Archytas

Archytas (Ἀρχύτας; 435/410–360/350 BC ) was an Ancient Greek mathematician, music theorist, statesman, and strategist from the ancient city of Taras (Tarentum) in Southern Italy. He was a scientist and philosopher affiliated with the Pythagorean school and famous for being the reputed founder of mathematical mechanics and a friend of Plato.

As a Pythagorean, Archytas believed that arithmetic (logistic), rather than geometry, provided the basis for satisfactory proofs, and developed the most famous argument for the infinity of the universe in antiquity.

Life
Archytas was born in Tarentum, a Greek city in the Italian Peninsula that was part of Magna Graecia, and was the son of Hestiaeus. He was presumably taught by Philolaus, and taught mathematics to Eudoxus of Cnidus and to Eudoxus' student, Menaechmus.

Politically and militarily, Archytas appears to have been the dominant figure in Tarentum in his generation, somewhat comparable to Pericles in Athens a half-century earlier. The Tarentines elected him strategos ("general") seven years in a row, a step that required them to violate their own rule against successive appointments. Archytas was allegedly undefeated as a general in Tarentine campaigns against their southern Italian neighbors.

In his public career, Archytas had a reputation for virtue as well as efficacy. The Seventh Letter, traditionally attributed to Plato, asserts that Archytas attempted to rescue Plato during his difficulties with Dionysius II of Syracuse. Some scholars have argued that Archytas may have served as one model for Plato's philosopher king, and that he influenced Plato's political philosophy as expressed in The Republic and other works.

Works
Archytas is said to be the first ancient Greek to have spoken of the sciences of arithmetic (logistic), geometry, astronomy, and harmonics as kin, which later became the medieval quadrivium. He is thought to have written a great number of works in the sciences, but only four fragments are generally believed to be authentic.

According to Eutocius, Archytas was the first to solve the problem of doubling the cube (the so-called Delian problem) with an ingenious geometric construction. Before this, Hippocrates of Chios had reduced this problem to the finding of two mean proportionals, equivalent to the extraction of cube roots. Archytas' demonstration uses lines generated by moving figures to construct the two proportionals between magnitudes and was, according to Diogenes Laërtius, the first in which mechanical motions entered geometry. The topic of proportions, which Archytas seems to have worked on extensively, is treated in Euclid's Elements, where the construction for two proportional means can also be found.

Archytas named the harmonic mean, important much later in projective geometry and number theory, though he did not discover it. He proved that supernummerary ratios cannot be divided by a mean proportional – an important result in ancient harmonics. Ptolemy considered Archytas the most sophisticated Pythagorean music theorist, and scholars believe Archytas gave a mathematical account of the musical scales used by practicing musicians of his day.

Later tradition regarded Archytas as the founder of mathematical mechanics. Vitruvius includes him in a list of twelve authors who wrote works on mechanics. T.N. Winter presents evidence that the pseudo-Aristotelian Mechanical Problems might have been authored by Archytas and later mis-attributed to Aristotle. Tradition also has it that Archytas built a mechanical flying dove. The sole mention of this from antiquity comes some five centuries after Archytas, when Aulus Gellius discusses a report by his mentor Favorinus :

"Archytas made a wooden model of a dove with such mechanical ingenuity and art that it flew; so nicely balanced was it, you see, with weights and moved by a current of air enclosed and hidden within it. About so improbable a story I prefer to give Favorinus' own words: 'Archytas the Tarentine, being in other lines also a mechanician, made a flying dove out of wood. Whenever it lit, it did not rise again.'"

Aulus Gellius views the reporting of the tradition as problematic, since it spreads implausible beliefs even if accompanied by skepticism.