Archytas

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.{{Cite journal |last=Despotopoulos |first=Constantin |date=2004-11-01 |title=Archytas' Logismos and Logistika |url=https://www.pdcnet.org/pdc/bvdb.nsf/purchase?openform&fp=philinquiry&id=philinquiry_2004_0026_0003_0001_0009 |journal=Philosophical Inquiry |language=en |volume=26 |issue=3 |pages=1–9 |doi=10.5840/philinquiry200426311|url-access=subscription }} 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.{{Cite web |last=Johnson |first=M. R. |date=2008 |title=Sources for the Philosophy of Archytas |url=https://philarchive.org/rec/JOHSFT |access-date=2023-10-30 |website=philarchive.org |language=en}}

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.{{Cite journal |last=Lloyd |first=G. E. R. |date=1990 |title=Plato and Archytas in the "Seventh Letter" |url=https://www.jstor.org/stable/4182355 |journal=Phronesis |volume=35 |issue=2 |pages=159–174 |doi=10.1163/156852890X00097 |jstor=4182355 |issn=0031-8868|url-access=subscription }} 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 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.

{{cite journal |last=Furner |first=J. |year=2021 |title=Classification of the sciences in Greco-Roman antiquity |url=https://www.isko.org/cyclo/greco-roman |journal=Knowledge Organization |volume=48 |issue=7–8 |pages=499–534 |doi=10.5771/0943-7444-7-8-499|url-access=subscription }}

{{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 |language = en

}}

He is thought to have written a great number of works in the sciences, but only four fragments are generally believed to be authentic.

{{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

}}

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.

{{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?id=nvpeCAAAQBAJ&dq=Archytas+double&pg=PA407

|via = Google books

}}

{{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 |language = en

}}

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.{{efn|

Plato blamed Archytas for his contamination of geometry with mechanics:

{{cite book

|author=Plutarch

|title = Symposiacs

|at = Book VIII, Question 2

|url = http://ebooks.adelaide.edu.au/p/plutarch/symposiacs/chapter8.html#section80

|url-status = dead

|archive-url = https://web.archive.org/web/20080815001514/http://ebooks.adelaide.edu.au/p/plutarch/symposiacs/chapter8.html#section80

|archive-date = 2008-08-15

}}

:

: 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.

}} 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.

{{cite book

|author = Euclid

|title = Elements {{grey|[of Geometry]}}

|title-link = Euclid's Elements

|at = book VIII

}}

Archytas named the harmonic mean, important much later in projective geometry and number theory, though he did not discover it.

{{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, Scotland

|url = http://www-history.mcs.st-andrews.ac.uk/Biographies/Archytas.html

|access-date = 11 August 2011

}}

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 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. 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.

{{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

|doi = 10.1163/156852894321052135

|jstor = 4182463 |issn = 0031-8868

|url-access = subscription

}}

File:The flying pigeon of Archytas, Kotsanas Museum of Ancient Greek Technology.jpg, Athens, Greece.]]

Later tradition regarded Archytas as the founder of mathematical mechanics.

{{cite LotEP|chapter=Archytas|loc=§ 83}}: Vitae philosophorum

Vitruvius includes him in a list of twelve authors who wrote works on mechanics.

{{cite book

|author = Vitruvius

|title = De architectura |language = la

|title-link = De architectura

|trans-title = On Architecture

|at = vii.14

}}

T.N. Winter presents evidence that the pseudo-Aristotelian Mechanical Problems might have been authored by Archytas and later mis-attributed to Aristotle.

{{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/

}}

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:

{{cite conference

| first1 = Marco

| last1 = Vespa

| first2 = Isabel

| last2 = Ruffell

| year = 2020

| title = Archytas' dove in context: an investigation of the non-human agency between paradoxography, encyclopedism, and mechanics in the ancient world

| url = https://ancientmedicineandtechnology.wordpress.com/archytas-dove-in-context/

| journal = Ancient Medicine and Technology Seminar Series

}}

{{ cite book

| title = Noctes Atticae (Attic Nights)

| author = A. Cornelius Gellius

| publisher = Loeb Classical Library

| volume = X

| chapter = 12

| year = 1927

| translator = J. C. Rolfe

| url = https://penelope.uchicago.edu/Thayer/E/Roman/Texts/Gellius/10*.html

}}

{{Blockquote|text=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.

{{cite book

| title = Aulus Gellius und die >Noctus Atticae< — die literarische Konstruktion einer Sammlung

| last = Beer

| first = Beate

| publisher = De Gruyter

| year = 2020

| isbn = 9783110695083

| page = 88

}}{{cite journal

| last = Beall

| first = Stephen M.

| title = Homo Fandi Dulcissimus: The Role of Favorinus in the "Attic Nights" of Aulus Gellius

| journal = The American Journal of Philology

| year = 2001

| volume = 122

| number = 1

| page = 87

| doi = 10.1353/ajp.2001.0001

}}

{{notelist}}

{{reflist|25em}}

• {{cite encyclopedia |last=von Fritz |first=Kurt |title=Archytas of Tarentum |encyclopedia=Dictionary of Scientific Biography |volume=1 |pages=231–233 |publisher=Charles Scribner's Sons |location=New York |year=1970 |isbn=0-684-10114-9 }} on line [http://www.encyclopedia.com/science/dictionaries-thesauruses-pictures-and-press-releases/archytas-tarentum]
• Huffman, Carl A. Archytas of Tarentum, Cambridge University Press, 2005, {{ISBN|0-521-83746-4}}

{{wikiquote}}

{{wikisource author}}

{{commons category|Archytas of Tarentum|Archytas}}

• {{cite SEP |url-id=archytas |title=Archytas |last=Huffman |first=Carl}}
• {{MacTutor Biography|id=Archytas}}
• [http://digitalcommons.unl.edu/classicsfacpub/68/ Pseudo-Aristotle, Mechanica] – Greek text and English translation
• [https://archive.org/details/ArquitasDeTarentoFragmentaEtTestimonia-ArchytasOfTarentum Complete fragments (Greek–Spanish bilingual edition)]
• [http://demonax.info/doku.php?id=classical:archytas Fragments and Life of Archytas]

{{Greek schools of philosophy}}

{{Greek mathematics}}

{{Authority control}}

{{DEFAULTSORT:Archytas}}

Category:428 BC births

Category:347 BC deaths

Category:4th-century BC Greek philosophers

Category:Ancient Greek generals

Category:Ancient Greek music theorists

Category:Ancient Greek physicists

Category:Ancient Greek inventors

Category:Ancient Tarantines

Category:Pythagoreans of Magna Graecia

Category:5th-century BC Greek mathematicians

Category:4th-century BC Greek mathematicians