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Hippasus of
Metapontum
Metapontum or Metapontium ( grc, Μεταπόντιον, Metapontion) was an important city of Magna Graecia, situated on the gulf of Tarentum, between the river Bradanus and the Casuentus (modern Basento). It was distant about 20 km from ...
(; grc-gre, Ἵππασος ὁ Μεταποντῖνος, ''Híppasos''; c. 530 – c. 450 BC) was a
Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece, a country in Southern Europe:
*Greeks, an ethnic group.
*Greek language, a branch of the Indo-European language family.
**Proto-Greek language, the assumed last common ancestor ...
philosopher
A philosopher is a person who practices or investigates philosophy. The term ''philosopher'' comes from the grc, φιλόσοφος, , translit=philosophos, meaning 'lover of wisdom'. The coining of the term has been attributed to the Greek th ...
and early follower of
Pythagoras
Pythagoras of Samos ( grc, Πυθαγόρας ὁ Σάμιος, Pythagóras ho Sámios, Pythagoras the Samos, Samian, or simply ; in Ionian Greek; ) was an ancient Ionians, Ionian Ancient Greek philosophy, Greek philosopher and the eponymou ...
. Little is known about his life or his beliefs, but he is sometimes credited with the discovery of the existence of
irrational numbers
In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integ ...
. 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
A deity or god is a supernatural being who is considered divine or sacred. The ''Oxford Dictionary of English'' defines deity as a god or goddess, or anything revered as divine. C. Scott Littleton defines a deity as "a being with powers greater ...
for divulging this. However, the few ancient sources which describe this story either do not mention Hippasus by name (e.g. Pappus) or alternatively tell that Hippasus drowned because he revealed how to construct a
dodecahedron
In geometry, a dodecahedron (Greek , from ''dōdeka'' "twelve" + ''hédra'' "base", "seat" or "face") or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagon ...
inside a
sphere
A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
. 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
Pythagoras of Samos ( grc, Πυθαγόρας ὁ Σάμιος, Pythagóras ho Sámios, Pythagoras the Samos, Samian, or simply ; in Ionian Greek; ) was an ancient Ionians, Ionian Ancient Greek philosophy, Greek philosopher and the eponymou ...
.
Metapontum
Metapontum or Metapontium ( grc, Μεταπόντιον, Metapontion) was an important city of Magna Graecia, situated on the gulf of Tarentum, between the river Bradanus and the Casuentus (modern Basento). It was distant about 20 km from ...
in
Italy
Italy ( it, Italia ), officially the Italian Republic, ) or the Republic of Italy, is a country in Southern Europe. It is located in the middle of the Mediterranean Sea, and its territory largely coincides with the homonymous geographical re ...
(
Magna Graecia
Magna Graecia (, ; , , grc, Μεγάλη Ἑλλάς, ', it, Magna Grecia) was the name given by the Romans to the coastal areas of Southern Italy in the present-day Italian regions of Calabria, Apulia, Basilicata, Campania and Sicily; these re ...
) is usually referred to as his birthplace,
although according to Iamblichus some claim Metapontum to be his birthplace, while others the nearby city of
Croton. Hippasus is recorded under the city of
Sybaris
Sybaris ( grc, Σύβαρις; it, Sibari) was an important city of Magna Graecia. It was situated in modern Calabria, in southern Italy, between two rivers, the Crathis (Crati) and the Sybaris (Coscile).
The city was founded in 720 BC ...
in
Iamblichus
Iamblichus (; grc-gre, Ἰάμβλιχος ; Aramaic: 𐡉𐡌𐡋𐡊𐡅 ''Yamlīḵū''; ) was a Syrian neoplatonic philosopher of Arabic origin. He determined a direction later taken by neoplatonism. Iamblichus was also the biographer of ...
list of each city's Pythagoreans. He also states that Hippasus was the founder of a sect of the Pythagoreans called the
''Mathematici'' () in opposition to the
''Acusmatici'' (); but elsewhere he makes him the founder of the ''Acusmatici'' in opposition to the ''Mathematici''.
Iamblichus
Iamblichus (; grc-gre, Ἰάμβλιχος ; Aramaic: 𐡉𐡌𐡋𐡊𐡅 ''Yamlīḵū''; ) was a Syrian neoplatonic philosopher of Arabic origin. He determined a direction later taken by neoplatonism. Iamblichus was also the biographer of ...
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).
According to Iamblichus (ca. 245-325 AD, 1918 translation) in ''The life of Pythagoras'', by
Thomas Taylor
: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
Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher and polymath during the Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatetic school of phil ...
speaks of Hippasus as holding the element of
fire
Fire is the rapid oxidation of a material (the fuel) in the exothermic chemical process of combustion, releasing heat, light, and various reaction Product (chemistry), products.
At a certain point in the combustion reaction, called the ignition ...
to be the cause of all things; and
Sextus Empiricus
Sextus Empiricus ( grc-gre, Σέξτος Ἐμπειρικός, ; ) was a Ancient Greece, Greek Pyrrhonism, Pyrrhonist philosopher and Empiric school physician. His philosophical works are the most complete surviving account of ancient Greek and ...
contrasts him with the Pythagoreans in this respect, that he believed the ''
arche
''Arche'' (; grc, ἀρχή; sometimes also transcribed as ''arkhé'') is a Greek word with primary senses "beginning", "origin" or "source of action" (: from the beginning, οr : the original argument), and later "first principle" or "element". ...
'' to be material, whereas they thought it was incorporeal, namely, number.
Diogenes Laërtius
Diogenes Laërtius ( ; grc-gre, Διογένης Λαέρτιος, ; ) was a biographer of the Ancient Greece, Greek philosophers. Nothing is definitively known about his life, but his surviving ''Lives and Opinions of Eminent Philosophers'' is a ...
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."
According to one statement, Hippasus left no writings,
according to another he was the author of the ''Mystic Discourse'', written to bring
Pythagoras
Pythagoras of Samos ( grc, Πυθαγόρας ὁ Σάμιος, Pythagóras ho Sámios, Pythagoras the Samos, Samian, or simply ; in Ionian Greek; ) was an ancient Ionians, Ionian Ancient Greek philosophy, Greek philosopher and the eponymou ...
into disrepute.
A
scholium
Scholia (singular scholium or scholion, from grc, σχόλιον, "comment, interpretation") are grammatical, critical, or explanatory comments – original or copied from prior commentaries – which are inserted in the margin of th ...
on
Plato
Plato ( ; grc-gre, Πλάτων ; 428/427 or 424/423 – 348/347 BC) was a Greek philosopher born in Athens during the Classical period in Ancient Greece. He founded the Platonist school of thought and the Academy, the first institution ...
's ''
Phaedo
''Phædo'' or ''Phaedo'' (; el, Φαίδων, ''Phaidōn'' ), also known to ancient readers as ''On The Soul'', is one of the best-known dialogues of Plato's middle period, along with the ''Republic'' and the ''Symposium.'' The philosophical ...
'' notes him as an early experimenter in
music theory
Music theory is the study of the practices and possibilities of music. ''The Oxford Companion to Music'' describes three interrelated uses of the term "music theory". The first is the "rudiments", that are needed to understand music notation (ke ...
, claiming that he made use of
bronze
Bronze is an alloy consisting primarily of copper, commonly with about 12–12.5% tin and often with the addition of other metals (including aluminium, manganese, nickel, or zinc) and sometimes non-metals, such as phosphorus, or metalloids such ...
disks to discover the fundamental musical ratios, 4:3, 3:2, and 2:1.
Irrational numbers
Hippasus is sometimes credited with the discovery of the existence of
irrational numbers
In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integ ...
, following which he was
drown
Drowning is a type of suffocation induced by the submersion of the mouth and nose in a liquid. Most instances of fatal drowning occur alone or in situations where others present are either unaware of the victim's situation or unable to offer as ...
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 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.
Iamblichus
Iamblichus (; grc-gre, Ἰάμβλιχος ; Aramaic: 𐡉𐡌𐡋𐡊𐡅 ''Yamlīḵū''; ) was a Syrian neoplatonic philosopher of Arabic origin. He determined a direction later taken by neoplatonism. Iamblichus was also the biographer of ...
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 geometry, a dodecahedron (Greek , from ''dōdeka'' "twelve" + ''hédra'' "base", "seat" or "face") or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagon ...
in the
sphere
A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
.
[Iamblichus, ''Vita Pythagorica'', 34 (246).] 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; but in another story this same punishment is meted out to the Pythagorean who divulged knowledge of the irrational. Iamblichus clearly states that the drowning at sea was a punishment from the
gods
A deity or god is a supernatural being who is considered divine or sacred. The ''Oxford Dictionary of English'' defines deity as a god or goddess, or anything revered as divine. C. Scott Littleton defines a deity as "a being with powers greater ...
for impious behaviour.
These stories are usually taken together to ascribe the discovery of irrationals to Hippasus, but whether he did or not is uncertain. 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
In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0,
where the Greek letter phi ( ...
of the regular
pentagon
In geometry, a pentagon (from the Greek πέντε ''pente'' meaning ''five'' and γωνία ''gonia'' meaning ''angle'') is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°.
A pentagon may be simpl ...
.
Some scholars in the early 20th century credited Hippasus with the discovery of the irrationality of
, the
square root of 2
The square root of 2 (approximately 1.4142) is a positive real number that, when multiplied by itself, equals the number 2. It may be written in mathematics as \sqrt or 2^, and is an algebraic number. Technically, it should be called the princip ...
.
Plato
Plato ( ; grc-gre, Πλάτων ; 428/427 or 424/423 – 348/347 BC) was a Greek philosopher born in Athens during the Classical period in Ancient Greece. He founded the Platonist school of thought and the Academy, the first institution ...
in his ''
Theaetetus'', describes how
Theodorus of Cyrene
Theodorus of Cyrene ( el, Θεόδωρος ὁ Κυρηναῖος) was an ancient Greek mathematician who lived during the 5th century BC. The only first-hand accounts of him that survive are in three of Plato's dialogues: the '' Theaetetus'', th ...
(c. 400 BC) proved the irrationality of
,
, etc. up to
, which implies that an earlier mathematician had already proved the irrationality of
.
Aristotle
Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher and polymath during the Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatetic school of phil ...
referred to the method for a proof of the irrationality of
, and a full proof along these same lines is set out in the proposition interpolated at the end of
Euclid
Euclid (; grc-gre, Wikt:Εὐκλείδης, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the ''Euclid's Elements, Elements'' trea ...
's Book X, which suggests that the proof was certainly ancient.
[Thomas Heath (1921) ''A History of Greek Mathematics, Volume 1, From Thales to Euclid'', p. 168.] The method is a proof by contradiction, or
reductio ad absurdum
In logic, (Latin for "reduction to absurdity"), also known as (Latin for "argument to absurdity") or ''apagogical arguments'', is the form of argument that attempts to establish a claim by showing that the opposite scenario would lead to absu ...
, which shows that if the diagonal of a
square
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length adj ...
is assumed to be
commensurable with the side, then the same number must be both odd and even.
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; while one writer even has
Pythagoras
Pythagoras of Samos ( grc, Πυθαγόρας ὁ Σάμιος, Pythagóras ho Sámios, Pythagoras the Samos, Samian, or simply ; in Ionian Greek; ) was an ancient Ionians, Ionian Ancient Greek philosophy, Greek philosopher and the eponymou ...
himself "to his eternal shame" sentencing Hippasus to death by drowning, for showing "that
is an irrational number".
Simon Singh
Simon Lehna Singh, (born 19 September 1964) is a British popular science author, theoretical and particle physicist. His written works include ''Fermat's Last Theorem'' (in the United States titled ''Fermat's Enigma: The Epic Quest to Solve th ...
(1998), '' Fermat's Enigma'', p. 50.
References
External links
Hippasus of Metapontumat scienceworld.wolfram.com
*
{{Authority control
5th-century BC Greek people
5th-century BC philosophers
Ancient Greek mathematicians
Ancient Greek physicists
Ancient Metapontines
Irrational numbers
Pythagoreans of Magna Graecia
Pythagoreans
5th-century BC mathematicians