Hippasus
Hippasus of Metapontum (; , "Híppasos"; c. 530 – c. 450 BC) was a Greek philosopher and early follower of Pythagoras. 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) 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, 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 in Iamblichus 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 says about the death of Hippasus: According to Iamblichus (ca. 245-325 AD, 1918 translation) in "The life of Pythagoras", by Thomas Taylor Doctrines. Aristotle 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. 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. 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 dodecahedra. Irrationality, by infinite reciprocal subtraction, can be easily seen in the Golden ratio of the regular pentagon. Some scholars in the early 20th century credited Hippasus with the discovery of the irrationality of . Plato in his "Theaetetus", describes how Theodorus of Cyrene (c. 400 BC) proved the irrationality of , , etc. up to , which implies that an earlier mathematician had already proved the irrationality of . Aristotle 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's Book X, which suggests that the proof was certainly ancient. 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. 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 is an irrational number."