Archytas: Difference between revisions

Archytas
Archytas
	Bust from Villa of the Papyri, Herculaneum, once identified as Archytas, now thought to be Pythagoras
Born	435/410 BC; Tarentum, Magna Graecia
Died	360/350 BC
Era	Classical Greek philosophy
Region	Western philosophy
School	Pythagoreanism
Notable ideas	Doubling the cube; Infinite universe

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Archytas (/ˈɑːrkɪtəs/; Template:Lang-el; 435/410–360/350 BC^[2]) was an Ancient Greek mathematician, music theorist,^[3] 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.^[4]

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]

Life

Archytas was born in Tarentum, a Greek city 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.^[6]

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.^[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 II of Syracuse.^[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]

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.^[10]^[11] He is thought to have written a great number of works in the sciences, but only four genuine fragments are known.^[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 [of Geometry], 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] 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&bsp;meters.^[22]^[23]^[24]^[25]^[26]

Notes

^ 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 $.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}⁠ n + 1 / n ⁠$ , 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 ⁠.

References

^ Archita; Pitagora, Sito ufficiale del Museo Archeologico Nazionale di Napoli, retrieved 25 September 2012
^ Philippa Lang, Science: Antiquity and its Legacy, Bloomsbury Academic, 2015, p. 154.
^ Barbera, André (2001). "Archytas of Tarentum". Grove Music Online. Oxford: Oxford University Press. doi:10.1093/gmo/9781561592630.article.01183. ISBN 978-1-56159-263-0. Retrieved 25 September 2021. (subscription or UK public library membership required)
^ Debra Nails, The People of Plato, ISBN 1603844031, p. 44
^ Morris Kline, Mathematical Thought from Ancient to Modern Times Oxford University Press, 1972 p. 49
^ ^a ^b ^c ^d Huffman, 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
^ Despotopoulos, Constantin (2004-11-01). "Archytas' Logismos and Logistika". Philosophical Inquiry. 26 (3): 1–9. doi:10.5840/philinquiry200426311.
^ Johnson, M. R. (2008). "Sources for the Philosophy of Archytas". philarchive.org. Retrieved 2023-10-30.
^ Lloyd, G. E. R. (1990). "Plato and Archytas in the "Seventh Letter"". Phronesis. 35 (2): 159–174. ISSN 0031-8868. JSTOR 4182355.
^ Furner, J. (2021). "Classification of the scieces in Greco-Roman antiquity". Knowledge Organization. 48 (7–8): 499–534.
^ Zhmud, L. (2008). The Origin of the History of Science in Classical Antiquity. Walter de Gruyter. pp. 62–63. ISBN 978-3-11-019432-6 – via Google books.
^ 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.
^ Masià, R. (2016). "A new reading of Archytas' doubling of the cube and its implications". Archive for History of Exact Sciences. 70 (2): 175–204. doi:10.1007/s00407-015-0165-9. ISSN 1432-0657.
^ Plutarch. Symposiacs. Book VIII, Question 2. Archived from the original on 2008-08-15.
^ Euclid. Elements [of Geometry]. book VIII.
^ O'Connor, J.J.; Robertson, E.F. Archytas of Tarentum. www-history.mcs.st-andrews.ac.uk/Biographies (Report). The MacTutor History of Mathematics archive. St. Andrews, Scotland: University of St. Andrews. Retrieved 11 August 2011.
^ Barker, A. (1994). "Ptolemy's Pythagoreans, Archytas, and Plato's conception of mathematics". Phronesis. 39 (2): 113–135. ISSN 0031-8868. JSTOR 4182463.
^ Laërtius 1925, § 83: Vitae philosophorum
^ Vitruvius. De architectura [On Archetecture] (in Latin). vii.14.
^ Winter, Thomas Nelson (2007). The Mechanical Problems in the Corpus of Aristotle (Report). Digital Commons. Lincoln, NB: University of Nebraska.
^ "Rocket history – B.C. era". grc.nasa.gov. Retrieved 2024-02-21.
^ Aulus Gellius. "Lacus Curtius". Attic Nights. University of Chicago. Book X, 12.9 – via penelope.uchicago.edu.
^ Archytas of Tarentum (Report). Thessaloniki, Macedonia, Greece: $Κέντρο Διάδοσης Επιστημών$ $και Μουσείο Τεχνολογίας$ [Thessaloniki Science Center and Technology Museum]. Archived from the original on December 26, 2008. Retrieved 23 March 2024.
^ Holford-Strevens, Leofranc (2005). Aulus Gellius: An Antonine author and his achievement (revised paperback ed.). Oxford, UK: Oxford University Press. ISBN 0-19-928980-8.
^ Chisholm, Hugh, ed. (1911). "Archytas" . Encyclopædia Britannica. Vol. 2 (11th ed.). Cambridge University Press. p. 446.

References

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

External links

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

[16] 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.

[19] Supernummerary ratios are integer ratios of the form $.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}⁠ n + 1 / n ⁠$ , 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 ⁠.

[1] Archita; Pitagora, Sito ufficiale del Museo Archeologico Nazionale di Napoli, retrieved 25 September 2012

[2] Philippa Lang, Science: Antiquity and its Legacy, Bloomsbury Academic, 2015, p. 154.

[3] Barbera, André (2001). "Archytas of Tarentum". Grove Music Online. Oxford: Oxford University Press. doi:10.1093/gmo/9781561592630.article.01183. ISBN 978-1-56159-263-0. Retrieved 25 September 2021. (subscription or UK public library membership required)

[4] Debra Nails, The People of Plato, ISBN 1603844031, p. 44

[5] Morris Kline, Mathematical Thought from Ancient to Modern Times Oxford University Press, 1972 p. 49

[:0-6] Huffman, 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

[7] Despotopoulos, Constantin (2004-11-01). "Archytas' Logismos and Logistika". Philosophical Inquiry. 26 (3): 1–9. doi:10.5840/philinquiry200426311.

[8] Johnson, M. R. (2008). "Sources for the Philosophy of Archytas". philarchive.org. Retrieved 2023-10-30.

[9] Lloyd, G. E. R. (1990). "Plato and Archytas in the "Seventh Letter"". Phronesis. 35 (2): 159–174. ISSN 0031-8868. JSTOR 4182355.

[10] Furner, J. (2021). "Classification of the scieces in Greco-Roman antiquity". Knowledge Organization. 48 (7–8): 499–534.

[11] Zhmud, L. (2008). The Origin of the History of Science in Classical Antiquity. Walter de Gruyter. pp. 62–63. ISBN 978-3-11-019432-6 – via Google books.

[12] 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.

[13] 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.

[14] Masià, R. (2016). "A new reading of Archytas' doubling of the cube and its implications". Archive for History of Exact Sciences. 70 (2): 175–204. doi:10.1007/s00407-015-0165-9. ISSN 1432-0657.

[15] Plutarch. Symposiacs. Book VIII, Question 2. Archived from the original on 2008-08-15.

[17] Euclid. Elements [of Geometry]. book VIII.

[18] O'Connor, J.J.; Robertson, E.F. Archytas of Tarentum. www-history.mcs.st-andrews.ac.uk/Biographies (Report). The MacTutor History of Mathematics archive. St. Andrews, Scotland: University of St. Andrews. Retrieved 11 August 2011.

[20] Barker, A. (1994). "Ptolemy's Pythagoreans, Archytas, and Plato's conception of mathematics". Phronesis. 39 (2): 113–135. ISSN 0031-8868. JSTOR 4182463.

[Diogenes_Laërtius-21] Laërtius 1925, § 83: Vitae philosophorum

[22] Vitruvius. De architectura [On Archetecture] (in Latin). vii.14.

[23] Winter, Thomas Nelson (2007). The Mechanical Problems in the Corpus of Aristotle (Report). Digital Commons. Lincoln, NB: University of Nebraska.

[24] "Rocket history – B.C. era". grc.nasa.gov. Retrieved 2024-02-21.

[25] Aulus Gellius. "Lacus Curtius". Attic Nights. University of Chicago. Book X, 12.9 – via penelope.uchicago.edu.

[26] Archytas of Tarentum (Report). Thessaloniki, Macedonia, Greece: $Κέντρο Διάδοσης Επιστημών$ $και Μουσείο Τεχνολογίας$ [Thessaloniki Science Center and Technology Museum]. Archived from the original on December 26, 2008. Retrieved 23 March 2024.

[27] Holford-Strevens, Leofranc (2005). Aulus Gellius: An Antonine author and his achievement (revised paperback ed.). Oxford, UK: Oxford University Press. ISBN 0-19-928980-8.

[28] Chisholm, Hugh, ed. (1911). "Archytas" . Encyclopædia Britannica. Vol. 2 (11th ed.). Cambridge University Press. p. 446.

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@@ Line 27: / Line 27: @@
 ==Works==
-Archytas is said to be the first ancient Greek to have spoken of the sciences of [[arithmetic]] (logistic), [[geometry]], [[astronomy]], and [[Pythagorean tuning|harmonics]] as kin, which later became the medieval [[quadrivium]].<ref>Furner, J. (2021). Classification of the scieces in Greco-Roman Antiquity. ''Knowledge Organization'', 48, 7-8: 499-534. https://www.isko.org/cyclo/greco-roman</ref><ref>{{Cite book |last=Zhmud |first=L. |url=https://books.google.com/books?id=fIeeA2fv5ZsC&dq=Leonid+Zhmud+Archytas&pg=PA63 |title=The Origin of the History of Science in Classical Antiquity |publisher=Walter de Gruyter |year=2008 |isbn=978-3-11-019432-6 |pages=62–63 |language=en}}</ref> He is thought to have written a number of works in the sciences but only four genuine fragments are extant.<ref>Horky, P. S. (2021). Archytas: Author and authenticator of Pythagoreanism. In C. Macris, T. Dorandi, & L. Brisson (Eds.), ''Pythagoras Redivivus: Studies on the Texts Attributed to Pythagoras and the Pythagoreans''. Academia.[https://durham-repository.worktribe.com/output/1648864]</ref>
+Archytas is said to be the first ancient Greek to have spoken of the sciences of [[arithmetic]] (logistic), [[geometry]], [[astronomy]], and [[Pythagorean tuning|harmonics]] as kin, which later became the medieval [[quadrivium]].<ref>
+{{cite journal
+ |last = Furner |first = J.
+ |year = 2021
+ |title = Classification of the scieces in Greco-Roman antiquity
+ |journal = [[Knowledge Organization]]
+ |volume = 48 |issue = 7-8 |pages = 499-534
+ |url = https://www.isko.org/cyclo/greco-roman
+}}
+</ref><ref>
+{{cite book
+ |last = Zhmud  |first = L.
+ |year = 2008
+ |title = The Origin of the History of Science in Classical Antiquity
+ |publisher = Walter de Gruyter
+ |isbn = 978-3-11-019432-6
+ |pages = 62–63
+ |url = https://books.google.com/books?id=fIeeA2fv5ZsC&dq=Leonid+Zhmud+Archytas&pg=PA63
+ |via = Google books |lang = en
+}}
+</ref>
+He is thought to have written a great number of works in the sciences, but only four genuine fragments are known.<ref>
+{{cite book
+ |last = Horky |first = P.S.
+ |year = 2021
+ |section = Archytas: Author and authenticator of Pythagoreanism
+ |editor1-first = C.  |editor1-last = Macris
+ |editor2-first = T.  |editor2-last = Dorandi
+ |editor3-first = L.  |editor3-last = Brisson
+ |title = Pythagoras Redivivus: Studies on the texts attributed to Pythagoras and the Pythagoreans
+ |publisher = Academia
+ |section-url = https://durham-repository.worktribe.com/output/1648864
+}}
+</ref>
+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.<ref>
-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.<ref>Menn, S. (2015). How Archytas doubled the cube. In B. Holmes & K-D Fischer (Eds.), ''The Frontiers of Ancient Science: Essays in Honor of Heinrich von Staden'' (pp. 407-436).[https://books.google.com/books?hl=en&lr=&id=nvpeCAAAQBAJ&oi=fnd&pg=PA407&dq=Archytas+double&ots=6wxxJM-zcJ&sig=Yl2pJftIsJJzRLV3IpmZh_dgel0#v=onepage&q&f=false]</ref><ref>{{Cite journal |last=Masià |first=R. |date=2016 |title=A new reading of Archytas' doubling of the cube and its implications |url=https://doi.org/10.1007/s00407-015-0165-9 |journal=Archive for History of Exact Sciences |language=en |volume=70 |issue=2 |pages=175–204 |doi=10.1007/s00407-015-0165-9 |issn=1432-0657}}</ref> [[Hippocrates of Chios]] before had reduced this problem to the finding of two mean [[Proportionality (mathematics)|proportionals]], equivalent to the extraction of [[cube root]]s. 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.<ref>[[Plato]] blamed Archytas for his contamination of geometry with mechanics ([[Plutarch]], [https://web.archive.org/web/20080815001514/http://ebooks.adelaide.edu.au/p/plutarch/symposiacs/chapter8.html#section80 ''Symposiacs'', Book VIII, Question 2] ): ''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.''</ref> The topic of proportions, which Archytas seems to have worked on extensively, is treated in book VIII of [[Euclid]]'s ''[[Euclid's Elements|Elements]]'', where the construction for two proportional means can also be found.
+{{cite book
+ |last = Menn  |first = S.
+ |year = 2015
+ |section = How Archytas doubled the cube
+ |editor1-first = B.    |editor1-last = Holmes
+ |editor2-first = K.-D. |editor2-last = Fischer
+ |title = The Frontiers of Ancient Science: Essays in honor of Heinrich von Staden
+ |pages = 407-436
+ |section-url = https://books.google.com/books?hl=en&lr=&id=nvpeCAAAQBAJ&oi=fnd&pg=PA407&dq=Archytas+double&ots=6wxxJM-zcJ&sig=Yl2pJftIsJJzRLV3IpmZh_dgel0#v=onepage&q&f=false
+ |via = Google books
+}}
+</ref><ref>
+{{cite journal
+ |last = Masià |first = R.
+ |year = 2016
+ |title = A new reading of Archytas' doubling of the cube and its implications
+ |journal = [[Archive for History of Exact Sciences]]
+ |volume = 70 |issue = 2 |pages = 175–204
+ |doi = 10.1007/s00407-015-0165-9  |issn = 1432-0657 |lang = en
+}}
+</ref>
+Before this, [[Hippocrates of Chios]] had reduced this problem to the finding of two mean [[proportionality (mathematics)|proportionals]], equivalent to the extraction of [[cube root]]s. 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.{{efn|
+[[Plato]] blamed Archytas for his contamination of geometry with mechanics:<ref>
+{{cite book
+ |author=[[Plutarch]]
+ |title = Symposiacs
+ |at = Book&nbsp;VIII, Question&nbsp;2
+ |url = http://ebooks.adelaide.edu.au/p/plutarch/symposiacs/chapter8.html#section80
+ |url-status = dead <!-- presumed -->
+ |archive-url = https://web.archive.org/web/20080815001514/http://ebooks.adelaide.edu.au/p/plutarch/symposiacs/chapter8.html#section80
+ |archive-date = 2008-08-15
+}}
+</ref>
+:
+: ''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.''
+}}
+The topic of proportions, which Archytas seems to have worked on extensively, is treated in [[Euclid's Elements|Euclid's ''Elements {{grey|[of Geometry]}}'']], where the construction for two proportional means can also be found.<ref>
+{{cite book
+ |author = [[Euclid]]
+ |title = Elements {{grey|[of Geometry]}}
+ |title-link = Euclid's Elements
+ |at = book&nbsp;VIII
+}}
+</ref>
+Archytas named the [[harmonic mean]], important much later in [[projective geometry]] and [[number theory]], though he did not discover it.<ref>
-Archytas named the [[harmonic mean]], important much later in [[projective geometry]] and [[number theory]], though he did not discover it.<ref>J. J. O'Connor and E. F. Robertson. [http://www-history.mcs.st-andrews.ac.uk/Biographies/Archytas.html Archytas of Tarentum]. The MacTutor History of Mathematics archive. Visited 11 August 2011.</ref> He proved that ratios of the form (''n'' + 1): ''n'' cannot be divided by a mean proportional, an important result in ancient harmonics.<ref name=":0" /> [[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.<ref>{{Cite journal |last=Barker |first=A. |date=1994 |title=Ptolemy's Pythagoreans, Archytas, and Plato's Conception of Mathematics |url=https://www.jstor.org/stable/4182463 |journal=Phronesis |volume=39 |issue=2 |pages=113–135 |jstor=4182463 |issn=0031-8868}}</ref>
+{{cite report
+ |first1 = J.J.  |last1 = O'Connor
+ |first2 = E.F.  |last2 = Robertson
+ |title = Archytas of Tarentum
+ |series = The MacTutor History of Mathematics archive
+ |website = www-history.mcs.st-andrews.ac.uk/Biographies
+ |publisher = [[University of St. Andrews]]
+ |place = [[St Andrews|St. Andrews, Scotland]]
+ |url = http://www-history.mcs.st-andrews.ac.uk/Biographies/Archytas.html
+ |access-date = 11 August 2011
+}}
+</ref>
+He proved that ''supernummerary ratios''{{efn|
+''Supernummerary ratios'' are integer ratios of the form {{math| {{sfrac| ''n'' + 1 | ''n'' }} }}, where {{mvar|n}} is some [[natural number]]; they are the "atoms" of mathematical theories of [[musical scales]] and [[musical tuning#tuning_systems_anchor|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  {{sfrac| 81 | 80 }}, {{sfrac| 25 | 24 }}, {{sfrac| 16 | 15 }}, {{sfrac| 10 | 9 }}, {{sfrac| 9 | 8 }}, {{sfrac| 6 | 5 }}, {{sfrac| 5 | 4 }}, {{sfrac| 4 | 3 }}, {{sfrac| 3 | 2 }}, and {{sfrac| 2 | 1 }}.
+}}
+cannot be divided by a mean proportional – an important result in ancient harmonics.<ref name=":0"/> [[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.<ref>
+{{cite journal
+ |last = Barker |first = A.
+ |year = 1994
+ |title = Ptolemy's Pythagoreans, Archytas, and Plato's conception of mathematics
+ |url = https://www.jstor.org/stable/4182463
+ |journal = Phronesis
+ |volume = 39  |issue=2  |pages=113–135
+ |jstor = 4182463  |issn = 0031-8868
+}}
+</ref>
+Later tradition regarded Archytas as the founder of mathematical [[mechanics]].<ref name="Diogenes Laërtius">
-Later tradition regarded Archytas as the founder of mathematical [[mechanics]].<ref name="Diogenes Laërtius">{{harvnb|Laërtius|1925|loc=§ 83}}: ''Vitae philosophorum''</ref> [[Vitruvius]] includes him in a list of twelve authors who wrote works on mechanics.<ref>[[Vitruvius]], ''De architectura'', vii.14.</ref> Thomas Nelson Winter presents evidence that the pseudo-Aristotelian ''[[Mechanics (Aristotle)|Mechanical Problems]]'' might have been authored by Archytas and later misattributed.<ref>Thomas Nelson Winter, "[http://digitalcommons.unl.edu/classicsfacpub/68/ The Mechanical Problems in the Corpus of Aristotle]," DigitalCommons@University of Nebraska - Lincoln, 2007.</ref> 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.<ref>{{Cite web |title=Rocket History - B.C. Era |url=https://www.grc.nasa.gov/www/k-12/rocket/BottleRocket/BCera.htm |access-date=2024-02-21 |website=www.grc.nasa.gov}}</ref><ref>[[Aulus Gellius]], "Attic Nights", Book X, 12.9 at [https://penelope.uchicago.edu/Thayer/E/Roman/Texts/Gellius/10*.html LacusCurtius]</ref><ref>[http://www.tmth.edu.gr/en/aet/1/14.html ARCHYTAS OF TARENTUM], Technology Museum of Thessaloniki, Macedonia, Greece. {{webarchive|url=https://web.archive.org/web/20081226181400/http://www.tmth.edu.gr/en/aet/1/14.html|date=December 26, 2008}}</ref><ref>{{cite book |author=Leofranc Holford-Strevens |title=Aulus Gellius: An Antonine Author and his Achievement |date=2005 |publisher=Oxford University Press |isbn=0-19-928980-8 |edition=Revised paperback}}</ref><ref>{{cite EB1911|wstitle=Archytas|volume=2|page=446}}</ref>
+{{harvnb|Laërtius|1925|loc=§ 83}}: ''Vitae philosophorum''
+</ref>
+[[Vitruvius]] includes him in a list of twelve authors who wrote works on mechanics.<ref>
+{{cite book
+ |author = [[Vitruvius]]
+ |title = De architectura  |lang = la
+ |title-link = De architectura
+ |trans-title = On Archetecture
+ |at = vii.14
+}}
+</ref>
+T.N. Winter presents evidence that the pseudo-Aristotelian ''[[Mechanics (Aristotle)|Mechanical Problems]]'' might have been authored by Archytas and later mis-attributed to [[Aristotle]].<ref>
+{{cite report
+ |first = Thomas Nelson |last = Winter
+ |year = 2007
+ |title = The Mechanical Problems in the Corpus of Aristotle
+ |series = Digital Commons
+ |publisher = [[University of Nebraska]]
+ |place = Lincoln, NB
+ |url = http://digitalcommons.unl.edu/classicsfacpub/68/
+}}
+</ref>
+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&bsp;meters.<ref>
+{{cite web
+ |title = Rocket history – B.C. era
+ |website = grc.nasa.gov
+ |url = https://www.grc.nasa.gov/www/k-12/rocket/BottleRocket/BCera.htm
+ |access-date = 2024-02-21
+}}
+</ref><ref>
+{{cite book
+ |author = [[Aulus Gellius]]
+ |section = Lacus Curtius
+ |title = Attic Nights
+ |at = Book&nbsp;X, 12.9
+ |publisher = [[University of Chicago]]
+ |section-url = https://penelope.uchicago.edu/Thayer/E/Roman/Texts/Gellius/10*.html
+ |via = penelope.uchicago.edu
+}}
+</ref><ref>
+{{cite report
+ |title = Archytas of Tarentum
+ |publisher = {{math|Κέντρο Διάδοσης Επιστημών}} {{math|και Μουσείο Τεχνολογίας}} [<nowiki/><i>[[Thessaloniki Science Center and Technology Museum]]</i><nowiki/>]
+ |place = Thessaloniki, Macedonia, Greece
+ |url = http://www.tmth.edu.gr/en/aet/1/14.html
+ |url-status = dead  |access-date = 23 March 2024
+ |archive-url = https://web.archive.org/web/20081226181400/http://www.tmth.edu.gr/en/aet/1/14.html
+ |archive-date = December 26, 2008
+}}
+</ref><ref>
+{{cite book
+ |first = Leofranc |last = Holford-Strevens
+ |year = 2005
+ |title = Aulus Gellius: An Antonine author and his achievement
+ |edition = revised paperback
+ |publisher = Oxford University Press
+ |place = Oxford, UK
+ |isbn = 0-19-928980-8
+}}
+</ref><ref>
+{{cite EB1911|wstitle=Archytas|volume=2|page=446}}
+</ref>
 ==Notes==

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