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f printable version in other projects wikimedia commons wikisource wikidata item appearance move to sidebar hide from wikipedia the free encyclopedia 5th century bc pythagorean philosopher this article is about the greek philosopher for the mythological characters called hippasus see hippasus mythology hippasus engraving by girolamo olgiati 1580 hippasus of metapontum ˈ h ɪ p ə s ə s ancient greek ἵππασος ὁ μεταποντῖνος híppasos c 530 c 450 bc 1 was a greek philosopher and early follower of pythagoras 2 3 little is known about his life or his beliefs but he is sometimes credited with the discovery of the existence of irrational numbers the discovery of irrational numbers is said to have been shocking to the pythagoreans and hippasus is supposed to have drowned at sea apparently as a punishment from the gods for divulging this and crediting it to himself instead of pythagoras which was the norm in pythagorean society the few ancient sources who describe this story however either do not mention hippasus by name e g pappus 4 or alternatively tell that hippasus drowned because he revealed how to construct a dodecahedron inside a sphere 5 the discovery of irrationality is not specifically ascribed to hippasus by any ancient writer life edit not much is known about the life of hippasus he may have lived in the late 5th century bc about a century after the time of pythagoras metapontum in magna graecia is usually referred to as his birthplace 6 7 8 9 10 although according to iamblichus 3rd century ad some claim metapontum to be his birthplace while others the nearby city of croton 11 hippasus is recorded under the city of sybaris in iamblichus s list of each city s pythagoreans 12 he also states that hippasus was the founder of a sect of the pythagoreans called the mathematici μαθηματικοί in opposition to the acusmatici ἀκουσματικοί 13 but elsewhere he makes him the founder of the acusmatici in opposition to the mathematici 14 iamblichus says about the death of hippasus it is related to hippasus that he was a pythagorean and that owing to his being the first to publish and describe the sphere from the twelve pentagons he perished at sea for his impiety but he received credit for the discovery though really it all belonged to him for in this way they refer to pythagoras and they do not call him by his name 15 according to iamblichus s the life of pythagoras 16 there were also two forms of philosophy for the two genera of those that pursued it the acusmatici and the mathematici the latter are acknowledged to be pythagoreans by the rest but the mathematici do not admit that the acusmatici derived their instructions from pythagoras but from hippasus the philosophy of the acusmatici consisted in auditions unaccompanied with demonstrations and a reasoning process because it merely ordered a thing to be done in a certain way and that they should endeavor to preserve such other things as were said by him as divine dogmas memory was the most valued faculty all these auditions were of three kinds some signifying what a thing is others what it especially is others what ought or ought not to be done p 61 doctrines edit aristotle speaks of hippasus as holding the element of fire to be the cause of all things 17 and sextus empiricus contrasts him with the pythagoreans in this respect that he believed the arche to be material whereas they thought it was incorporeal namely number 18 diogenes laërtius tells us that hippasus believed that there is a definite time which the changes in the universe take to complete and that the universe is limited and ever in motion 7 according to one statement hippasus left no writings 7 according to another he was the author of the mystic discourse written to bring pythagoras into disrepute 19 a scholium on plato s phaedo notes him as an early experimenter in music theory claiming that he made use of bronze disks to discover the fundamental musical ratios 4 3 3 2 and 2 1 20 irrational numbers edit hippasus is sometimes credited with the discovery of the existence of irrational numbers following which he was drowned at sea pythagoreans preached that all numbers could be expressed as the ratio of integers and the discovery of irrational numbers is said to have shocked them however the evidence linking the discovery to hippasus is unclear pappus 4th century ad merely says that the knowledge of irrational numbers originated in the pythagorean school and that the member who first divulged the secret perished by drowning 21 iamblichus 3rd century ad gives a series of inconsistent reports in one story he explains how a pythagorean was merely expelled for divulging the nature of the irrational but he then cites the legend of the pythagorean who drowned at sea for making known the construction of the regular dodecahedron in the sphere 22 in another account he tells how it was hippasus who drowned at sea for betraying the construction of the dodecahedron and taking credit for this construction himself 23 but in another story this same punishment is meted out to the pythagorean who divulged knowledge of the irrational 24 iamblichus clearly states that the drowning at sea was a punishment from the gods for impious behaviour 22 these stories are usually taken together to ascribe the discovery of irrationals to hippasus but whether he did or not is uncertain 25 in principle the stories can be combined since it is possible to discover irrational numbers when constructing dodecahedra irrationality by infinite reciprocal subtraction can be easily seen in the golden ratio of the regular pentagon 26 some scholars in the early 20th century credited hippasus with the discovery of the irrationality of 2 displaystyle sqrt 2 the square root of 2 plato in his theaetetus 27 describes how theodorus of cyrene c 400 bc proved the irrationality of 3 displaystyle sqrt 3 5 displaystyle sqrt 5 etc up to 17 displaystyle sqrt 17 which implies that an earlier mathematician had already proved the irrationality of 2 displaystyle sqrt 2 28 aristotle referred to the method for a proof of the irrationality of 2 displaystyle sqrt 2 29 and a full proof along these same lines is set out in the proposition interpolated at the end of euclid s book x 30 which suggests that the proof was certainly ancient 31 the method is a proof by contradiction or reductio ad absurdum which shows that if the diagonal of a square is assumed to be commensurable with the side then the same number must be both odd and even 31 in the hands of modern writers this combination of vague ancient reports and modern guesswork has sometimes evolved into a more emphatic and colourful tale some writers have hippasus making his discovery while on board a ship as a result of which his pythagorean shipmates toss him overboard 32 while one writer even has pythagoras himself to his eternal shame sentencing hippasus to death by drowning for showing that 2 displaystyle sqrt 2 is an irrational number 33 references edit huffman carl a 1993 philolaus of croton pythagorean and presocratic cambridge university press p 8 hippasus of metapontum greek philosopher encyclopedia britannica retrieved 2021 09 20 iamblichus 1918 the life of pythagoras 1918 translation ed p 327 william thompson 1930 the commentary of pappus on book x of euclid s elements pdf harvard university press p 64 couprie dirk l 2011 the dodecahedron or the shape of the earth according to plato heaven and earth in ancient greek cosmology from thales to heraclides ponticus astrophysics and space science library vol 374 springer pp 201 212 doi 10 1007 978 1 4419 8116 5_17 isbn 9781441981165 aristotle metaphysics i 3 984a7 1 2 3 diogenes laertius lives of eminent philosophers viii 84 simplicius physica 23 33 aetius i 5 5 dox 292 clement of alexandria protrepticus 64 2 iamblichus vita pythagorica 18 81 iamblichus vita pythagorica 34 267 iamblichus de communi mathematica scientia 76 iamblichus vita pythagorica 18 81 cf iamblichus in nic 10 20 de anima ap stobaeus i 49 32 iamblichus thomas ed 1939 18 on the pythagorean life p 88 iamblichus 1918 the life of pythagoras translated by thomas taylor aristotle metaphysics english translation sextus empiricus ad phys i 361 diogenes laertius lives of eminent philosophers viii 7 scholium on plato s phaedo 108d pappus commentary on book x of euclid s elements a similar story is quoted in a greek scholium to the tenth book 1 2 iamblichus vita pythagorica 34 246 iamblichus vita pythagorica 18 88 de communi mathematica scientia 25 iamblichus vita pythagorica 34 247 wilbur richard knorr 1975 the evolution of the euclidean elements a study of the theory of incommensurable magnitudes and its significance for early greek geometry pages 21 22 50 51 springer walter burkert 1972 lore and science in ancient pythagoreanism page 459 harvard university press plato theaetetus 147d ff thomas heath 1921 a history of greek mathematics volume 1 from thales to euclid p 155 aristotle prior analytics i 23 thomas heath 1921 a history of greek mathematics volume 1 from thales to euclid p 157 1 2 thomas heath 1921 a history of greek mathematics volume 1 from thales to euclid p 168 morris kline 1990 mathematical thought from ancient to modern times page 32 oxford university press simon singh 1998 fermat s last theorem p 54 external links edit wikisource has the text of the 1911 encyclopædia britannica article hippasus of metapontum hippasus of metapontum at scienceworld wolfram com laërtius diogenes pythagoreans hippasus lives of the eminent philosophers vol 2 8 translated by hicks robert drew two volume ed loeb classical library v t e ancient greek schools of philosophy proto philosophy epimenides pherecydes seven sages thales solon pittacus bias periander cleobulus chilon pre socratic list ionian milesian thales anaximander anaximenes heraclitean heraclitus cratylus diogenes italian pythagorean pythagoras hippasus philolaus archytas lamiskos alcmaeon brontinus theano damo calliphon hermotimus metrodorus of cos arignote myia eurytus more skeptic xenophanes xeniades eleatic parmenides zeno melissus hippo pluralist ionian anaxagoras archelaus metrodorus of lampsacus italian empedocles acron pausanias atomist leucippus democritus sophist ionian protagoras prodicus hippias thrasymachus damon more italian gorgias callicles lycophron dionysodorus euthydemus more classical socrates students xenophon cynic antisthenes diogenes crates menippus more cyrenaic aristippus aristippus the younger theodorus the atheist hegesias of cyrene anniceris more eretrian phaedo of elis menedemus asclepiades of phlius megarian euclid of megara ichthyas thrasymachus of corinth eubulides stilpo nicarete pasicles bryson dialecticians clinomachus apollonius cronus euphantus dionysius diodorus cronus philo alexinus panthoides platonic plato speusippus xenocrates more peripatetic aristotle aristotelianism theophrastus strato of lampsacus lyco of troas alexander of aphrodisias more hellenistic pyrrhonist pyrrho timon of phlius aenesidemus agrippa the skeptic sextus empiricus more stoic zeno of citium cleanthes chrysippus panaetius posidonius seneca lucius annaeus cornutus musonius rufus epictetus arrian marcus aurelius more epicurean epicurus metrodorus zeno of sidon philodemus lucretius diogenes of oenoanda more academic skeptic arcesilaus carneades philo of larissa cicero more middle platonist antiochus eudorus of alexandria philo of alexandria plutarch gaius apuleius alcinous galen more neopythagorean nigidius figulus apollonius of tyana moderatus of gades nicomachus numenius of apamea more neoplatonist ammonius saccas plotinus students origen porphyry iamblichus julian plutarch of athens syrianus proclus ammonius hermiae damascius simplicius more second sophistic nicetes of smyrna dio chrysostom favorinus lucian philostratus more v t e ancient greek mathematics mathematicians timeline anaxagoras anthemius apollonius archimedes archytas aristaeus the elder aristarchus autolycus bion bryson callippus carpus chrysippus cleomedes conon ctesibius democritus dicaearchus dinostratus diocles dionysodorus of caunus dionysodorus of amisene diophantus domninus eratosthenes euclid eudemus eudoxus eutocius geminus heliodorus heron hipparchus hippasus hippias hippocrates hypatia hypsicles isidore of miletus leon marinus menaechmus menelaus metrodorus nicomachus nicomedes nicoteles oenopides pandrosion pappus perseus philolaus philon philonides porphyry of tyre posidonius proclus ptolemy pythagoras serenus sosigenes sporus thales theaetetus theodorus theodosius theon of alexandria theon of smyrna thymaridas xenocrates zeno of elea zeno of sidon zenodorus treatises almagest arithmetica conics apollonius catoptrics data euclid elements euclid little astronomy measurement of a circle on conoids and spheroids on the sizes and distances aristarchus on sizes and distances hipparchus on the moving sphere autolycus optics euclid on spirals on the sphere and cylinder ostomachion phaenomena euclid planisphaerium spherics theodosius spherics menelaus the quadrature of the parabola the sand reckoner concepts and definitions chord circles of apollonius apollonian circles apollonian gasket problem of apollonius commensurability diophantine equation euclidean geometry golden ratio lune of hippocrates method of exhaustion parallel postulate platonic solid regular polygon straightedge and compass construction angle trisection doubling the cube squaring the circle quadratrix of hippias neusis construction results in elements angle bisector theorem exterior angle theorem euclidean algorithm euclid s theorem geometric mean theorem hinge theorem inscribed angle theorem intercept theorem intersecting chords theorem intersecting secants theorem law of cosines pons asinorum pythagorean theorem tangent secant theorem thales s theorem theorem of the gnomon apollonius s theorem aristarchus s inequality heron s formula law of sines menelaus s theorem pappus s area theorem problem ii 8 of arithmetica ptolemy s inequality ptolemy s table of chords ptolemy s theorem spiral of theodorus centers schools cyrene platonic academy pythagoreanism school of chios related ancient greek astronomy attic numerals greek numerals history of a history of greek mathematics by thomas heath archimedes palimpsest algebra timeline arithmetic timeline calculus timeline geometry timeline logic timeline mathematics timeline numbers prehistoric counting numeral systems list other cultures arabian islamic babylonian chinese egyptian inca indian japanese ancient greece portal mathematics portal authority control databases international isni viaf 2 gnd worldcat national united states people deutsche biographie retrieved from https en wikipedia org w index php title hippasus oldid 1356349479 categories 5th century bc greek mathematicians 5th century bc greek philosophers ancient greek physicists ancient metapontines irrational numbers pythagoreans of magna graecia hidden 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