Hippasus

 ( 5th-century Bc Greek Mathematicians ) 

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.Scholium on Plato's Phaedo, 108d

Pappus (4th century AD) 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.Pappus, Commentary on Book X of Euclid's Elements. A similar story is quoted in a Greek scholium to the tenth book. Iamblichus (3rd century AD) 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.Iamblichus, Vita Pythagorica, 34 (246). 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;Iamblichus, Vita Pythagorica, 18 (88), De Communi Mathematica Scientia, 25. but in another story this same punishment is meted out to the Pythagorean who divulged knowledge of the irrational.Iamblichus, Vita Pythagorica, 34 (247). Iamblichus clearly states that the drowning at sea was a punishment from the Greek 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.Wilbur Richard Knorr (1975), The Evolution of the Euclidean Elements: A Study of the Theory of Incommensurable Magnitudes and its Significance for Early Greek Geometry, pages 21–22, 50–51. Springer. 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.Walter Burkert (1972), Lore and Science in Ancient Pythagoreanism, page 459. Harvard University Press.

Some scholars in the early 20th century credited Hippasus with the discovery of the irrationality of $\backslash sqrt2$, the square root of 2. Plato in his Theaetetus,Plato, Theaetetus, 147d ff. describes how Theodorus of Cyrene (c. 400 BC) proved the irrationality of $\backslash sqrt3$, $\backslash sqrt5$, etc. up to $\backslash sqrt\{17\}$, which implies that an earlier mathematician had already proved the irrationality of $\backslash sqrt2$.Thomas Heath (1921) A History of Greek Mathematics, Volume 1, From Thales to Euclid, p. 155. Aristotle referred to the method for a proof of the irrationality of $\backslash sqrt2$,Aristotle, Prior Analytics, I-23. and a full proof along these same lines is set out in the proposition interpolated at the end of Euclid's Book X,Thomas Heath (1921) A History of Greek Mathematics, Volume 1, From Thales to Euclid, p. 157. which suggests that the proof was certainly ancient.Thomas Heath (1921) A History of Greek Mathematics, Volume 1, From Thales to Euclid, p. 168. 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 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;Morris Kline (1990), Mathematical Thought from Ancient to Modern Times, page 32. Oxford University Press. while one writer even has Pythagoras himself "to his eternal shame" sentencing Hippasus to death by drowning, for showing "that $\backslash sqrt2$ is an irrational number".Simon Singh (1998), Fermat's Last Theorem, p. 54.

• Hippasus of Metapontum at scienceworld.wolfram.com