Hippasus ofMetapontum (/ˈhɪpəsəs/;Ancient Greek:Ἵππασος ὁ Μεταποντῖνος,Híppasos; c. 530 – c. 450 BC)[1] was aGreekphilosopher and early follower ofPythagoras.[2][3] Little is known about his life or his beliefs, but he is sometimes credited with the discovery of the existence ofirrational 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 thegods 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 adodecahedron inside asphere.[5] The discovery of irrationality is not specifically ascribed to Hippasus by any ancient writer.
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 ofPythagoras.Metapontum inMagna Graecia is usually referred to as his birthplace,[6][7][8][9][10] although according toIamblichus (3rd century AD) some claim Metapontum to be his birthplace, while others the nearby city ofCroton.[11] Hippasus is recorded under the city ofSybaris 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 theMathematici (μαθηματικοί) in opposition to theAcusmatici (ἀκουσματικοί);[13] but elsewhere he makes him the founder of theAcusmatici in opposition to theMathematici.[14]
Iamblichus says about the death of Hippasus:
According to Iamblichus'sThe life of Pythagoras,[16]
Aristotle speaks of Hippasus as holding the element offire to be the cause of all things;[17] andSextus Empiricus contrasts him with the Pythagoreans in this respect, that he believed thearche 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 theMystic Discourse, written to bringPythagoras into disrepute.[19]
Ascholium onPlato'sPhaedo notes him as an early experimenter inmusic theory, claiming that he made use ofbronze disks to discover the fundamental musical ratios, 4:3, 3:2, and 2:1.[20]
Hippasus is sometimes credited with the discovery of the existence ofirrational numbers, following which he wasdrowned 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 regulardodecahedron in thesphere.[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 thegods 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 thegolden ratio of the regularpentagon.[26]
Some scholars in the early 20th century credited Hippasus with the discovery of the irrationality of, thesquare root of 2.Plato in hisTheaetetus,[27] describes howTheodorus of Cyrene (c. 400 BC) proved the irrationality of,, etc. up to, which implies that an earlier mathematician had already proved the irrationality of.[28]Aristotle referred to the method for a proof of the irrationality of,[29] and a full proof along these same lines is set out in the proposition interpolated at the end ofEuclid's Book X,[30] which suggests that the proof was certainly ancient.[31] The method is a proof by contradiction, orreductio ad absurdum, which shows that if the diagonal of asquare is assumed to becommensurable 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 is an irrational number".[33]
Laërtius, Diogenes."Pythagoreans: Hippasus" .Lives of the Eminent Philosophers. Vol. 2:8. Translated byHicks, Robert Drew (Two volume ed.). Loeb Classical Library.| International | |
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