided killing and killers that he not only abstained from animal foods, but even kept his distance from cooks and hunters."Eudoxus, frg. 325{{sfn|Zhmud|2012|page=235}} Other authorities contradict this statement.Aristo ap. Diogenes Laërtius, viii. 20; comp. Porphyry, ''Vit. Pyth.'' 7; Iamblichus, ''Vit. Pyth.'' 85, 108 According to [[Aristoxenus]],Aristoxenus ap. Diogenes Laërtius, viii. 20 Pythagoras allowed the use of all kinds of animal food except the flesh of [[oxen]] used for [[plough]]ing, and [[sheep|rams]].{{sfn|Zhmud|2012|page=235}}comp. Porphyry, ''Vit. Pyth.'' 7; Iamblichus, ''Vit. Pyth.'' 85, 108 According to Heraclides Ponticus, Pythagoras ate the meat from sacrifices{{sfn|Zhmud|2012|page=235}} and established a diet for athletes dependent on meat.{{sfn|Zhmud|2012|page=235}} [158] => [159] => == Legends == [160] => [[File:Salvator Rosa - Pythagoras Emerging from the Underworld - Google Art Project.jpg|thumb|upright=1.35|alt=Oil painting showing, at the far left, a cloaked and hooded Pythagoras emerging from a cave in the woods as a large crowd of adoring followers wait outside to greet him.|''Pythagoras Emerging from the Underworld'' (1662) by [[Salvator Rosa]]]] [161] => [162] => Within his own lifetime, Pythagoras was already the subject of elaborate [[hagiography|hagiographic]] legends.{{sfn|Kahn|2001|page=5}}{{sfn|Riedweg|2005|page=1}} Aristotle described Pythagoras as a [[wonder-worker]] and somewhat of a supernatural figure.{{sfn|Riedweg|2005|page=2}}{{sfn|Gregory|2015|pages=30–31}} In a fragment, Aristotle writes that Pythagoras had a golden thigh,{{sfn|Riedweg|2005|page=2}}{{sfn|Gregory|2015|page=30}}{{sfn|Kenny|2004|page=11}} which he publicly exhibited at the [[Ancient Olympic Games|Olympic Games]]{{sfn|Riedweg|2005|page=2}}{{sfn|Ferguson|2008|page=60}} and showed to [[Abaris the Hyperborean]] as proof of his identity as the "Hyperborean Apollo".{{sfn|Riedweg|2005|page=2}}Porphyry, ''Vit. Pyth.'' 20; Iamblichus, ''Vit. Pyth.'' 31, 140; Aelian, ''Varia Historia'', ii. 26; Diogenes Laërtius, viii. 36. Supposedly, the priest of Apollo gave Pythagoras a magic arrow, which he used to fly over long distances and perform ritual purifications.{{sfn|McKeown|2013|page=155}} He was supposedly once seen at both Metapontum and Croton [[bilocation|at the same time]].Comp. Herodian, iv. 94, etc.{{sfn|Kahn|2001|page=5}}{{sfn|Ferguson|2008|page=60}}{{sfn|Gregory|2015|page=30}}{{sfn|Kenny|2004|page=11}} When Pythagoras crossed the river Kosas (the modern-day [[Basento]]), "several witnesses" reported that they heard it greet him by name.{{sfn|Burkert|1972|page=144}}{{sfn|Ferguson|2008|page=60}}{{sfn|Gregory|2015|page=30}} In Roman times, a legend claimed that Pythagoras was the son of Apollo.{{sfn|Ferguson|2008|page=10}}{{sfn|Kenny|2004|page=11}} According to [[Hermes Trismegistus#In Islamic tradition|Muslim tradition]], Pythagoras was said to have been initiated by [[Hermes Trismegistus|Hermes]] (Egyptian [[Thoth]]).See Antoine Faivre, in ''The Eternal Hermes'' (1995) [163] => [164] => Pythagoras was said to have dressed all in white.{{sfn|Riedweg|2005|page=2}}{{sfn|Joost-Gaugier|2006|page=47}} He is also said to have borne a golden [[wreath]] atop his head{{sfn|Riedweg|2005|page=2}} and to have worn [[trousers]] after the fashion of the [[Thrace|Thracians]].{{sfn|Riedweg|2005|page=2}} Diogenes Laërtius presents Pythagoras as having exercised remarkable [[self-control]];{{sfn|Ferguson|2008|pages=58–59}} he was always cheerful,{{sfn|Ferguson|2008|pages=58–59}} but "abstained wholly from laughter, and from all such indulgences as jests and idle stories".{{sfn|Ferguson|2008|page=59}} Pythagoras was said to have had extraordinary success in dealing with animals.{{sfn|Kahn|2001|page=5}}{{sfn|Cornelli|McKirahan|2013|page=160}}{{sfn|Ferguson|2008|page=60}} A fragment from Aristotle records that, when a deadly snake bit Pythagoras, he bit it back and killed it.{{sfn|McKeown|2013|page=155}}{{sfn|Ferguson|2008|page=60}}{{sfn|Gregory|2015|page=30}} Both Porphyry and Iamblichus report that Pythagoras once persuaded a bull not to eat fava beans{{sfn|Kahn|2001|page=5}}{{sfn|Cornelli|McKirahan|2013|page=160}} and that he once convinced a notoriously destructive bear to swear that it would never harm a living thing again, and that the bear kept its word.{{sfn|Kahn|2001|page=5}}{{sfn|Cornelli|McKirahan|2013|page=160}} [165] => [166] => Riedweg suggests that Pythagoras may have personally encouraged these legends,{{sfn|Riedweg|2005|page=1}} but Gregory states that there is no direct evidence of this.{{sfn|Gregory|2015|page=31}} Anti-Pythagorean legends were also circulated.{{sfn|Ferguson|2008|pages=60–61}} Diogenes Laërtes retells a story told by Hermippus of Samos, which states that Pythagoras had once gone into an underground room, telling everyone that he was descending to the underworld.{{sfn|Ferguson|2008|page=61}} He stayed in this room for months, while his mother secretly recorded everything that happened during his absence.{{sfn|Ferguson|2008|page=61}} After he returned from this room, Pythagoras recounted everything that had happened while he was gone,{{sfn|Ferguson|2008|page=61}} convincing everyone that he had really been in the underworld{{sfn|Ferguson|2008|page=61}} and leading them to trust him with their wives.{{sfn|Ferguson|2008|page=61}} [167] => [168] => == Attributed discoveries == [169] => === In mathematics === [170] => [[File:Pythagorean.svg|thumb|upright=1.1|alt=Diagram illustrating the Pythagorean theorem|'''[[Pythagorean theorem|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'').]] [171] => [172] => Although Pythagoras is most famous today for his alleged mathematical discoveries,{{sfn|Kahn|2001|pages=1–2}}{{sfn|Gregory|2015|pages=21–22}} classical historians dispute whether he himself ever actually made any significant contributions to the field.{{sfn|Burkert|1972|pages=428–433}}{{sfn|Kahn|2001|pages=2–3}} Many mathematical and scientific discoveries were attributed to Pythagoras, including [[Pythagorean theorem|his famous theorem]],Diogenes Laërtius, viii. 12; Plutarch, ''Non posse suav. vivi sec. Ep.'' p. 1094 as well as discoveries in the fields of [[Music of ancient Greece|music]],Porphyry, in ''Ptol. Harm.'' p. 213; Diogenes Laërtius, viii. 12. [[Greek astronomy|astronomy]],Diogenes Laërtius, viii. 14; Pliny, ''Hist. Nat.'' ii. 8. and [[Ancient Greek medicine|medicine]].Diogenes Laërtius, viii. 12, 14, 32. Since at least the first century BC, Pythagoras has commonly been given credit for discovering the Pythagorean theorem,{{sfn|Kahn|2001|pages=32–33}}{{sfn|Riedweg|2005|pages=26–27}} a theorem in geometry that states that "in a right-angled triangle the square of the hypotenuse is equal [to the sum of] the squares of the two other sides"{{sfn|Riedweg|2005|page=27}}—that is, a^2 + b^2 = c^2. According to a popular legend, after he discovered this theorem, Pythagoras sacrificed an ox, or possibly even a whole ''[[hecatomb]]'', to the gods.{{sfn|Riedweg|2005|page=27}}{{sfn|Burkert|1972|page=428}} Cicero rejected this story as spurious{{sfn|Riedweg|2005|page=27}} because of the much more widely held belief that Pythagoras forbade blood sacrifices.{{sfn|Riedweg|2005|page=27}} Porphyry attempted to explain the story by asserting that the ox was actually made of [[dough]].{{sfn|Riedweg|2005|page=27}} [173] => [174] => The Pythagorean theorem was known and used by the [[Babylonian mathematics|Babylonians]] and [[Indian mathematics|Indians]] centuries before Pythagoras,{{sfn|Burkert|1972|pages=429, 462}}{{sfn|Riedweg|2005|page=27}}{{sfn|Kahn|2001|page=32}}{{sfn|Ferguson|2008|pages=6–7}} but he may have been the first to introduce it to the Greeks.{{sfn|Burkert|1972|page=429}}{{sfn|Kahn|2001|page=32}} Some historians of mathematics have even suggested that he—or his students—may have constructed the first [[Mathematical proof|proof]].{{sfn|Kahn|2001|page=33}} Burkert rejects this suggestion as implausible,{{sfn|Burkert|1972|page=429}} noting that Pythagoras was never credited with having proved any theorem in antiquity.{{sfn|Burkert|1972|page=429}} Furthermore, the manner in which the Babylonians employed 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.{{efn|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).}} Pythagoras's biographers state that he also was the first to identify the [[Platonic solid|five regular solids]]{{sfn|Kahn|2001|pages=1–2}} and that he was the first to discover the [[Proportionality (mathematics)|Theory of Proportions]].{{sfn|Kahn|2001|pages=1–2}} [175] => [176] => === In music === [177] => [[File:Gaffurio Pythagoras.png|thumb|alt=Woodcut showing four scenes. In the upper right scene, blacksmiths are pounding with hammers. In the upper left scene, a man labelled "Pitagora" is shown playing different-sized bells and glasses with different amounts of liquid in them. Both the bells and glasses are labelled. In the bottom left scene, "Pitagora" is striking chords of different length laid out across a table, once again, all of which have numbers labels. In the bottom right scene, "Pitagora" and another man labeled "Phylolavs" are shown playing auloi.|Late medieval woodcut from [[Franchinus Gaffurius|Franchino Gafurio]]'s ''Theoria musice'' (1492), showing Pythagoras with bells and other instruments in Pythagorean tuning{{sfn|Gregory|2015|page=28}}]] [178] => {{See also|Pythagorean tuning |Pythagorean hammers}} [179] => According to legend, Pythagoras discovered that musical notes could be translated into mathematical equations when he passed blacksmiths at work one day and heard the sound of their [[Pythagorean hammers|hammers]] clanging against the anvils.{{sfn|Riedweg|2005|pages=27–28}}{{sfn|Gregory|2015|page=27}} Thinking that the sounds of the hammers were beautiful and harmonious, except for one,{{sfn|Riedweg|2005|page=28}} he rushed into the [[blacksmith]] shop and began testing the hammers.{{sfn|Riedweg|2005|page=28}} He then realized that the tune played when the hammer struck was directly proportional to the size of the hammer and therefore concluded that music was mathematical.{{sfn|Gregory|2015|page=27}}{{sfn|Riedweg|2005|page=28}} [180] => [181] => === In astronomy === [182] => In ancient times, Pythagoras and his contemporary [[Parmenides|Parmenides of Elea]] were both credited with having been the first to teach that the [[Spherical Earth|Earth was spherical]],{{sfn|Burkert|1972|page=306}} the first to divide the globe into [[Geographical zone|five climatic zones]],{{sfn|Burkert|1972|page=306}} and the first to identify the [[Phosphorus (morning star)|morning star]] and the [[Hesperus|evening star]] as the same celestial object (now known as [[Venus]]).{{sfn|Burkert|1972|pages=307–308}} Of the two philosophers, Parmenides has a much stronger claim to having been the first{{sfn|Burkert|1972|pages=306–308}} and the attribution of these discoveries to Pythagoras seems to have possibly originated from a [[pseudepigrapha]]l poem.{{sfn|Burkert|1972|pages=307–308}} [[Empedocles]], who lived in Magna Graecia shortly after Pythagoras and Parmenides, knew that the earth was spherical.{{sfn|Kahn|2001|page=53}} By the end of the fifth century BC, this fact was universally accepted among Greek intellectuals.{{sfn|Dicks|1970|page=68}} The identity of the morning star and evening star was known to the [[Babylonia]]ns over a thousand years earlier.{{sfn|Langdon|Fotheringham|1928}} [183] => [184] => == Later influence in antiquity == [185] => === On Greek philosophy === [186] => [[File:Timaeus trans calcidius med manuscript.jpg|thumb|upright=1.3|alt=Old manuscript with writing in a thin, fancy script with colored geometric diagrams illustrating the text.|Medieval manuscript of [[Calcidius]]'s Latin translation of Plato's ''[[Timaeus (dialogue)|Timaeus]]'', which is one of the Platonic dialogues with the most overt Pythagorean influences{{sfn|Kahn|2001|pages=55–62}}]] [187] => {{See also|Timaeus (dialogue)}} [188] => [189] => Sizeable Pythagorean communities existed in Magna Graecia, [[Phlius]], and [[Thebes, Greece|Thebes]] during the early fourth century BC.{{sfn|Kahn|2001|pages=48–49}} Around the same time, the Pythagorean philosopher [[Archytas]] was highly influential on the politics of the city of [[Taranto|Tarentum]] in Magna Graecia.{{sfn|Kahn|2001|page=39}} According to later tradition, Archytas was elected as ''[[strategos]]'' ("general") seven times, even though others were prohibited from serving more than a year.{{sfn|Kahn|2001|page=39}} Archytas was also a renowned mathematician and musician.{{sfn|Kahn|2001|pages=39–43}} He was a close friend of Plato{{sfn|Kahn|2001|pages=39–40}} and he is quoted in Plato's ''Republic''.{{sfn|Kahn|2001|pages=40, 44–45}}Plato, ''Republic'' VII, 530d Aristotle states that the philosophy of Plato was heavily dependent on the teachings of the Pythagoreans.Metaphysics, 1.6.1 (987a){{sfn|Kahn|2001|page=1}} Cicero repeats this statement, remarking that ''Platonem ferunt didicisse Pythagorea omnia'' ("They say Plato learned all things Pythagorean").Tusc. Disput. 1.17.39. According to [[Charles H. Kahn]], Plato's middle dialogues, including ''[[Meno]]'', ''[[Phaedo]]'', and ''[[Republic (Plato)|The Republic]]'', have a strong "Pythagorean coloring",{{sfn|Kahn|2001|page=55}} and his last few dialogues (particularly ''[[Philebus]]'' and ''[[Timaeus (dialogue)|Timaeus]]''){{sfn|Kahn|2001|pages=55–62}} are extremely Pythagorean in character.{{sfn|Kahn|2001|pages=55–62}} [190] => [191] => According to [[R. M. Hare]], Plato's ''[[Republic (Plato)|Republic]]'' may be partially based on the "tightly organised community of like-minded thinkers" established by Pythagoras at Croton.{{sfn|Hare|1999|pages=117–119}} Additionally, Plato may have borrowed from Pythagoras the idea that mathematics and abstract thought are a secure basis for philosophy, science, and morality.{{sfn|Hare|1999|pages=117–119}} Plato and Pythagoras shared a "mystical approach to the [[Soul (spirit)|soul]] and its place in the material world"{{sfn|Hare|1999|pages=117–119}} and both were probably influenced by [[Orphicism|Orphism]].{{sfn|Hare|1999|pages=117–119}} The historian of philosophy [[Frederick Copleston]] states that Plato probably borrowed [[Plato's tripartite theory of soul|his tripartite theory of the soul]] from the Pythagoreans.{{sfn|Copleston|2003|page=37}} [[Bertrand Russell]], in his ''[[A History of Western Philosophy]]'', contends that the influence of Pythagoras on Plato and others was so great that he should be considered the most influential philosopher of all time.{{sfn|Russell|2008|pages=33–37}} He concludes that "I do not know of any other man who has been as influential as he was in the school of thought."{{sfn|Russell|2008|page=37}} [192] => [193] => A revival of Pythagorean teachings occurred in the first century BC{{sfn|Riedweg|2005|pages=123–124}} when [[Middle Platonism|Middle Platonist]] philosophers such as [[Eudorus of Alexandria|Eudorus]] and [[Philo of Alexandria]] hailed the rise of a "new" Pythagoreanism in [[Alexandria]].{{sfn|Riedweg|2005|page=124}} At around the same time, [[Neopythagoreanism]] became prominent.{{sfn|Riedweg|2005|pages=125–126}} The first-century AD philosopher [[Apollonius of Tyana]] sought to emulate Pythagoras and live by Pythagorean teachings.{{sfn|Riedweg|2005|page=125}} The later first-century Neopythagorean philosopher [[Moderatus of Gades]] expanded on Pythagorean number philosophy{{sfn|Riedweg|2005|page=125}} and probably understood the soul as a "kind of mathematical harmony".{{sfn|Riedweg|2005|page=125}} The Neopythagorean mathematician and musicologist [[Nicomachus]] likewise expanded on Pythagorean numerology and music theory.{{sfn|Riedweg|2005|pages=125–126}} [[Numenius of Apamea]] interpreted Plato's teachings in light of Pythagorean doctrines.{{sfn|Riedweg|2005|pages=126–127}} [194] => [195] => === On art and architecture === [196] => [[File:Giovanni Paolo Panini - Interior of the Pantheon, Rome - Google Art Project.jpg|thumb|alt=Painting showing a massive room with a high, domed ceiling. A hole is open at the top of the dome. Columns and statues line the walls.|[[Hadrian]]'s [[Pantheon, Rome|Pantheon]] in [[Rome]], depicted in this eighteenth-century painting by [[Giovanni Paolo Panini]], was built according to Pythagorean teachings.{{sfn|Joost-Gaugier|2006|pages=166–181}}]] [197] => [198] => Greek sculpture sought to represent the permanent reality behind superficial appearances.{{sfn|Homann-Wedeking|1968|page=63}} Early [[Archaic Greece|Archaic]] sculpture represents life in simple forms, and may have been influenced by the earliest Greek natural philosophies.{{efn|"For [[Thales]], the origin was water, and for [[Anaximander]] the infinite ([[apeiron (cosmology)|apeiron]]), which must be considered a material form"{{sfn|Homann-Wedeking|1968|page=63}}}} The Greeks generally believed that nature expressed itself in ideal forms and was represented by a type ({{lang|grc|εἶδος}}), which was mathematically calculated.{{sfn|Homann-Wedeking|1968|page=62}}{{sfn|Carpenter|1921|pages=107, 122, 128}} When dimensions changed, architects sought to relay permanence through mathematics.{{sfn|Homann-Wedeking|1968|pages=62–63}}{{sfn|Bowra|1994|page=166}} [[Maurice Bowra]] believes that these ideas influenced the theory of Pythagoras and his students, who believed that "all things are numbers".{{sfn|Bowra|1994|page=166}} [199] => [200] => During the sixth century BC, the number philosophy of the Pythagoreans triggered a revolution in Greek sculpture.{{sfn|Homann-Wedeking|1968|pages=62–65}} Greek sculptors and architects attempted to find the mathematical relation ([[Canon of proportions|canon]]) behind aesthetic perfection.{{sfn|Carpenter|1921|pages=107, 122, 128}} Possibly drawing on the ideas of Pythagoras,{{sfn|Carpenter|1921|pages=107, 122, 128}} the sculptor [[Polykleitos]] wrote in [[Polykleitos#Canon of Polykleitos|his ''Canon'']] that beauty consists in the proportion, not of the elements (materials), but of the interrelation of parts with one another and with the whole.{{sfn|Carpenter|1921|pages=107, 122, 128}}{{efn|"Each part (finger, palm, arm, etc) transmitted its individual existence to the next, and then to the whole": ''Canon of Polykleitos'', also [[Plotinus]], ''Ennead'' I.vi.i: Nigel Spivey, pp. 290–294.}} In the Greek architectural orders, every element was calculated and constructed by mathematical relations. [[Rhys Carpenter]] states that the ratio 2:1 was "the generative ratio of the [[Doric order]], and in Hellenistic times an ordinary Doric colonnade, beats out a rhythm of notes."{{sfn|Carpenter|1921|pages=107, 122, 128}} [201] => [202] => The oldest known building designed according to Pythagorean teachings is the [[Porta Maggiore Basilica]],{{sfn|Joost-Gaugier|2006|page=154}} a subterranean basilica which was built during the reign of the Roman emperor [[Nero]] as a secret place of worship for Pythagoreans.{{sfn|Joost-Gaugier|2006|pages=154–156}} The basilica was built underground because of the Pythagorean emphasis on secrecy{{sfn|Joost-Gaugier|2006|pages=157–158}} and also because of the legend that Pythagoras had sequestered himself in a cave on Samos.{{sfn|Joost-Gaugier|2006|page=158}} The basilica's apse is in the east and its atrium in the west out of respect for the rising sun.{{sfn|Joost-Gaugier|2006|pages=158–159}} It has a narrow entrance leading to a small pool where the initiates could purify themselves.{{sfn|Joost-Gaugier|2006|page=159}} The building is also designed according to Pythagorean numerology,{{sfn|Joost-Gaugier|2006|pages=159–161}} with each table in the sanctuary providing seats for seven people.{{sfn|Joost-Gaugier|2006|page=161}} Three aisles lead to a single altar, symbolizing the three parts of the soul approaching the unity of Apollo.{{sfn|Joost-Gaugier|2006|page=161}} The apse depicts a scene of the poet [[Sappho]] leaping off the [[Lefkada|Leucadian cliffs]], clutching her lyre to her breast, while Apollo stands beneath her, extending his right hand in a gesture of protection,{{sfn|Joost-Gaugier|2006|page=162}} symbolizing Pythagorean teachings about the immortality of the soul.{{sfn|Joost-Gaugier|2006|page=162}} The interior of the sanctuary is almost entirely white because the color white was regarded by Pythagoreans as sacred.{{sfn|Joost-Gaugier|2006|pages=162–164}} [203] => [204] => The emperor [[Hadrian]]'s [[Pantheon, Rome|Pantheon]] in [[Rome]] was also built based on Pythagorean numerology.{{sfn|Joost-Gaugier|2006|pages=166–181}} The temple's circular plan, central axis, hemispherical [[dome]], and alignment with the four cardinal directions symbolize Pythagorean views on the order of the universe.{{sfn|Joost-Gaugier|2006|pages=167–168}} The single [[Oculus (architecture)|oculus]] at the top of the dome symbolizes the monad and the sun-god Apollo.{{sfn|Joost-Gaugier|2006|page=168}} The twenty-eight ribs extending from the oculus symbolize the moon, because twenty-eight was the same number of months on the Pythagorean lunar calendar.{{sfn|Joost-Gaugier|2006|pages=169–170}} The five coffered rings beneath the ribs represent the marriage of the sun and moon.{{sfn|Joost-Gaugier|2006|pages=170–172}} [205] => [206] => === In early Christianity === [207] => Many early Christians had a deep respect for Pythagoras.{{sfn|Joost-Gaugier|2006|pages=57–65}} [[Eusebius]] ({{circa}} 260 – {{circa}} 340 AD), bishop of [[Caesarea Maritima|Caesarea]], praises Pythagoras in his ''Against Hierokles'' for his rule of silence, his frugality, his "extraordinary" morality, and his wise teachings.{{sfn|Joost-Gaugier|2006|page=57}} In another work, Eusebius compares Pythagoras to [[Moses]].{{sfn|Joost-Gaugier|2006|page=57}} In one of his letters, the [[Church Fathers|Church Father]] [[Jerome]] ({{circa}} 347 – 420 AD) praises Pythagoras for his wisdom{{sfn|Joost-Gaugier|2006|page=57}} and, in another letter, he credits Pythagoras for his belief in the immortality of the soul, which he suggests Christians inherited from him.{{sfn|Joost-Gaugier|2006|pages=57–58}} [[Augustine of Hippo]] (354 – 430 AD) rejected Pythagoras's teaching of metempsychosis without explicitly naming him, but otherwise expressed admiration for him.{{sfn|Joost-Gaugier|2006|pages=58–59}} In ''[[On the Trinity]]'', Augustine lauds the fact that Pythagoras was humble enough to call himself a ''philosophos'' or "lover of wisdom" rather than a "sage".{{sfn|Joost-Gaugier|2006|page=59}} In another passage, Augustine defends Pythagoras's reputation, arguing that Pythagoras certainly never taught the doctrine of metempsychosis.{{sfn|Joost-Gaugier|2006|page=59}} [208] => [209] => == Influence after antiquity == [210] => === In the Middle Ages === [211] => [[File:Pythagore-chartres.jpg|thumb|right|alt=Medieval carving of a man with long hair and a long beard hunched over a musical instrument he is working on|Pythagoras appears in a relief sculpture on one of the [[archivolts]] over the right door of the west portal at [[Chartres Cathedral]].{{sfn|Celenza|2010|page=798}}]] [212] => [213] => During the [[Middle Ages]], Pythagoras was revered as the founder of mathematics and music, two of the [[Seven Liberal Arts]].{{sfn|Celenza|2010|page=798}} He appears in numerous medieval depictions, in illuminated manuscripts and in the relief sculptures on the portal of the [[Cathedral of Chartres]].{{sfn|Celenza|2010|page=798}} The ''Timaeus'' was the only dialogue of Plato to survive in Latin translation in western Europe,{{sfn|Celenza|2010|page=798}} which led [[William of Conches]] (c. 1080–1160) to declare that Plato was Pythagorean.{{sfn|Celenza|2010|page=798}} A large-scale translation movement emerged during the Abbasid Caliphate, translating many Greek texts into Arabic. Works ascribed to Pythagoras included the "Golden Verses" and snippets of his scientific and mathematical theories.{{Cite book |last=Lindberg |first=David C. |title=Science in the Middle Ages |publisher=University of Chicago Press |date=March 15, 1978 |isbn=978-0226482330}} By translating and disseminating Pythagorean texts, Islamic scholars ensured their survival and wider accessibility. This preserved knowledge that might have otherwise been lost through the decline of the Roman Empire and the neglect of classical learning in Europe.{{Cite book |last=Lindberg |first=David C. |title=The Cambridge History of Science: Volume 2, Medieval Science |publisher=Cambridge University Press |date=October 7, 2013 |isbn=978-0521594486}} In the 1430s, the Camaldolese friar [[Ambrose Traversari]] translated Diogenes Laërtius's ''Lives and Opinions of Eminent Philosophers'' from Greek into Latin{{sfn|Celenza|2010|page=798}} and, in the 1460s, the philosopher [[Marsilio Ficino]] translated Porphyry and Iamblichus's ''Lives of Pythagoras'' into Latin as well,{{sfn|Celenza|2010|page=798}} thereby allowing them to be read and studied by western scholars.{{sfn|Celenza|2010|page=798}} In 1494, the Greek Neopythagorean scholar [[Constantine Lascaris]] published ''[[The Golden Verses of Pythagoras]]'', translated into Latin, with a printed edition of his ''Grammatica'',{{sfn|Russo|2004|pages=5–87, especially 51–53}} thereby bringing them to a widespread audience.{{sfn|Russo|2004|pages=5–87, especially 51–53}} In 1499, he published the first Renaissance biography of Pythagoras in his work ''Vitae illustrium philosophorum siculorum et calabrorum'', issued in [[Messina]].{{sfn|Russo|2004|pages=5–87, especially 51–53}} [214] => [215] => === On modern science === [216] => In his preface to his book ''[[On the Revolution of the Heavenly Spheres]]'' (1543), [[Nicolaus Copernicus]] cites various Pythagoreans as the most important influences on the development of his [[Heliocentrism|heliocentric model]] of the universe,{{sfn|Celenza|2010|page=798}}{{sfn|Kahn|2001|page=160}} deliberately omitting mention of [[Aristarchus of Samos]], a non-Pythagorean astronomer who had developed a fully heliocentric model in the fourth century BC, in effort to portray his model as fundamentally Pythagorean.{{sfn|Kahn|2001|page=160}} [[Johannes Kepler]] considered himself to be a Pythagorean.{{sfn|Celenza|2010|page=798}}{{sfn|Kahn|2001|pages=161–171}}{{sfn|Ferguson|2008|page=265}} He believed in the Pythagorean doctrine of ''musica universalis''{{sfn|Ferguson|2008|pages=264–274}} and it was his search for the mathematical equations behind this doctrine that led to his discovery of the [[Kepler's laws of planetary motion|laws of planetary motion]].{{sfn|Ferguson|2008|pages=264–274}} Kepler titled his book on the subject ''[[Harmonices Mundi]]'' (''Harmonics of the World''), after the Pythagorean teaching that had inspired him.{{sfn|Celenza|2010|page=798}}{{sfn|Kahn|2001|page=162}} Near the conclusion of the book, Kepler describes himself falling asleep to the sound of the heavenly music, "warmed by having drunk a generous draught{{nbsp}}... from the cup of Pythagoras."{{sfn|Ferguson|2008|page=274}} He also called Pythagoras the "grandfather" of all Copernicans.Jamie James, The Music of the Spheres: Music, Science, and the Natural Order of the Universe, p 142. [217] => [218] => [[Isaac Newton]] firmly believed in the Pythagorean teaching of the mathematical harmony and order of the universe.{{sfn|Ferguson|2008|page=279}} Though Newton was notorious for rarely giving others credit for their discoveries,{{sfn|Ferguson|2008|pages=279–280}} he attributed the discovery of the [[Newton's law of universal gravitation|Law of Universal Gravitation]] to Pythagoras.{{sfn|Ferguson|2008|pages=279–280}} [[Albert Einstein]] believed that a scientist may also be "a Platonist or a Pythagorean insofar as he considers the viewpoint of logical simplicity as an indispensable and effective tool of his research."{{sfn|Kahn|2001|page=172}} The English philosopher [[Alfred North Whitehead]] argued that "In a sense, Plato and Pythagoras stand nearer to modern physical science than does Aristotle. The two former were mathematicians, whereas Aristotle was the son of a doctor".{{sfn|Whitehead|1953|pages=36–37}} By this measure, Whitehead declared that Einstein and other modern scientists like him are "following the pure Pythagorean tradition."{{sfn|Kahn|2001|page=172}}{{sfn|Whitehead|1953|page=36}} [219] => [220] => === On vegetarianism === [221] => [[File:Pythagoras advocating vegetarianism (1618-20); Peter Paul Rubens.jpg|thumb|upright=1.4|alt=Painting showing Pythagoras on the far left quizzically stroking his beard as he gazes upon a massive pile of fruits and vegetables. Two followers stand behind him, fully clothed. A man with a greying beard sits at the base of a tree gesturing to the pile of produce. Next to him, a fleshy, nude woman with blonde hair plucks fruits from it. Slightly behind her, two other women, one partially clothed and the other nude but obscured by the tree branch, are also plucking fruits. At the far right end of the painting, two nude, faun-like men with beards and pointed ears hurl more fruits upon the pile.|''Pythagoras Advocating Vegetarianism'' (1618–1630) by [[Peter Paul Rubens]] was inspired by Pythagoras's speech in Ovid's ''Metamorphoses''.{{sfn|Borlik|2011|page=192}} The painting portrays the Pythagoreans with corpulent bodies, indicating a belief that vegetarianism was healthful and nutritious.{{sfn|Borlik|2011|page=192}}]] [222] => [223] => A fictionalized portrayal of Pythagoras appears in Book XV of [[Ovid]]'s ''[[Metamorphoses]]'',{{sfn|Borlik|2011|page=189}} in which he delivers a speech imploring his followers to adhere to a strictly vegetarian diet.{{sfn|Borlik|2011|pages=189–190}} It was through [[Arthur Golding]]'s 1567 English translation of Ovid's ''Metamorphoses'' that Pythagoras was best known to English-speakers throughout the early modern period.{{sfn|Borlik|2011|pages=189–190}} [[John Donne]]'s ''Progress of the Soul'' discusses the implications of the doctrines expounded in the speech,{{sfn|Borlik|2011|page=190}} and [[Michel de Montaigne]] quoted the speech no less than three times in his treatise "Of Cruelty" to voice his moral objections against the mistreatment of animals.{{sfn|Borlik|2011|page=190}} [[William Shakespeare]] references the speech in his play ''[[The Merchant of Venice]]''.{{sfn|Ferguson|2008|page=282}} [[John Dryden]] included a translation of the scene with Pythagoras in his 1700 work ''[[Fables, Ancient and Modern]]'',{{sfn|Borlik|2011|page=190}} and [[John Gay]]'s 1726 fable "Pythagoras and the Countryman" reiterates its major themes, linking carnivorism with tyranny.{{sfn|Borlik|2011|page=190}} [[Lord Chesterfield]] records that his conversion to vegetarianism had been motivated by reading Pythagoras's speech in Ovid's ''Metamorphoses''.{{sfn|Borlik|2011|page=190}} Until the word ''vegetarianism'' was coined in the 1840s, vegetarians were referred to in English as "Pythagoreans".{{sfn|Borlik|2011|page=190}} [[Percy Bysshe Shelley]] wrote an ode entitled "To the Pythagorean Diet",{{sfn|Ferguson|2008|page=294}} and [[Leo Tolstoy]] adopted the Pythagorean diet himself.{{sfn|Ferguson|2008|page=294}} [224] => [225] => === On Western esotericism === [226] => Early modern [[Western esotericism|European esotericism]] drew heavily on the teachings of Pythagoras.{{sfn|Celenza|2010|page=798}} The German [[Renaissance humanism|humanist]] scholar [[Johannes Reuchlin]] (1455–1522) synthesized Pythagoreanism with [[Christian theology]] and Jewish [[Kabbalah]],{{sfn|Riedweg|2005|pages=127–128}} arguing that Kabbalah and Pythagoreanism were both inspired by [[Moses|Mosaic]] tradition{{sfn|Riedweg|2005|page=128}} and that Pythagoras was therefore a kabbalist.{{sfn|Riedweg|2005|page=128}} In his dialogue ''De verbo mirifico'' (1494), Reuchlin compared the Pythagorean tetractys to the [[Ineffability|ineffable]] divine name [[Tetragrammaton|YHWH]],{{sfn|Riedweg|2005|pages=127–128}} ascribing each of the four letters of the tetragrammaton a symbolic meaning according to Pythagorean mystical teachings.{{sfn|Riedweg|2005|page=128}} [227] => [228] => [[Heinrich Cornelius Agrippa]]'s popular and influential three-volume treatise ''[[Three Books of Occult Philosophy|De Occulta Philosophia]]'' cites Pythagoras as a "religious magi"{{sfn|French|2002|page=30}} and advances the idea that Pythagoras's mystical numerology operates on a [[Allegory in Renaissance literature#Three world theory|supercelestial]] level,{{sfn|French|2002|page=30}} a religious term used to describe a high heavenly realm used during his time. The [[Freemasonry|freemasons]] deliberately modeled their society on the community founded by Pythagoras at Croton.{{sfn|Riedweg|2005|page=133}} [[Rosicrucianism]] used Pythagorean symbolism,{{sfn|Celenza|2010|page=798}} as did [[Robert Fludd]] (1574–1637),{{sfn|Celenza|2010|page=798}} who believed his own musical writings to have been inspired by Pythagoras.{{sfn|Celenza|2010|page=798}} [[John Dee]] was heavily influenced by Pythagorean ideology,{{sfn|Sherman|1995|page=15}}{{sfn|French|2002|page=30}} particularly the teaching that all things are made of numbers.{{sfn|Sherman|1995|page=15}}{{sfn|French|2002|page=30}} [[Adam Weishaupt]], the founder of the [[Illuminati]], was a strong admirer of Pythagoras{{sfn|Ferguson|2008|pages=284–288}} and, in his book ''Pythagoras'' (1787), he advocated that society should be reformed to be more like Pythagoras's commune at Croton.{{sfn|Ferguson|2008|pages=287–288}} [[Wolfgang Amadeus Mozart]] incorporated Masonic and Pythagorean symbolism into his opera ''[[The Magic Flute]]''.{{sfn|Ferguson|2008|pages=286–287}} [[Sylvain Maréchal]], in his six-volume 1799 biography ''The Voyages of Pythagoras'', declared that all revolutionaries in all time periods are the "heirs of Pythagoras".{{sfn|Ferguson|2008|page=288}} [229] => [230] => === On literature === [231] => [[File:Paradiso Canto 31.jpg|thumb|upright=1.2|right|alt=Engraving showing two small silhouettes standing before a tunnel of innumerably manifold circling angels leading to a bright, beautiful light at the end.|[[Dante Alighieri]]'s description of Heaven in his ''[[Paradiso (Dante)|Paradiso]]'' incorporates Pythagorean numerology.{{sfn|Haag|2013|page=89}}]] [232] => [233] => [[Dante Alighieri]] was fascinated by Pythagorean numerology{{sfn|Haag|2013|page=89}} and based his descriptions of [[Hell]], [[Purgatory]], and [[Heaven]] on Pythagorean numbers.{{sfn|Haag|2013|page=89}} Dante wrote that Pythagoras saw Unity as Good and Plurality as Evil{{sfn|Haag|2013|page=90}} and, in ''[[Paradiso (Dante)|Paradiso]]'' XV, 56–57, he declares: "five and six, if understood, ray forth from unity".{{sfn|Haag|2013|pages=90–91}} The number eleven and its multiples are found throughout the ''[[Divine Comedy]]'', each book of which has thirty-three [[canto]]s, except for the ''[[Inferno (Dante)|Inferno]]'', which has thirty-four, the first of which serves as a general introduction.{{sfn|Haag|2013|page=91}} Dante describes the ninth and tenth bolgias in the Eighth Circle of Hell as being twenty-two miles and eleven miles respectively,{{sfn|Haag|2013|page=91}} which correspond to the fraction {{sfrac|22|7}}, which was the Pythagorean approximation of [[pi]].{{sfn|Haag|2013|page=91}} [234] => [235] => The [[Transcendentalism|Transcendentalists]] read the ancient ''Lives of Pythagoras'' as guides on how to live a model life.{{sfn|Bregman|2002|page=186}} [[Henry David Thoreau]] was impacted by [[Thomas Taylor (neoplatonist)|Thomas Taylor]]'s translations of Iamblichus's ''Life of Pythagoras'' and [[Stobaeus]]'s ''Pythagoric Sayings''{{sfn|Bregman|2002|page=186}} and his views on nature may have been influenced by the Pythagorean idea of images corresponding to archetypes.{{sfn|Bregman|2002|page=186}} The Pythagorean teaching of ''musica universalis'' is a recurring theme throughout Thoreau's ''[[magnum opus]]'', ''[[Walden]]''.{{sfn|Bregman|2002|page=186}} [236] => [237] => == See also == [238] => * [[List of things named after Pythagoras]] [239] => * [[Ex pede Herculem|''Ex pede Herculem'']], "from his foot, [we can measure] Hercules" – a maxim based on the apocryphal story that Pythagoras estimated Hercules's stature based on the length of a racecourse at Pisae [240] => * [[Pythagorean cup]] – a prank cup with a hidden siphon built in, attributed to Pythagoras [241] => * [[Pythagorean means]] – the arithmetic mean, the geometric mean, and the harmonic mean, claimed to have been studied by Pythagoras [242] => [243] => == Notes == [244] => {{notelist}} [245] => [246] => == Citations == [247] => {{Reflist|20em}} [248] => [249] => == References == [250] => === Classical sources === [251] => {{refbegin}} [252] => *[[Porphyry (philosopher)|Porphyry]], ''Vita Pythagorae'' (''Life of Pythagoras''), {{Circa|270 AD}} — [http://www.ccel.org/ccel/pearse/morefathers/files/porphyry_life_of_pythagoras_02_text.htm ''Porphyry, Life of Pythagoras''], translated by [[Kenneth Sylvan Guthrie]] (1920) [253] => *[[Iamblichus (philosopher)|Iamblichus]], ''De Vita Pythagorica'' (''On the Pythagorean Life''), {{Circa|300 AD}} — [http://www.completepythagoras.net/mainframeset.html ''Iamblichus, Life of Pythagoras''], translated by [[Kenneth Sylvan Guthrie]] (1920) [254] => *[[Apuleius]], following [[Aristoxenus]], writes about Pythagoras in ''Apologia'', {{Circa|150 AD}}, including a story of his being taught by [[Zoroaster]]—a story also found in [[Clement of Alexandria]]. {{harv|Vasunia|2007|p=246}} [255] => *[[Hierocles of Alexandria]], ''Golden Verses of Pythagoras'', {{Circa|430 AD}} [256] => {{refend}} [257] => [258] => === Modern secondary sources === [259] => {{refbegin|30em}} [260] => * {{cite book |last=Acevedo |first=Juan |url=https://books.google.com/books?id=cavvDwAAQBAJ |title=Alphanumeric Cosmology From Greek into Arabic |date=2020 |publisher=[[Mohr Siebeck]] |isbn=978-3-16-159245-4 |language=en}} [261] => * {{citation |last1=Asheri |first1=David |author1-link=David Asheri|last2=Lloyd |first2=Alan |last3=Corcella |first3=Aldo |year=2007 |title=A Commentary on Herodotus, Books 1–4 |publisher=Oxford University Press |isbn=978-0-19-814956-9}} [262] => * {{cite book |last=Barry |first=Kieren |url=https://books.google.com/books?id=w70rWN_-ANAC |title=The Greek Qabalah: Alphabetic Mysticism and Numerology in the Ancient World |date=1999 |publisher=[[Samuel Weiser, Inc.]] |isbn=978-1-57863-110-0 |language=en}} [263] => * {{citation|last=Bowra|first=C. 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K.|date=1979|title=A History of Greek Philosophy: Earlier Presocratics and the Pythagoreans|location=Cambridge, England|publisher=Cambridge University Press|isbn=978-0-521-29420-1}} [310] => * {{citation|last=Hermann|first=Arnold|date=2005|title=To Think Like God: Pythagoras and Parmenides—the Origins of Philosophy|location=Las Vegas, Nevada|publisher=Parmenides Publishing|isbn=978-1-930972-00-1|url-access=registration|url=https://archive.org/details/tothinklikegodpy0000herm}} [311] => * {{citation|last=Horky|first=Philip Sydney|date=2013|title=Plato and Pythagoreanism|url=https://books.google.com/books?id=EXkRDAAAQBAJ&q=Pythagoreanism|location=Oxford, England|publisher=Oxford University Press|isbn=978-0-19-989822-0}} [312] => * {{citation|last=Kingsley|first=Peter|author-link=Peter Kingsley|date=1995|title=Ancient Philosophy, Mystery, and Magic: Empedocles and the Pythagorean Tradition|location=Oxford, England|publisher=Oxford University Press}} [313] => * {{citation|last=O'Meara|first=Dominic J.|date=1989|title=Pythagoras Revived|location=Oxford, England|publisher=Oxford University Press|isbn=978-0-19-823913-0}} [314] => * {{citation|last=Schofield|first=Malcolm|date=2013|title=Aristotle, Plato and Pythagoreanism in the First Century BC: New Directions for Philosophy|url=https://books.google.com/books?id=VZR7AAAAQBAJ&q=Pythagoreanism|location=Cambridge, England|publisher=Cambridge University Press|isbn=978-1-107-02011-5}} [315] => [316] => == External links == [317] => {{sister project links}} [318] => {{EB1911 poster|Pythagoras}} [319] => *{{In Our Time|Pythagoras|b00p693b|Pythagoras}} [320] => *{{cite SEP |url-id=pythagoras |title=Pythagoras |last=Huffman |first=Carl|mode=cs2}} [321] => *[http://www-history.mcs.st-andrews.ac.uk/Mathematicians/Pythagoras.html "Pythagoras of Samos"], [[The MacTutor History of Mathematics archive]], School of Mathematics and Statistics, [[University of St Andrews]], Scotland [322] => *[http://history.hanover.edu/texts/presoc/pythagor.html "Pythagoras and the Pythagoreans, Fragments and Commentary"], [[Arthur Fairbanks]] Hanover Historical Texts Project, [[Hanover College]] Department of History [323] => *[http://www.math.tamu.edu/~don.allen/history/pythag/pythag.html "Pythagoras and the Pythagoreans"], Department of Mathematics, [[Texas A&M University]] [324] => * [http://www.newadvent.org/cathen/12587b.htm "Pythagoras and Pythagoreanism"], ''[[The Catholic Encyclopedia]]'' [325] => * {{Internet Archive author |sname=Pythagoras}} [326] => * {{Librivox author |id=9944}} [327] => [328] => {{Ancient Greek mathematics}} [329] => {{Greek schools of philosophy}} [330] => {{Ancient Greece topics}} [331] => {{Portal bar|Mathematics|Music|Astronomy|Stars|Society|Philosophy}} [332] => {{Authority control}} [333] => [334] => [[Category:Pythagoras| ]] [335] => [[Category:570s BC births]] [336] => [[Category:490s BC deaths]] [337] => [[Category:6th-century BC mathematicians]] [338] => [[Category:6th-century BC Greek philosophers]] [339] => [[Category:5th-century BC mathematicians]] [340] => [[Category:5th-century BC Greek philosophers]] [341] => [[Category:Ancient Greek geometers]] [342] => [[Category:Ancient Greek metaphysicians]] [343] => [[Category:Ancient Greek music theorists]] [344] => [[Category:Ancient Greek philosophers of mind]] [345] => [[Category:Ancient Greek political refugees]] [346] => [[Category:Ancient Greek shamans]] [347] => [[Category:Ancient occultists]] [348] => [[Category:Ancient Samians]] [349] => [[Category:Esotericists]] [350] => [[Category:Founders of religions]] [351] => [[Category:Numerologists]] [352] => [[Category:Pythagoreans of Magna Graecia]] [353] => [[Category:6th-century BC religious leaders]] [354] => [[Category:5th-century BC religious leaders]] [] => )
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Pythagoras

Pythagoras was an ancient Greek philosopher and mathematician, known for his contributions to the fields of mathematics, music, and philosophy. He established a religious and philosophical school in Croton, Italy, which became known as the Pythagorean school.

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He established a religious and philosophical school in Croton, Italy, which became known as the Pythagorean school. Although little is known about his life, his teachings and discoveries have had a lasting impact on the development of Western mathematics and philosophy. Pythagoras is best known for the Pythagorean theorem, which states that the square of the hypotenuse of a right-angled triangle is equal to the sum of the squares of the other two sides. This theorem revolutionized the understanding of geometry and had wide-ranging applications in various fields, from architecture to engineering. In addition to his mathematical achievements, Pythagoras also made important contributions to the study of music. He believed in the harmonious nature of the universe and saw music as a means to connect with the divine. Pythagoras is credited with discovering the mathematical relationship between musical intervals, which laid the foundation for the development of musical theory. Pythagoras' philosophical beliefs centered around the concept of the soul and its immortality. He believed in reincarnation and that the pursuit of knowledge and wisdom was essential for the soul's liberation from the cycle of rebirth. Pythagorean philosophy emphasized moral purity, vegetarianism, and the importance of cultivating a balanced and harmonious life. Although Pythagoras himself did not leave behind any written works, his teachings and ideas were preserved and expanded upon by his followers. Many of his ideas were later integrated into the philosophical systems of Plato and Aristotle. Despite the challenges in piecing together an accurate biography of Pythagoras, his contributions to mathematics, music, and philosophy continue to be recognized and appreciated to this day.

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