Hippasus

Hippasos of Metapontum was a Greek mathematician, music theorist and philosopher from among the Pythagoreans. He lived in the late 6th and early 5th century BC in southern Italy and is one of the most famous Pythagoreans of the early period. Are attributed to him three discoveries: the construction of a sphere inscribed dodecahedron, the discovery of incommensurability and the determination of the relative numbers of Grundkonsonanzen through sound experiments.

Life

Hippasos came from Metapontum (now Metaponto in Basilicata, southern Italy ). This city, spent his last years in the Pythagoras was one of the most important centers of the Pythagorean movement, the Hippasos joined.

From the messages delivered to the late ancient philosopher Iamblichus, it appears that Hippasos one in the alleged, controversial in the research division of the Pythagoreans in two directions, mathematical and natural scientific research-based " mathematician " and living according to traditional rules of conduct " acousmatics " to have played a role. However, the information in Iamblichus are contradictory: He writes in several places, Hippasos was acousmatics, and reported that he had been considered for the mathematicians as the founder of acousmatic direction; at other points it counts Hippasos to the mathematicians and announced that the acousmatics have taken him for the founder of the mathematician. The original tradition is that which designates Hippasos as a mathematician.

Apparently Hippasos was a prominent, among the Pythagoreans highly controversial personality. One legend, handed down in several versions, he revealed a secret of the Pythagoreans, was then excluded from their community and later fatally crashed in the sea, which was interpreted as divine punishment. The legend is in this shape though unbelievable, but it reflects the considerable tensions that were related to its occurrence, and the attitude of opposing circles him falsification of the Pythagorean tradition accused.

A historic core of the legend is that it is actually among the Pythagoreans came to a rift and Hippasos played a prominent role. The cause of the conflict but was not in the handling of mathematical knowledge, but in the political opposition between revolutionary democrats and conservative forces. Hippasos supported a democratic party, while the majority of politically conservative Pythagoreans was close to the leading families and thus got into a conflict with folk speakers. Striking the independent attitude that Hippasos is occupied among the Pythagoreans. Maybe he was only loosely connected with them.

Teaching

Aristotle informs Hippasos have kept the fire for the raw material ( arché ) in the world. He is said to have held the soul to fiery. In ancient sources it is often named along with Heraclitus, in the nature of philosophy, the fire also plays a central role. Hippasos taught that the universe was conceived and finally in constant motion and change his full draw in a specified time frame.

Diogenes Laertius reports that Demetrius of Magnesia ( a grammarian of the 1st century BC ) claimed Hippasos have left no writing. According to another, also reported by Diogenes Laertius tradition Hippasos however, a work authored entitled "Mystic Logos", with which he intended to vilify Pythagoras. This news is, even if not true, an indication that Hippasos opposed the authority of the school's founders, Pythagoras, or at least this was subordinate to him by his opponents.

The incommensurability

Whether Hippasos discovered the incommensurability of side and diagonal of a square or a regular pentagon, is not known. In the 4th century Plato showed in his dialogue Meno, that the inner, transverse square, which is limited by the small diagonals (left), half as large as the whole square. But on the ratio of side length and diagonal he was not one.

A geometric proof of incommensurability can be performed at the pentagon as follows:

Task is to find for two distances, a common measure, so a small part of route, from both routes are an integer multiple of. A method to find this measure, the later named after Euclid exchange removal: Man pulls off the smaller distance so often from the larger until it no longer works. Now take the remaining residue and pulls it from the smaller from. The new remainder is removed from the old waste, etc. When one comes to a close with this method, since no residual is, it has the desired part line. Incommensurability of the routes is proved if it can be shown that this is impossible.

In the case of the regular pentagon it comes to the question of whether the side with the diagonal has a common level. One pulls the first Fünfeckseite (or the same long distance AC, right) of a diagonal ( line AD ) from; it results in a residual ( line CD). This is withdrawn from the side. The new radical has the length of the segment BC. Here we come back to the starting line ratio, because the distance BC is the side of an inner pentagon and the track is the CD as long as its small diagonal CC '. The smaller pentagon is geometrically similar to Ausgangsfünfeck because it is a regular pentagon. So You stand back at the beginning, because AD to AC behaves as CD ( or CC ') to BC ( " Golden Section "). In geometrically similar figures, the length ratios of analog lines are still the same. The process thus can be repeated infinitely often and always leads to a smaller pentagon. One can represent this process as infinite continued fraction. So there is no common measure.

" Foundational crisis "

In ancient sources is talk of a secret betrayal of Hippasos. In the secret betrayed, it is said to have acted either the dodecahedron or the incommensurability. It is, Hippasos have published his discovery and was subsequently excluded from the community of the Pythagoreans. Later, he was drowned in the sea, which was interpreted as divine punishment for his crime.

In the earlier research has been suggested as a background to this legend, a " foundational crisis " of Pythagoreanism. It was assumed that Pythagoras claimed that all phenomena are expressible as integer ratios and there can be no incommensurability. The basis of Pythagoreanism thus had been refuted by the discovery of Hippasos, and this would have the Pythagoreans held against him.

From this interpretation, the research is, however, strayed. Walter Burkert and Leonid Zhmud - represent the otherwise completely contrary positions - agree that there is no convincing evidence for the claim that Pythagoras had dogmatically committed to a worldview that excluded any incommensurability principle. There is also no evidence that the discovery of incommensurability was considered scandalous and philosophically was a problem; Rather, it was regarded as a brilliant performance of the Pythagoreans. Played a significant role in the development of the legend of secret betrayal probably the fact that the Greek word árrhētos (literally " unspeakable "), which in mathematics means " irrational" had was ambiguous; " Ineffable " could also mean " secret," and in this sense the word outside mathematics for religious teachings secret ( mystery ) was used. Thus, the notion that irrationality was a secret was, probably from a misunderstanding.

Music Theory

One of the main areas of interest of the Pythagoreans was the music theory, specifically the question of how the harmonic intervals can be expressed mathematically. Hippasos is attributed to an experiment in which he and four bronze discs of the same diameter and different thickness intervals generated and thus showed that the Grundkonsonanzen can be expressed by ratios. The thicknesses of the four slices behaved like 1: 1 ⅓: 1 ½: 2

According to the late antique scholars Boëthius have Hippasos and Eubulides, another Pythagorean, the already known three consonant intervals two more added to the double octave and the twelfth.

Swell

• Maria Timpanaro Cardini: Pitagorici. Testimonianze e frammenti. Volume 1, La Nuova Italia, Florence 1958, p 78-105 ( Greek and Latin source texts with Italian translation and commentary )