invented "the most compelling argument ever produced for the infinity of space" Richard Sorabji
Archytas (/ˈɑːrkɪtəs/; Greek: Ἀρχύτας; 435/410–360/350 BC[2]) was an Ancient Greek philosopher, mathematician, astronomer, statesman, and strategist. He was a scientist of the Pythagorean school and famous for being the reputed founder of mathematical mechanics, as well as a good friend of Plato.[3]
Life and work
Archytas was born in Tarentum, Magna Graecia and was the son of Mnesagoras or Histiaeus. For a while, he was taught by Philolaus, and was a teacher of mathematics to Eudoxus of Cnidus. Archytas and Eudoxus' student was Menaechmus. As a Pythagorean, Archytas believed that only arithmetic, not geometry, could provide a basis for satisfactory proofs.[4]
Archytas is believed to be the founder of mathematical mechanics.[5] As only described in the writings of Aulus Gellius five centuries after him, he was reputed to have designed and built the first artificial, selfpropelled flying device, a birdshaped model propelled by a jet of what was probably steam, said to have actually flown some 200 meters.[6][7] This machine, which its inventor called The pigeon, may have been suspended on a wire or pivot for its flight.[8][9] Archytas also wrote some lost works, as he was included by Vitruvius in the list of the twelve authors of works of mechanics.[10] Thomas Nelson Winter presents evidence that the pseudoAristotelian Mechanical Problems was actually authored by Archytas and misattributed.[11]
Archytas named the harmonic mean, important much later in projective geometry and number theory, though he did not invent it.[12] According to Eutocius, Archytas solved the problem of doubling the cube (the socalled Delian problem) in his manner (though he believed "that only arithmetic, not geometry", could provide a basis for satisfactory proofs) with a geometric construction.[13] Hippocrates of Chios before, reduced this problem to finding mean proportionals. Archytas' theory of proportions is treated in book VIII of Euclid's Elements, where is the construction for two proportional means, equivalent to the extraction of the cube root. According to Diogenes Laërtius, this demonstration, which uses lines generated by moving figures to construct the two proportionals between magnitudes, was the first in which geometry was studied with concepts of mechanics.[14] The Archytas curve, which he used in his solution of the doubling the cube problem, is named after him.
Politically and militarily, Archytas appears to have been the dominant figure in Tarentum in his generation, somewhat comparable to Pericles in Athens a halfcentury 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. He was allegedly undefeated as a general, in Tarentine campaigns against their southern Italian neighbors. The Seventh Letter of Plato asserts that Archytas attempted to rescue Plato during his difficulties with Dionysius II of Syracuse. In his public career, Archytas had a reputation for virtue as well as efficacy. 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 (i.e., how does a society obtain good rulers like Archytas, instead of bad ones like Dionysius II?).
Archytas may have drowned in a shipwreck in the shore of Mattinata, where his body laid unburied on the shore until a sailor humanely cast a handful of sand on it. Otherwise, he would have had to wander on this side of the Styx for a hundred years, such the virtue of a little dust, munera pulveris, as Horace calls it in Ode 1.28 on which this information on his death is based. The poem, however, is difficult to interpret and it is not certain that the shipwrecked and Archytas are in fact the same person.
The Flammarion Woodcut
The crater Archytas on the Moon is named in his honour.
Archytas curve
The Archytas curve
The Archytas curve is created by placing a semicircle (with a diameter of d) on the diameter of one of the two circles of a cylinder (which also has a diameter of d) such that the plane of the semicircle is at right angles to the plane of the circle and then rotating the semicircle about one of its ends in the plane of the cylinder's diameter. This rotation will cut out a portion of the cylinder forming the Archytas curve.[15]
Another way of thinking of this construction is that the Archytas curve is basically the result of cutting out a torus formed by rotating a hemisphere of diameter d out of a cylinder also of diameter d. A cone can go through the same procedures also producing the Archytas curve. Archytas used his curve to determine the construction of a cube with a volume of one third of that of a given cube.
Notes
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.
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
Laërtius 1925, § 83: Vitae philosophorum
Aulus Gellius, "Attic Nights", Book X, 12.9 at LacusCurtius[permanent dead link]
ARCHYTAS OF TARENTUM, Technology Museum of Thessaloniki, Macedonia, Greece Archived December 26, 2008, at the Wayback Machine
Modern rocketry[permanent dead link]
"Automata history". Archived from the original on 20021205. Retrieved 20181128.
Vitruvius, De architectura, vii.14.
Thomas Nelson Winter, "The Mechanical Problems in the Corpus of Aristotle," DigitalCommons@University of Nebraska  Lincoln, 2007.
J. J. O'Connor and E. F. Robertson. Archytas of Tarentum. The MacTutor History of Mathematics archive. Visited 11 August 2011.
Eutocius, commentary on Archimedes' On the sphere and cylinder.
Plato blamed Archytas for his contamination of geometry with mechanics (Plutarch, 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.
"Archived copy". Archived from the original on 20080718. Retrieved 20150329.
References
Laërtius, Diogenes (1925). "Pythagoreans: Archytas" . Lives of the Eminent Philosophers. 2:8. Translated by Hicks, Robert Drew (Two volume ed.). Loeb Classical Library.
Further reading
von Fritz, Kurt (1970). "Archytas of Tarentum". Dictionary of Scientific Biography. 1. New York: Charles Scribner's Sons. pp. 231–233. ISBN 0684101149. on line [1]
Huffman, Carl A. Archytas of Tarentum, Cambridge University Press, 2005, ISBN 0521837464
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.
PseudoAristotle, Mechanica – Greek text and English translation
Complete fragments (Greek–Spanish bilingual edition)
Fragments and Life of Archytas
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Ancient Greek schools of philosophy
PreSocratic
Ionian
Epimenides of Knossos Pherecydes of Syros Diogenes Metrodorus of Lampsacus Xenophanes Xeniades Theodorus of Cyrene Anacharsis
Milesian
Ephesian
Heraclitus Cratylus Antisthenes
Atomist
Leucippus Democritus
Italian
Hippo Musaeus of Athens Themistoclea
Pythagorean
Pythagoras Hippasus Philolaus Archytas Alcmaeon Brontinus Theano Arignote Myia Damo Calliphon Hermotimus Metrodorus of Cos Eurytus
Eleatic
Parmenides Zeno Melissus
Pluralist
Anaxagoras Archelaus Empedocles
Sophist
Protagoras Gorgias Prodicus Hippias Antiphon Lycophron Damon Callicles Thrasymachus Euthydemus Dionysodorus Euenus Critias
Socratic
Socrates Xenophon Cebes of Thebes Simmias of Thebes
Cynic
Antisthenes Diogenes Diodorus Zoilus Onesicritus Philiscus Crates Hipparchia Metrocles Monimus Cleomenes Bion Sotades Menippus Menedemus Cercidas Teles Meleager Favonius Demetrius Dio Chrysostom Agathobulus Secundus Demonax Peregrinus Proteus Theagenes Oenomaus Pancrates Crescens Heraclius Horus Asclepiades Sallustius
Cyrenaic
Aristippus Arete of Cyrene Aristippus the Younger Theodorus the Atheist Antipater of Cyrene Aristotle of Cyrene Hegesias of Cyrene Anniceris Dionysius the Renegade Euhemerus
Eretrian
Phaedo of Elis Menedemus Asclepiades of Phlius
Megarian
Euclid of Megara Ichthyas Thrasymachus Eubulides Stilpo Nicarete Pasicles Bryson
Dialectical
Clinomachus Apollonius Cronus Euphantus Dionysius Diodorus Cronus Philo Alexinus Panthoides
Peripatetic
Aristotle Aristoxenus Clearchus of Soli Dicaearchus Eudemus of Rhodes Theophrastus Strato of Lampsacus Lyco of Troas Aristo of Ceos Critolaus Diodorus of Tyre Erymneus Andronicus of Rhodes Cratippus Andronicus of Rhodes Boethus of Sidon Aristocles of Messene Aspasius Adrastus Alexander of Aphrodisias Themistius Olympiodorus the Elder
Platonic
Plato Eudoxus Philip of Opus Aristonymus Coriscus Erastus of Scepsis Demetrius of Amphipolis Euaeon of Lampsacus Heraclides Python of Aenus Hestiaeus of Perinthus Lastheneia of Mantinea Timolaus of Cyzicus Speusippus Axiothea of Phlius Heraclides Ponticus Menedemus of Pyrrha Xenocrates Crantor Polemon Crates of Athens
Hellenistic
Academic Skeptic
Middle
Arcesilaus Diocles of Cnidus Lacydes Telecles Evander Hegesinus
New
Carneades Hagnon of Tarsus Metrodorus of Stratonicea Clitomachus Charmadas Aeschines of Neapolis Philo of Larissa Cicero Dio of Alexandria
Epicurean
Epicurus Polyaenus Metrodorus Batis Leontion Carneiscus Idomeneus Hermarchus Colotes Themista Leonteus Polystratus Dionysius of Lamptrai Basilides Philonides Diogenes of Tarsus Alcaeus and Philiscus Apollodorus Demetrius Lacon Zeno of Sidon Amafinius Rabirius Titus Albucius Phaedrus Philodemus Lucretius Patro Catius Siro Diogenes of Oenoanda
Middle Platonic
Antiochus Philo of Alexandria Plutarch Justin Martyr Gaius Albinus Alcinous Apuleius Atticus Maximus of Tyre Numenius of Apamea Longinus Clement of Alexandria Origen the Pagan Calcidius
Neoplatonist
Ammonius Saccas Plotinus Disciples Origen Amelius Porphyry Iamblichus Sopater Eustathius of Cappadocia Sosipatra Aedesius Dexippus Chrysanthius Theodorus of Asine Julian Sallustius Maximus of Ephesus Eusebius of Myndus Priscus of Epirus Antoninus Gregory of Nyssa Hypatia Augustine Macrobius Plutarch of Athens Hierius Asclepigenia Hierocles Syrianus Hermias Aedesia Proclus Ammonius Hermiae Asclepiodotus Hegias Zenodotus Marinus Agapius Isidore Damascius Simplicius Priscian
Neopythagorean
Nigidius Figulus Apollonius of Tyana Moderatus of Gades Nicomachus Alexicrates Anaxilaus Bolus of Mendes Cronius Damis Numenius of Apamea Secundus the Silent Quintus Sextius Sotion Theon of Smyrna
Pyrrhonist
Pyrrho Aenesidemus Agrippa the Skeptic Arcesilaus Hecataeus of Abdera Heraclides of Tarentum Herodotus of Tarsus Menodotus of Nicomedia Nausiphanes Sextus Empiricus Theodas of Laodicea Timon of Phlius
Stoic
Greek
Zeno of Citium Persaeus Aratus of Soli Athenodorus of Soli Aristo of Chios Apollophanes of Antioch Dionysius the Renegade Sphaerus Herillus of Carthage Cleanthes Eratosthenes Hermagoras of Amphipolis Chrysippus Dioscorides Aristocreon Zeno of Tarsus Eudromus Crates of Mallus Diogenes of Babylon Zenodotus Apollodorus of Seleucia Basilides Antipater of Tarsus Apollodorus of Athens Archedemus of Tarsus Panaetius of Rhodes Boethus of Sidon Polemon of Athens Marcus Vigellius Heraclides of Tarsus Dardanus Mnesarchus Publius Rutilius Rufus Stilo Dionysius of Cyrene Quintus Lucilius Balbus Hecato of Rhodes Diotimus the Stoic Posidonius Crinis Proclus of Mallus Diodotus the Stoic Geminus of Rhodes Athenodoros Cordylion Apollonius of Tyre Cato the Younger Antipater of Tyre Porcia Catonis Apollonides Jason of Nysa Athenodoros Cananites Quintus Sextius Arius Didymus
Roman
Attalus Papirius Fabianus Seneca Thrasea Paetus Lucius Annaeus Cornutus Chaeremon of Alexandria Paconius Agrippinus Publius Egnatius Celer Persius Helvidius Priscus Arulenus Rusticus Musonius Rufus Fannia Euphrates the Stoic Cleomedes Epictetus Hierocles Flavius Arrianus Basilides Apollonius of Chalcedon Claudius Maximus Junius Rusticus Marcus Aurelius

Ancient Greek and Hellenistic mathematics (Euclidean geometry)
Mathematicians
(timeline)
Anaxagoras Anthemius Archytas Aristaeus the Elder Aristarchus Apollonius Archimedes Autolycus Bion Bryson Callippus Carpus Chrysippus Cleomedes Conon Ctesibius Democritus Dicaearchus Diocles Diophantus Dinostratus Dionysodorus Domninus Eratosthenes Eudemus Euclid Eudoxus Eutocius Geminus Heliodorus Heron Hipparchus Hippasus Hippias Hippocrates Hypatia Hypsicles Isidore of Miletus Leon Marinus Menaechmus Menelaus Metrodorus Nicomachus Nicomedes Nicoteles Oenopides Pappus Perseus Philolaus Philon Philonides Porphyry Posidonius Proclus Ptolemy Pythagoras Serenus Simplicius Sosigenes Sporus Thales Theaetetus Theano Theodorus Theodosius Theon of Alexandria Theon of Smyrna Thymaridas Xenocrates Zeno of Elea Zeno of Sidon Zenodorus
Treatises
Almagest Archimedes Palimpsest Arithmetica Conics (Apollonius) Catoptrics Data (Euclid) Elements (Euclid) Measurement of a Circle On Conoids and Spheroids On the Sizes and Distances (Aristarchus) On Sizes and Distances (Hipparchus) On the Moving Sphere (Autolycus) Euclid's Optics On Spirals On the Sphere and Cylinder Ostomachion Planisphaerium Sphaerics The Quadrature of the Parabola The Sand Reckoner
Problems
Angle trisection Doubling the cube Squaring the circle Problem of Apollonius
Concepts/definitions
Circles of Apollonius
Apollonian circles Apollonian gasket Circumscribed circle Commensurability Diophantine equation Doctrine of proportionality Golden ratio Greek numerals Incircle and excircles of a triangle Method of exhaustion Parallel postulate Platonic solid Lune of Hippocrates Quadratrix of Hippias Regular polygon Straightedge and compass construction Triangle center
Results
In Elements
Angle bisector theorem Exterior angle theorem Euclidean algorithm Euclid's theorem Geometric mean theorem Greek geometric algebra Hinge theorem Inscribed angle theorem Intercept theorem Pons asinorum Pythagorean theorem Thales's theorem Theorem of the gnomon
Apollonius
Apollonius's theorem
Other
Aristarchus's inequality Crossbar theorem Heron's formula Irrational numbers Menelaus's theorem Pappus's area theorem Problem II.8 of Arithmetica Ptolemy's inequality Ptolemy's table of chords Ptolemy's theorem Spiral of Theodorus
Centers
Cyrene Library of Alexandria Platonic Academy
Other
Ancient Greek astronomy Greek numerals Latin translations of the 12th century Neusis construction
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