How did hippasus discover irrational numbers

Hippasus of Metapontum (/ˈhɪpəsəs/; Greek: Ἵππασος ὁ Μεταποντῖνος, Híppasos; c. – c. BC)[1] was a Pythagorean philosopher.[2] 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.

However, the few ancient sources which describe this story either do not mention Hippasus by name (e.g. Pappus)[3] or alternatively tell that Hippasus drowned because he revealed how to construct a dodecahedron inside a sphere. 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 Italy (Magna Graecia) is usually referred to as his birthplace,[4][5][6][7][8] although according to Iamblichus some claim Metapontum to be his birthplace, while others the nearby city of Croton.[9] Hippasus is recorded under the city of Sybaris in Iamblichus list of each city's Pythagoreans.[10] He also states that Hippasus was the founder of a sect of the Pythagoreans called the Mathematici (μαθηματικοί) in opposition to the Acusmatici (ἀκουσματικοί);[11] but elsewhere he makes him the founder of the Acusmatici in opposition to the Mathematici.[12]

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).[13]

According to Iamblichus (ca.

What happened to hippasus

Hippasus death date

Pythagorean hippasus of metapontum

How did hippasus die

What did hippasus discover

AD, translation) in The life of Pythagoras, by Thomas Taylor[14]

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 to be the cause of all things;[15] 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.[16] 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."[5] According to one statement, Hippasus left no writings,[5] according to another he was the author of the Mystic Discourse, written to bring Pythagoras into disrepute.[17]

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, , , and [18]
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 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.[19] Iamblichus gives a series of inconsistent reports.

Biography of william shakespeare Thales Anaximander Anaximenes. There were also two forms of philosophy, for the two genera of those that pursued it: the Acusmatici and the Mathematici. Socrates students Xenophon. The life of Pythagoras.

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.[20] 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;[21] but in another story this same punishment is meted out to the Pythagorean who divulged knowledge of the irrational.[22] Iamblichus clearly states that the drowning at sea was a punishment from the gods for impious behaviour.[20]

These stories are usually taken together to ascribe the discovery of irrationals to Hippasus, but whether he did or not is uncertain.[23] 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.[24]

Some scholars in the early 20th century credited Hippasus with the discovery of the irrationality of √2. Plato in his Theaetetus,[25] describes how Theodorus of Cyrene (c. BC) proved the irrationality of √3, √5, etc. up to √17, which implies that an earlier mathematician had already proved the irrationality of √2.[26] Aristotle referred to the method for a proof of the irrationality of √2,[27] and a full proof along these same lines is set out in the proposition interpolated at the end of Euclid's Book X,[28] which suggests that the proof was certainly ancient.[29] 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.[29]

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.

Hippasus of metapontum biography of william hamilton

Hippasus of metapontum biography of william blake

Hippasus of metapontum biography of william hudson

Some writers have Hippasus making his discovery while on board a ship, as a result of which his Pythagorean shipmates toss him overboard;[30] 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."[31]
See also

Incommensurable magnitudes

References

Huffman, Carl A.

(). Philolaus of Croton: Pythagorean and Presocratic. Cambridge University Press. p.

Hippasus of metapontum biography of william hamilton: Life [ edit ]. Cross-references from this page 1 : Aristotle, Metaphysics , 1. Iamblichus Hippasus of Metapontum gale.

8.
Iamblichus (). The life of Pythagoras ( translation ed.). p.

Aristotle, Metaphysics I a7
Diogenes Laertius, Lives of Eminent Philosophers VIII,84
Simplicius, Physica
Aetius I (Dox. )
Clement of Alexandria, Protrepticus
Iamblichus, Vita Pythagorica, 18 (81)
Iamblichus, Vita Pythagorica, 34 ()
Iamblichus, De Communi Mathematica Scientia, 76
Iamblichus, Vita Pythagorica, 18 (81); cf.

Iamblichus, In Nic. ; De anima ap. Stobaeus, i
Iamblichus, Thomas, ed.

Hippasus of metapontum biography of william shakespeare Hermo'genes - Heron. Apollonius's theorem. There were also two forms of philosophy, for the two genera of those that pursued it: the Acusmatici and the Mathematici. Wikimedia Commons Wikisource Wikidata item.

(). "18". On the Pythagorean Life. p.

Hippasus of metapontum biography of william Ionian Anaxagoras Archelaus Metrodorus of Lampsacus. Hipolite Egg Company v. Hippasus apparently discovered that the length of an isosceles triangle's shorter side must be expressed as an irrational number if the length of the two equal legs is a whole one. Greek Physician.

Iamblichus (). The life of Pythagoras.
Aristotle, Metaphysics (English translation)
Sextus Empiricus, ad Phys. i.
Diogenes Laertius, Lives of Eminent Philosophers, viii. 7
Scholium on Plato's Phaedo, d
Pappus, Commentary on Book X of Euclid's Elements. A similar story is quoted in a Greek scholium to the tenth book.
Iamblichus, Vita Pythagorica, 34 ()
Iamblichus, Vita Pythagorica, 18 (88), De Communi Mathematica Scientia, 25
Iamblichus, Vita Pythagorica, 34 ()
Wilbur Richard Knorr (), The Evolution of the Euclidean Elements: A Study of the Theory of Incommensurable Magnitudes and its Significance for Early Greek Geometry, pages , Springer.
Walter Burkert (), Lore and Science in Ancient Pythagoreanism, page Harvard University Press
Plato, Theaetetus, d ff
Thomas Heath () A History of Greek Mathematics, Volume 1, From Thales to Euclid, p.

Hippasus of metapontum biography of william hurt Epimenides Pherecydes. Hippo Regius. 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. Retrieved January 11, from Encyclopedia.

Aristotle, Prior Analytics, I
Thomas Heath () A History of Greek Mathematics, Volume 1, From Thales to Euclid, p.
Thomas Heath () A History of Greek Mathematics, Volume 1, From Thales to Euclid, p.
Morris Kline (), Mathematical Thought from Ancient to Modern Times, page Oxford University Press
Simon Singh (), Fermat's Enigma, p.

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