Irrational numbers7/5/2023 ![]() These stories are usually taken together to ascribe the discovery of irrationals to Hippasus, but whether he did or not is uncertain. Iamblichus clearly states that the drowning at sea was a punishment from the gods for impious behaviour. 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. 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. Iamblichus gives a series of inconsistent reports. ![]() 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. However, the evidence linking the discovery to Hippasus is unclear. 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. Hippasus is sometimes credited with the discovery of the existence of irrational numbers, following which he was drowned at sea. Ī 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. 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. 61) Doctrines Īristotle 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. 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. ![]() 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. 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. 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. 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). Iamblichus says about the death of Hippasus: He also states that Hippasus was the founder of a sect of the Pythagoreans called the Mathematici ( μαθηματικοί) in opposition to the Acusmatici ( ἀκουσματικοί) but elsewhere he makes him the founder of the Acusmatici in opposition to the Mathematici. Hippasus is recorded under the city of Sybaris in Iamblichus list of each city's Pythagoreans. Metapontum in Italy ( Magna Graecia) 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. He may have lived in the late 5th century BC, about a century after the time of Pythagoras. Little is known about the life of Hippasus. The discovery of irrationality is not specifically ascribed to Hippasus by any ancient writer. Pappus) or alternatively tell that Hippasus drowned because he revealed how to construct a dodecahedron inside a sphere. However, the few ancient sources which describe this story either do not mention Hippasus by name (e.g. 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. ![]() Little is known about his life or his beliefs, but he is sometimes credited with the discovery of the existence of irrational numbers. 450 BC) was a Greek philosopher and early follower of Pythagoras. Hippasus, engraving by Girolamo Olgiati, 1580 ![]()
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