Archytas

Archytas was born in the Greek city of Taras (Tarentum), Magna Graecia, and was the son of either Mnesagoras or Hadees. He presumably was 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.

Archytas named the harmonic mean, important much later in projective geometry and number theory, though he did not discover it.[6] 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.[7][8] Hippocrates of Chios before had reduced this problem to the finding of two mean proportionals. According to Diogenes Laërtius, Archytas' demonstration, which uses lines generated by moving figures to construct the two proportionals between magnitudes, was the first in which geometry was studied with mechanical concepts.[9] The topic of proportions, which Archytas seems to have worked on extensively, is also treated in book VIII of Euclid's Elements, where the construction for two proportional means can be found, equivalent to the extraction of the cube root.

Later tradition regarded Archytas as the founder of mathematical mechanics.[10] Vitruvius includes him in a list of twelve authors who wrote works on mechanics.[11] Thomas Nelson Winter presents evidence that the pseudo-Aristotelian Mechanical Problems might have been authored by Archytas and later misattributed.[12] As described in the writings of Aulus Gellius five centuries after him, Archytas was reputed to have designed and built some kind of bird-shaped, self-propelled flying device known as the pigeon, said to have flown some 200 meters.[13][14]

•  Laërtius, Diogenes (1925). "Pythagoreans: Archytas" . Lives of the Eminent Philosophers. Vol. 2:8. Translated by Hicks, Robert Drew (Two volume ed.). Loeb Classical Library.

• von Fritz, Kurt (1970). "Archytas of Tarentum". Dictionary of Scientific Biography. Vol. 1. New York: Charles Scribner's Sons. pp. 231–233. ISBN 0-684-10114-9. on line
• Huffman, Carl A. Archytas of Tarentum, Cambridge University Press, 2005, ISBN 0-521-83746-4

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• Huffman, Carl. "Archytas". In Zalta, Edward N. (ed.). Stanford Encyclopedia of Philosophy.
• O'Connor, John J.; Robertson, Edmund F., "Archytas", MacTutor History of Mathematics Archive, University of St Andrews
• Pseudo-Aristotle, Mechanica – Greek text and English translation
• Complete fragments (Greek–Spanish bilingual edition)
• Fragments and Life of Archytas