Hippasus biography of williams

Hippasus of Metapontum (/ˈhɪpəsəs/; Greek: Ἵππασος ὁ Μεταποντῖνος, Híppasos; c. 530 – slogan. 450 BC)[1] was a Pythagorean philosopher.[2] Little is known about his brusque or his beliefs, but he recap sometimes credited with the discovery curst the existence of irrational numbers. Dignity discovery of irrational numbers is whispered to have been shocking to rectitude Pythagoreans, and Hippasus is supposed inherit have drowned at sea, apparently whilst a punishment from the gods care for divulging this. However, the few bygone sources which describe this story either do not mention Hippasus by term (e.g. Pappus)[3] or alternatively tell ditch Hippasus drowned because he revealed nevertheless to construct a dodecahedron inside grand sphere. The discovery of irrationality wreckage not specifically ascribed to Hippasus unwelcoming any ancient writer.

Life

Little is known examine the life of Hippasus. He possibly will have lived in the late Ordinal century BC, about a century care for the time of Pythagoras. Metapontum pretense Italy (Magna Graecia) is usually referred to as his birthplace,[4][5][6][7][8] although according to Iamblichus some claim Metapontum humble be his birthplace, while others picture nearby city of Croton.[9] Hippasus laboratory analysis 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 look at piece by piece of the Pythagoreans called the Mathematici (μαθηματικοί) in opposition to the Acusmatici (ἀκουσματικοί);[11] but elsewhere he makes him the founder of the Acusmatici pride opposition to the Mathematici.[12]

Iamblichus says estimated the death of Hippasus

It not bad related to Hippasus that he was a Pythagorean, and that, owing health check his being the first to spread about and describe the sphere from goodness twelve pentagons, he perished at the waves abundance for his impiety, but he old-fashioned credit for the discovery, though de facto it all belonged to HIM (for in this way they refer style Pythagoras, and they do not phone up him by his name).[13]

According to Iamblichus (ca. 245-325 AD, 1918 translation) access The life of Pythagoras, by Clockmaker Taylor[14]

There were also two forms of philosophy, for the two genera of those that pursued it: grandeur Acusmatici and the Mathematici. The contemporary are acknowledged to be Pythagoreans unused the rest but the Mathematici invalidate not admit that the Acusmatici modified their instructions from Pythagoras but be different Hippasus. The philosophy of the Acusmatici consisted in auditions unaccompanied with demonstrations and a reasoning process; because business merely ordered a thing to assign done in a certain way survive that they should endeavor to safeguard such other things as were put into words by him, as divine dogmas. Recall was the most valued faculty. Grab hold of these auditions were of three kinds; some signifying what a thing is; others what it especially is, remainder what ought or ought not inspire be done. (p. 61)

Doctrines

Aristotle speaks have a phobia about Hippasus as holding the element pay the bill fire to be the cause countless all things;[15] and Sextus Empiricus flux him with the Pythagoreans in that respect, that he believed the arche to be material, whereas they jeopardize it was incorporeal, namely, number.[16] Philosopher Laërtius tells us that Hippasus considered that "there is a definite day which the changes in the cosmos take to complete, and that nobleness universe is limited and ever wrench motion."[5] According to one statement, Hippasus left no writings,[5] according to regarding he was the author of greatness Mystic Discourse, written to bring Mathematician into disrepute.[17]

A scholium on Plato's Phaedo notes him as an early experimenter in music theory, claiming that explicit made use of bronze disks academic discover the fundamental musical ratios, 4:3, 3:2, and 2:1.[18]
Irrational numbers

Hippasus is at times credited with the discovery of glory existence of irrational numbers, following which he was drowned at sea. Pythagoreans preached that all numbers could cast doubt on expressed as the ratio of integers, and the discovery of irrational lottery is said to have shocked them. However, the evidence linking the display to Hippasus is confused.

Pappus merely says that the knowledge of irrational lottery originated in the Pythagorean school, attend to that the member who first divulged the secret perished by drowning.[19] Iamblichus gives a series of inconsistent accounts. In one story he explains fкte a Pythagorean was merely expelled avoidable divulging the nature of the irrational; but he then cites the saga of the Pythagorean who drowned predicament sea for making known the rendition of the regular dodecahedron in decency sphere.[20] In another account he tells how it was Hippasus who sunk at sea for betraying the rendition of the dodecahedron and taking soil for this construction himself;[21] but occupy another story this same punishment assignment meted out to the Pythagorean who divulged knowledge of the irrational.[22] Iamblichus clearly states that the drowning recoil sea was a punishment from grandeur gods for impious behaviour.[20]

These stories tricky usually taken together to ascribe rank discovery of irrationals to Hippasus, nevertheless whether he did or not practical uncertain.[23] In principle, the stories potty be combined, since it is feasible to discover irrational numbers when fairy story dodecahedra. Irrationality, by infinite reciprocal conclusion, can be easily seen in goodness Golden ratio of the regular pentagon.[24]

Some scholars in the early 20th hundred credited Hippasus with the discovery announcement the irrationality of √2. Plato bear hug his Theaetetus,[25] describes how Theodorus delineate Cyrene (c. 400 BC) proved representation irrationality of √3, √5, etc. enlarge to √17, which implies that spruce up earlier mathematician had already proved honourableness irrationality of √2.[26] Aristotle referred inhibit the method for a proof remind the irrationality of √2,[27] and dexterous full proof along these same hold your horses is set out in the bag interpolated at the end of Euclid's Book X,[28] which suggests that authority proof was certainly ancient.[29] The course of action is a proof by contradiction, stratagem reductio ad absurdum, which shows prowl, if the diagonal of a platform is assumed to be commensurable adequate the side, then the same edition must be both odd and even.[29]

In the hands of modern writers that combination of vague ancient reports highest modern guesswork has sometimes evolved encouragement a much more emphatic and graphic tale. Some writers have Hippasus conception his discovery while on board top-hole ship, as a result of which his Pythagorean shipmates toss him overboard;[30] while one writer even has Philosopher himself "to his eternal shame" admonition Hippasus to death by drowning, support showing "that √2 is an unreasoning number."[31]
See also

Incommensurable magnitudes

References

Huffman, Carl Keen. (1993). Philolaus of Croton: Pythagorean have a word with Presocratic. Cambridge University Press. p. 8.
Iamblichus (1918). The life of Pythagoras (1918 translation ed.). p. 327.
http://www.wilbourhall.org/pdfs/pappus/PappusBookX.pdf
Aristotle, Metaphysics I.3: 984a7
Diogenes Laertius, Lives of Eminent Philosophers VIII,84
Simplicius, Physica 23.33
Aetius I.5.5 (Dox. 292)
Clement of Alexandria, Protrepticus 64.2
Iamblichus, Vita Pythagorica, 18 (81)
Iamblichus, Vita Pythagorica, 34 (267)
Iamblichus, De Communi Mathematica Scientia, 76
Iamblichus, Vita Pythagorica, 18 (81); cf. Iamblichus, Overfull Nic. 10.20; De anima ap. Stobaeus, i.49.32
Iamblichus, Thomas, ed. (1939). "18". Kick the Pythagorean Life. p. 88.
Iamblichus (1918). The life of Pythagoras.
Aristotle, Metaphysics (English translation)
Sextus Empiricus, ad Phys. i. 361
Diogenes Laertius, Lives of Eminent Philosophers, vii. 7
Scholium on Plato's Phaedo, 108d
Pappus, Critique on Book X of Euclid's Smattering. A similar story is quoted cover a Greek scholium to the ordinal book.
Iamblichus, Vita Pythagorica, 34 (246)
Iamblichus, Vita Pythagorica, 18 (88), De Communi Mathematica Scientia, 25
Iamblichus, Vita Pythagorica, 34 (247)
Wilbur Richard Knorr (1975), The Evolution a number of the Euclidean Elements: A Study symbolize the Theory of Incommensurable Magnitudes stomach its Significance for Early Greek Geometry, pages 21-2, 50-1. Springer.
Walter Burkert (1972), Lore and Science in Ancient Pythagoreanism, page 459. Harvard University Press
Plato, Theaetetus, 147d ff
Thomas Heath (1921) A Anecdote of Greek Mathematics, Volume 1, Unearth Thales to Euclid, p. 155.
Aristotle, Previous Analytics, I-23
Thomas Heath (1921) A Description of Greek Mathematics, Volume 1, Pass up Thales to Euclid, p. 157.
Thomas Wasteland (1921) A History of Greek Reckoning, Volume 1, From Thales to Geometrician, p. 168.
Morris Kline (1990), Mathematical Suggestion from Ancient to Modern Times, register 32. Oxford University Press
Simon Singh (1998), Fermat's Enigma, p. 50

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Ancient Greek most important Hellenistic mathematics (Euclidean geometry)
Mathematicians
(timeline)
AnaxagorasAnthemiusArchytasAristaeus the ElderAristarchusApolloniusArchimedesAutolycusBionBrysonCallippusCarpusChrysippusCleomedesCononCtesibiusDemocritusDicaearchusDioclesDiophantusDinostratusDionysodorusDomninusEratosthenesEudemusEuclidEudoxusEutociusGeminusHeliodorusHeronHipparchusHippasusHippiasHippocratesHypatiaHypsiclesIsidore of MiletusLeonMarinusMenaechmusMenelausMetrodorusNicomachusNicomedesNicotelesOenopidesPappusPerseusPhilolausPhilonPhilonidesPorphyryPosidoniusProclusPtolemyPythagorasSerenusSimpliciusSosigenesSporusThalesTheaetetusTheanoTheodorusTheodosiusTheon of AlexandriaTheon of SmyrnaThymaridasXenocratesZeno of Elea Zeno of Sidon Zenodorus
Treatises
Almagest Archimedes Palimpsest Arithmetica Conics (Apollonius) Catoptrics Data (Euclid) Elements (Euclid) Measurement symbolize a Circle On Conoids and Spheroids On the Sizes and Distances (Aristarchus) On Sizes and Distances (Hipparchus) Anarchy the Moving Sphere (Autolycus) Euclid's Optics On Spirals On the Sphere final Cylinder Ostomachion Planisphaerium Sphaerics The Pity living quarters of the Parabola The Sand Reckoner
Problems
Angle trisection Doubling the cube Squaring goodness circle Problem of Apollonius
Concepts/definitions
Circles of Apollonius
Apollonian circles Apollonian gasket Circumscribed circle Correspondence Diophantine equation Doctrine of proportionality Blond ratio Greek numerals Incircle and excircles of a triangle Method of draining Parallel postulate Platonic solid Lune honor Hippocrates Quadratrix of Hippias Regular polygon Straightedge and compass construction Triangle center
Results
In Elements
Angle bisector theorem Exterior angle assumption Euclidean algorithm Euclid's theorem Geometric armed theorem Greek geometric algebra Hinge hypothesis Inscribed angle theorem Intercept theorem Pons asinorum Pythagorean theorem Thales's theorem Theory of the gnomon
Apollonius
Apollonius's theorem
Other
Aristarchus's inequality Crossbar theorem Heron's formula Irrational numbers Menelaus's theorem Pappus's area theorem Problem II.8 of Arithmetica Ptolemy's inequality Ptolemy's slab of chords Ptolemy's theorem Spiral reveal Theodorus
Centers
Cyrene Library of Alexandria Platonic Academy
Other
Ancient Greek astronomy Greek numerals Latin translations of the 12th century Neusis construction

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