As a Pythagorean, Archytas believed that arithmetic (logistic), rather than geometry, provided the basis for satisfactory proofs,[5] and developed the most famous argument for the infinity of the universe in antiquity.[6]
Politically and militarily, Archytas appears to have been the dominant figure in Tarentum in his generation, somewhat comparable to PericlesinAthens a half-century earlier.[7] 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.[8]
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 IIofSyracuse.[9] 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.[6]
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.[10][11] He is thought to have written a great number of works in the sciences, but only four fragments are generally believed to be authentic.[12]
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.[13][14] 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.[a] 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.[16]
Archytas named the harmonic mean, important much later in projective geometry and number theory, though he did not discover it.[17] He proved that supernummerary ratios[b] cannot be divided by a mean proportional – an important result in ancient harmonics.[6]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.[18]
Later tradition regarded Archytas as the founder of mathematical mechanics.[19]Vitruvius includes him in a list of twelve authors who wrote works on mechanics.[20] T.N. Winter presents evidence that the pseudo-Aristotelian Mechanical Problems might have been authored by Archytas and later mis-attributed to Aristotle.[21] 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[22][23]:
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.[24][25]
^Plato blamed Archytas for his contamination of geometry with mechanics:[15]
And therefore Plato himself dislikes Eudoxus, Archytas, and Menaechmus for endeavoring to bring down the doubling the cube to mechanical operations; for by this means all that was good in geometry would be lost and corrupted, it falling back again to sensible things, and not rising upward and considering immaterial and immortal images, in which God being versed is always God.
^Supernummerary ratios are integer ratios of the form n + 1/n, where n is some natural number; they are the "atoms" of mathematical theories of musical scales and tuning, and were extensively used by musicologists of the Greek classical period, of which Archytas was one among several. Examples of supernummerary ratios seen frequently in musical analysis of intonation even to the present day are 81 /80, 25 /24, 16 /15, 10 /9, 9 /8, 6 /5, 5 /4, 4 /3, 3 /2, and 2/ 1 .
^Morris Kline, Mathematical Thought from Ancient to Modern Times Oxford University Press, 1972 p. 49
^ abcdHuffman, Carl (2020), "Archytas", in Zalta, Edward N. (ed.), The Stanford Encyclopedia of Philosophy (Winter 2020 ed.), Metaphysics Research Lab, Stanford University, retrieved 2023-10-28
^
Horky, P.S. (2021). "Archytas: Author and authenticator of Pythagoreanism". In Macris, C.; Dorandi, T.; Brisson, L. (eds.). Pythagoras Redivivus: Studies on the texts attributed to Pythagoras and the Pythagoreans. Academia.
^
Menn, S. (2015). "How Archytas doubled the cube". In Holmes, B.; Fischer, K.-D. (eds.). The Frontiers of Ancient Science: Essays in honor of Heinrich von Staden. pp. 407–436 – via Google books.
^
A. Cornelius Gellius (1927). "12". Noctes Atticae (Attic Nights). Vol. X. Translated by J. C. Rolfe. Loeb Classical Library.
^
Beer, Beate (2020). Aulus Gellius und die >Noctus Atticae< — die literarische Konstruktion einer Sammlung. De Gruyter. p. 88. ISBN9783110695083.
^Beall, Stephen M. (2001). "Homo Fandi Dulcissimus: The Role of Favorinus in the "Attic Nights" of Aulus Gellius". The American Journal of Philology. 122 (1): 87.
Laërtius, Diogenes (1925). "Pythagoreans: Archytas" . Lives of the Eminent Philosophers. Vol. 2:8. Translated by Hicks, Robert Drew (Two volume ed.). Loeb Classical Library.
