Hippasus
Encyclopedia
Hippasus of Metapontum
in Magna Graecia
, was a Pythagorean
philosopher. Little is known about his life or his beliefs, but he is sometimes credited with the discovery of the existence of irrational numbers.
in Italy
(Magna Graecia
), although the nearby city of Croton
is also mentioned as his birthplace. Iamblichus states that he 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.
speaks of Hippasus as holding the element of fire
to be the cause of all things; 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. 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." According to one statement, Hippasus left no writings, according to another he was the author of the Mystic Discourse, written to bring Pythagoras
into disrepute.
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.
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 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
. 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 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 dodecahedrons. Irrationality, by infinite reciprocal subtraction, can be easily seen in the Golden ratio
of the regular pentagon
.
Some modern scholars prefer to credit Hippasus with the discovery of the irrationality of √2
. Plato
in his Theaetetus
, describes how Theodorus of Cyrene
(c. 400 BC) proved the irrationality of √3, √5, etc. up to √17, which implies that an earlier mathematician had already proved the irrationality of √2. A simple proof of the irrationality of √2 is indicated by Aristotle
, and it is set out in the proposition interpolated at the end of Euclid
's Book X, which suggests that the proof was certainly ancient. The proof is one of reductio ad absurdum
, and the method is to show that, if the diagonal of a square
is 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
himself "to his eternal shame" sentencing Hippasus to death by drowning, for showing "that √2 is an irrational number."
Metapontum
Metapontum, Metapontium or Metapontion , was an important city of Magna Graecia, situated on the gulf of Tarentum, between the river Bradanus and the Casuentus . It was distant about 20 km from Heraclea and 40 from Tarentum...
in Magna Graecia
Magna Graecia
Magna Græcia is the name of the coastal areas of Southern Italy on the Tarentine Gulf that were extensively colonized by Greek settlers; particularly the Achaean colonies of Tarentum, Crotone, and Sybaris, but also, more loosely, the cities of Cumae and Neapolis to the north...
, was a Pythagorean
Pythagoreanism
Pythagoreanism was the system of esoteric and metaphysical beliefs held by Pythagoras and his followers, the Pythagoreans, who were considerably influenced by mathematics. Pythagoreanism originated in the 5th century BCE and greatly influenced Platonism...
philosopher. Little is known about his life or his beliefs, but he is sometimes credited with the discovery of the existence of irrational numbers.
Life
Little is known about the life of Hippasus. He may have lived in the late 5th century BC. He probably came from MetapontumMetapontum
Metapontum, Metapontium or Metapontion , was an important city of Magna Graecia, situated on the gulf of Tarentum, between the river Bradanus and the Casuentus . It was distant about 20 km from Heraclea and 40 from Tarentum...
in Italy
Italy
Italy , officially the Italian Republic languages]] under the European Charter for Regional or Minority Languages. In each of these, Italy's official name is as follows:;;;;;;;;), is a unitary parliamentary republic in South-Central Europe. To the north it borders France, Switzerland, Austria and...
(Magna Graecia
Magna Graecia
Magna Græcia is the name of the coastal areas of Southern Italy on the Tarentine Gulf that were extensively colonized by Greek settlers; particularly the Achaean colonies of Tarentum, Crotone, and Sybaris, but also, more loosely, the cities of Cumae and Neapolis to the north...
), although the nearby city of Croton
Crotone
Crotone is a city and comune in Calabria, southern Italy, on the Ionian Sea. Founded circa 710 BC as the Achaean colony of Croton , it was known as Cotrone from the Middle Ages until 1928, when its name was changed to the current one. In 1994 it became the capital of the newly established...
is also mentioned as his birthplace. Iamblichus states that he 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.
Doctrines
AristotleAristotle
Aristotle was a Greek philosopher and polymath, a student of Plato and teacher of Alexander the Great. His writings cover many subjects, including physics, metaphysics, poetry, theater, music, logic, rhetoric, linguistics, politics, government, ethics, biology, and zoology...
speaks of Hippasus as holding the element of fire
Fire (classical element)
Fire has been an important part of all cultures and religions from pre-history to modern day and was vital to the development of civilization. It has been regarded in many different contexts throughout history, but especially as a metaphysical constant of the world.-Greek and Roman tradition:Fire...
to be the cause of all things; and Sextus Empiricus
Sextus Empiricus
Sextus Empiricus , was a physician and philosopher, and has been variously reported to have lived in Alexandria, Rome, or Athens. His philosophical work is the most complete surviving account of ancient Greek and Roman skepticism....
contrasts him with the Pythagoreans in this respect, that he believed the arche
Arche
Arche is a Greek word with primary senses 'beginning', 'origin' or 'first cause' and 'power', 'sovereignty', 'domination' as extended meanings. This list is extended to 'ultimate underlying substance' and 'ultimate undemonstrable principle'...
to be material, whereas they thought it was incorporeal, namely, number. Diogenes Laërtius
Diogenes Laertius
Diogenes Laertius was a biographer of the Greek philosophers. Nothing is known about his life, but his surviving Lives and Opinions of Eminent Philosophers is one of the principal surviving sources for the history of Greek philosophy.-Life:Nothing is definitively known about his life...
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
Pythagoras of Samos was an Ionian Greek philosopher, mathematician, and founder of the religious movement called Pythagoreanism. Most of the information about Pythagoras was written down centuries after he lived, so very little reliable information is known about him...
into disrepute.
A scholium
Scholium
Scholia , are grammatical, critical, or explanatory comments, either original or extracted from pre-existing commentaries, which are inserted on the margin of the manuscript of an ancient author, as glosses. One who writes scholia is a scholiast...
on Plato
Plato
Plato , was a Classical Greek philosopher, mathematician, student of Socrates, writer of philosophical dialogues, and founder of the Academy in Athens, the first institution of higher learning in the Western world. Along with his mentor, Socrates, and his student, Aristotle, Plato helped to lay the...
's Phaedo
Phaedo
Plato's Phaedo is one of the great dialogues of his middle period, along with the Republic and the Symposium. The Phaedo, which depicts the death of Socrates, is also Plato's seventh and last dialogue to detail the philosopher's final days .In the dialogue, Socrates...
notes him as an early experimenter in music theory
Music theory
Music theory is the study of how music works. It examines the language and notation of music. It seeks to identify patterns and structures in composers' techniques across or within genres, styles, or historical periods...
, claiming that he made use of bronze
Bronze
Bronze is a metal alloy consisting primarily of copper, usually with tin as the main additive. It is hard and brittle, and it was particularly significant in antiquity, so much so that the Bronze Age was named after the metal...
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, 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 confused.Pappus
Pappus of Alexandria
Pappus of Alexandria was one of the last great Greek mathematicians of Antiquity, known for his Synagoge or Collection , and for Pappus's Theorem in projective geometry...
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 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
Sphere
A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...
. 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 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 dodecahedrons. Irrationality, by infinite reciprocal subtraction, can be easily seen in the Golden ratio
Golden ratio
In mathematics and the arts, two quantities are in the golden ratio if the ratio of the sum of the quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one. The golden ratio is an irrational mathematical constant, approximately 1.61803398874989...
of the regular pentagon
Pentagon
In geometry, a pentagon is any five-sided polygon. A pentagon may be simple or self-intersecting. The sum of the internal angles in a simple pentagon is 540°. A pentagram is an example of a self-intersecting pentagon.- Regular pentagons :In a regular pentagon, all sides are equal in length and...
.
Some modern scholars prefer to credit Hippasus with the discovery of the irrationality of √2
Square root of 2
The square root of 2, often known as root 2, is the positive algebraic number that, when multiplied by itself, gives the number 2. It is more precisely called the principal square root of 2, to distinguish it from the negative number with the same property.Geometrically the square root of 2 is the...
. Plato
Plato
Plato , was a Classical Greek philosopher, mathematician, student of Socrates, writer of philosophical dialogues, and founder of the Academy in Athens, the first institution of higher learning in the Western world. Along with his mentor, Socrates, and his student, Aristotle, Plato helped to lay the...
in his Theaetetus
Theaetetus (dialogue)
The Theaetetus is one of Plato's dialogues concerning the nature of knowledge. The framing of the dialogue begins when Euclides tells his friend Terpsion that he had written a book many years ago based on what Socrates had told him of a conversation he'd had with Theaetetus when Theaetetus was...
, describes how Theodorus of Cyrene
Theodorus of Cyrene
Theodorus of Cyrene was a Greek mathematician of the 5th century BC. The only first-hand accounts of him that we have are in two of Plato's dialogues: the Theaetetus and the Sophist...
(c. 400 BC) proved the irrationality of √3, √5, etc. up to √17, which implies that an earlier mathematician had already proved the irrationality of √2. A simple proof of the irrationality of √2 is indicated by Aristotle
Aristotle
Aristotle was a Greek philosopher and polymath, a student of Plato and teacher of Alexander the Great. His writings cover many subjects, including physics, metaphysics, poetry, theater, music, logic, rhetoric, linguistics, politics, government, ethics, biology, and zoology...
, and it is set out in the proposition interpolated at the end of Euclid
Euclid
Euclid , fl. 300 BC, also known as Euclid of Alexandria, was a Greek mathematician, often referred to as the "Father of Geometry". He was active in Alexandria during the reign of Ptolemy I...
's Book X, which suggests that the proof was certainly ancient. The proof is one of reductio ad absurdum
Reductio ad absurdum
In logic, proof by contradiction is a form of proof that establishes the truth or validity of a proposition by showing that the proposition's being false would imply a contradiction...
, and the method is to show that, if the diagonal of a square
Square (geometry)
In geometry, a square is a regular quadrilateral. This means that it has four equal sides and four equal angles...
is commensurable
Commensurability (mathematics)
In mathematics, two non-zero real numbers a and b are said to be commensurable if a/b is a rational number.-History of the concept:...
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
Pythagoras of Samos was an Ionian Greek philosopher, mathematician, and founder of the religious movement called Pythagoreanism. Most of the information about Pythagoras was written down centuries after he lived, so very little reliable information is known about him...
himself "to his eternal shame" sentencing Hippasus to death by drowning, for showing "that √2 is an irrational number."
Proof of the irrationality of √2
The reductio ad absurdum proof of the irrationality of √2 is as follows:- Take a right triangle whose short sides are 1 unit in length
- By the Pythagorean theorem, the diagonal is √2
- Suppose that √2 is the ratio of two natural numbers, √2=m/n
- Suppose that m/n has been reduced to its lowest common form by division
- It follows that either m and n are both odd, or that m is odd and n is even, or that m is even and n is odd (if not, we could reduce m/n even further by dividing both numbers by 2)
- Square both sides of √2=m/n, so that 2=m2/n2
- Then 2n2=m2, so that m2 is even, and therefore m is even
- If m is even, then m=2x, where x is some other natural number
- Squaring this, it follows that m2=4x2=2n2
- It follows that n2=2x2, and therefore n2 is even, which means that n, being a natural number, must be even
- So we've reached a contradiction: although we assumed that m and n cannot both be even, it now turns out they both are. It therefore follows that √2 cannot be expressed as the ratio of two natural numbers, and must therefore be in another class of numbers
External links
- Hippasus of Metapontum at scienceworld.wolfram.com