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{{Short description|5th-century BC Pythagorean philosopher}}
{{About|the Greek philosopher|the mythological characters called Hippasus|Hippasus (mythology)}}
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[[File:Hippasus Metapontinus - Illustrium philosophorum et sapientum effigies ab eorum numistatibus extractae.png|right|thumb|Hippasus, engraving by Girolamo Olgiati, 1580]]
'''Hippasus of [[Metapontum]]''' ({{IPAc-en|ˈ|h|ɪ|p|ə|s|ə|s}}; {{langx|grc|Ἵππασος ὁ Μεταποντῖνος}}, ''Híppasos''; c. 530 – c. 450 BC)<ref>{{cite book |last1=Huffman |first1=Carl A. |title=Philolaus of Croton: Pythagorean and Presocratic |date=1993 |publisher=Cambridge University Press |page=8}}</ref> was a [[Greeks|Greek]] [[philosopher]] and early follower of [[Pythagoras]].<ref>{{Cite web|title=Hippasus of Metapontum {{!}} Greek philosopher|url=https://www.britannica.com/biography/Hippasus-of-Metapontum|access-date=2021-09-20|website=Encyclopedia Britannica|language=en}}</ref><ref>{{cite book |author=Iamblichus |edition=1918 translation |title=The life of Pythagoras |year=1918 |url=https://archive.org/details/lifeofpythagoras00iamb |page=327}}</ref> 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 [[Greek gods|gods]] for divulging this and crediting it to himself instead of Pythagoras which was the norm in Pythagorean society. However, the few ancient sources who describe this story either do not mention Hippasus by name (e.g. Pappus)<ref>{{cite book|title=The Commentary of Pappus on Book X of Euclid's Elements|url=http://www.wilbourhall.org/pdfs/pappus/PappusBookX.pdf|author=William Thompson|publisher=Harvard University Press|year=1930|page=64}}</ref> or alternatively tell that Hippasus drowned because he revealed how to construct a [[dodecahedron]] inside a [[sphere]].<ref>{{cite book
 | last = Couprie | first = Dirk L.
 | contribution = The dodecahedron, or the shape of the earth according to Plato
 | doi = 10.1007/978-1-4419-8116-5_17
 | isbn = 9781441981165
 | pages = 201–212
 | publisher = Springer
 | series = Astrophysics and Space Science Library
 | title = Heaven and Earth in Ancient Greek Cosmology: From Thales to Heraclides Ponticus
 | year = 2011| volume = 374
 }}</ref> The discovery of irrationality is not specifically ascribed to Hippasus by any ancient writer.

==Life==
Little 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,<ref>{{citation|url=https://www.perseus.tufts.edu/hopper/text?doc=Perseus:text:1999.01.0052:book=1:section=984a&highlight=Hippasus%2CMetapontum|title=Aristotle, Metaphysics I.3: 984a7}}</ref><ref name="tufts1">{{citation|url=https://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.01.0258%3Abook%3D8%3Achapter%3D6|title=Diogenes Laertius, Lives of Eminent Philosophers VIII,84}}</ref><ref>[[Simplicius of Cilicia|Simplicius]], ''Physica'' 23.33</ref><ref>[[Aetius (philosopher)|Aetius]] I.5.5 (Dox. 292)</ref><ref>[[Clement of Alexandria]], [[Protrepticus (Clement)|Protrepticus]] 64.2</ref> although according to [[Iamblichus]] (3rd century AD) some claim Metapontum to be his birthplace, while others the nearby city of [[Crotone|Croton]].<ref>Iamblichus, ''Vita Pythagorica'', 18 (81)</ref> Hippasus is recorded under the city of [[Sybaris]] in Iamblichus's list of each city's Pythagoreans.<ref>Iamblichus, Vita Pythagorica, 34 (267)</ref> He also states that Hippasus was the founder of a sect of the Pythagoreans called the [[Pythagoreanism#The mathēmatikoi|''Mathematici'']] ({{lang|grc|μαθηματικοί}}) in opposition to the [[Pythagoreanism#The akousmatikoi|''Acusmatici'']] ({{lang|grc|ἀκουσματικοί}});<ref>Iamblichus, ''De Communi Mathematica Scientia'', 76</ref> but elsewhere he makes him the founder of the ''Acusmatici'' in opposition to the ''Mathematici''.<ref>Iamblichus, ''Vita Pythagorica'', 18 (81); cf. Iamblichus, ''In Nic.'' 10.20; ''De anima'' ap. [[Stobaeus]], i.49.32</ref>

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 [[dodecahedron|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).<ref>{{cite book |editor1-last=Iamblichus |editor1-first=Thomas |title=On the Pythagorean Life |date=1939 |page=88 |chapter=18}}</ref>

According to Iamblichus's ''The life of Pythagoras'',<ref>{{cite book |author=Iamblichus |title=The life of Pythagoras |year=1918 |translator=Thomas Taylor |translator-link=Thomas Taylor (neoplatonist) |url=https://archive.org/details/lifeofpythagoras00iamb}}</ref>
: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==
[[Aristotle]]  speaks of Hippasus as holding the element of [[fire (classical element)|fire]] to be the cause of all things;<ref>{{citation|url=https://www.perseus.tufts.edu/hopper/text?doc=Perseus:text:1999.01.0052:book=1:section=984a&highlight=Hippasus%2Cfire|title=Aristotle, Metaphysics (English translation)}}</ref> 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.<ref>Sextus Empiricus, ''ad Phys.'' i. 361</ref> [[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."<ref name="tufts1"/> According to one statement, Hippasus left no writings,<ref name="tufts1"/> according to another he was the author of the ''Mystic Discourse'', written to bring [[Pythagoras]] into disrepute.<ref>{{citation|url=https://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.01.0258%3Abook%3D8%3Achapter%3D1&highlight=Hippasus|title=Diogenes Laertius, Lives of Eminent Philosophers, viii. 7}}</ref>

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.<ref>Scholium on Plato's ''Phaedo'', 108d</ref>

===Irrational numbers===
Hippasus is sometimes credited with the discovery of the existence of [[irrational numbers]], following which he was [[drown]]ed 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 of Alexandria|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.<ref>Pappus, ''Commentary on Book X of Euclid's Elements''. A similar story is quoted in a Greek [[scholium]] to the tenth book.</ref> [[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]].<ref name="iamblichus246">Iamblichus, ''Vita Pythagorica'', 34 (246).</ref> 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;<ref>Iamblichus, ''Vita Pythagorica'', 18 (88), ''De Communi Mathematica Scientia'', 25.</ref> but in another story this same punishment is meted out to the Pythagorean who divulged knowledge of the irrational.<ref>Iamblichus, ''Vita Pythagorica'', 34 (247).</ref> Iamblichus clearly states that the drowning at sea was a punishment from the [[Greek gods|gods]] for impious behaviour.<ref name="iamblichus246"/>

These stories are usually taken together to ascribe the discovery of irrationals to Hippasus, but whether he did or not is uncertain.<ref>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.</ref> 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]].<ref>[[Walter Burkert]] (1972), ''Lore and Science in Ancient Pythagoreanism'', page 459. Harvard University Press.</ref>

Some scholars in the early 20th century credited Hippasus with the discovery of the irrationality of <math>\sqrt2</math>, the [[square root of 2]]. [[Plato]] in his ''[[Theaetetus (dialogue)|Theaetetus]]'',<ref>Plato, ''Theaetetus'', 147d&nbsp;ff.</ref> describes how [[Theodorus of Cyrene]] (c. 400 BC) proved the irrationality of <math>\sqrt3</math>, <math>\sqrt5</math>, etc. up to <math>\sqrt{17}</math>, which implies that an earlier mathematician had already proved the irrationality of <math>\sqrt2</math>.<ref>[[T. L. Heath|Thomas Heath]] (1921) ''A History of Greek Mathematics, Volume&nbsp;1, From Thales to Euclid'', p.&nbsp;155.</ref> [[Aristotle]] referred to the method for a proof of the irrationality of <math>\sqrt2</math>,<ref>Aristotle, ''Prior Analytics'', I-23.</ref> and a full proof along these same lines is set out in the proposition interpolated at the end of [[Euclid]]'s Book&nbsp;X,<ref>Thomas Heath (1921) ''A History of Greek Mathematics, Volume&nbsp;1, From Thales to Euclid'', p.&nbsp;157.</ref> which suggests that the proof was certainly ancient.<ref name="heath168">Thomas Heath (1921) ''A History of Greek Mathematics, Volume&nbsp;1, From Thales to Euclid'', p.&nbsp;168.</ref> The method is a proof by contradiction, or [[reductio ad absurdum]], which shows that if the diagonal of a [[Square (geometry)|square]] is assumed to be [[commensurability (mathematics)|commensurable]] with the side, then the same number must be both odd and even.<ref name="heath168"/>

In the hands of modern writers this combination of vague ancient reports and modern guesswork has sometimes evolved into a much 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;<ref>[[Morris Kline]] (1990), ''Mathematical Thought from Ancient to Modern Times'', page 32. Oxford University Press.</ref> while one writer even has [[Pythagoras]] himself "to his eternal shame" sentencing Hippasus to death by drowning, for showing "that <math>\sqrt2</math> is an irrational number".<ref>[[Simon Singh]] (1998), ''[[Fermat's Enigma]]'', p.&nbsp;50.</ref>

==References==
{{reflist|2}}

==External links==
{{Wikisource1911Enc|Hippasus of Metapontum}}
*[http://scienceworld.wolfram.com/biography/Hippasus.html Hippasus of Metapontum] at scienceworld.wolfram.com
*{{cite LotEP |chapter=Hippasus}}

{{Greek schools of philosophy}}
{{Greek mathematics}}

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[[Category:5th-century BC Greek philosophers]]
[[Category:Ancient Greek mathematicians]]
[[Category:Ancient Greek physicists]]
[[Category:Ancient Metapontines]]
[[Category:Irrational numbers]]
[[Category:Pythagoreans of Magna Graecia]]
[[Category:5th-century BC mathematicians]]