Archytas

Archytas (/ˈɑːrkɪtəs/; Greek: Ἀρχύτας; 428–347 BC) 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.

Archytas
Archytas of Tarentum MAN Napoli Inv5607
Bust from the Villa of the Papyri in Herculaneum, once identified as Archytas, now thought to be Pythagoras[1]
Born435 - 410 BC
Died347 BC (aged 80–81)
EraPre-Socratic philosophy
RegionWestern philosophy
SchoolPythagoreanism
Notable ideas
Archytas curve

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.[2]

Archytas is believed to be the founder of mathematical mechanics.[3] As only described in the writings of Aulus Gellius five centuries after him, he was reputed to have designed and built the first artificial, self-propelled flying device, a bird-shaped model propelled by a jet of what was probably steam, said to have actually flown some 200 meters.[4][5] This machine, which its inventor called The pigeon, may have been suspended on a wire or pivot for its flight.[6][7] Archytas also wrote some lost works, as he was included by Vitruvius in the list of the twelve authors of works of mechanics.[8] Thomas Winter has suggested that the pseudo-Aristotelian Mechanical Problems is an important mechanical work by Archytas, not lost after all, but misattributed.[9]

Archytas named the harmonic mean, important much later in projective geometry and number theory, though he did not invent it.[10] According to Eutocius, Archytas solved the problem of doubling the cube in his manner (though he believed "that only arithmetic, not geometry", could provide a basis for satisfactory proofs) with a geometric construction.[11] 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.[12] 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 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. 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 lay 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 crater Archytas on the Moon is named in his honour.

Archytas curve

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.[13]

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.

The Delian Problem

Dup cubo
Doubling the volume of a cube, given a single side. Solved by Archytas.

One of Archytas' most notable accomplishments comes in the form of a mathematical solution to The Delian Problem, more informally known as doubling the cube. The problem is as follows: given a cube that a side is known, construct a cube with double the original volume. The proof of his model comes from Eudemus, who in the late 4th century wrote a history of geometry, including solutions to this problem from multiple mathematicians and philosophers before him- namely Eudoxus and Menaechmus. Although Eudemus' work did not survive to current day, a transmission of his geometric solution does survive in the form of Eutocius' commentary on Archimedes' De Sphaera et Cylindro. Archytas' solution begins with the concept of mean proportionality and the construction of four similar triangles. Each triangle's hypotenuse and long leg are proportionally similar as the triangle increase in size, which is essentially today's version of similarity of triangles. Archytas then applied the mean proportionals for a given length of a cube. If the volume of the original cube is written as V1 = x3, where x represents the length of a side, we let k1 and k2 represent the proportionality constants, and the cube is then doubled so that a side length is now 2x, a mean proportional between the two can be written as . With the proportionals finished, Archytas completed the solution to his similar triangles as follows: If you cube the proportion of the original length of the side and solve using the mean proportional set, the solution comes to After using light algebra, where the k1 variable represents the edge of the newly doubled cube.

Harmonic Theory

Archytas developed an impactful physical theory for pitch when a string instrument is strum. His theory was that the pitch of the ensuing sound depends on the speed of the wave moving through its medium (usually air). Although incorrect, as pitch is correlated with frequency rather than wave velocity, Archytas' theory was the first of its kind that was both adopted and adapted by Plato and later Aristotle. By the time of his analysis, it was known from the Pythagorean diatonic scale that whole numbers alone accounted for musical intervals on a scale. Archytas's work on musical scales included a thorough proof that no mean proportional numbers, like the ones used in his solving of the double cube problem, exist between basic music intervals (the difference in pitch between two sounds). This is to say that the basic interval, does not include any mean proportional number, and cannot then be divided in half. The octave can be doubled without violating this rule, as multiplying a whole number by 2 will always result in a whole number, and can therefore be equated by two mean proportional ratios.

Cosmology

Due to the severe scarcity of resources for Archytas' direct work, it is difficult to pinpoint his exact thoughts on the universe. Through Eudemus and later Simplicius' commentary, however, his thought experiment in regards to the size of the universe remain intact to current day. The experiment is credited as being an influential spark though the early ages, even though Plato nor Aristotle bought the argument. In his experiment for others to participate and decide for themselves, Archytas tells of a scenario in which he is at the effective edge of the fixed stars. He says that if he outreaches his arm, or his stick (staff), that his hand will push the limit of what the edge is. He is then free to move into the newly created space and outstretch his staff once more, thus increasing the limit of space. With his argument, he attested that space, the region of the fixed stars, is infinite.[14] This thought persisted even through modern day, although it is important to note that his model has a defined edge, whereas some current models do not account for a defined edge of space.

Notes

  1. ^ Archita; Pitagora, Sito ufficiale del Museo Archeologico Nazionale di Napoli, retrieved 25 September 2012
  2. ^ Morris Kline, Mathematical Thought from Ancient to Modern Times Oxford University Press, 1972 p. 49
  3. ^ Laërtius 1925, § 83: Vitae philosophorum
  4. ^ Aulus Gellius, "Attic Nights", Book X, 12.9 at LacusCurtius
  5. ^ ARCHYTAS OF TARENTUM, Technology Museum of Thessaloniki, Macedonia, Greece Archived December 26, 2008, at the Wayback Machine
  6. ^ Modern rocketry
  7. ^ "Automata history". Archived from the original on 2002-12-05. Retrieved 2018-11-28.
  8. ^ Vitruvius, De architectura, vii.14.
  9. ^ Thomas Nelson Winter, "The Mechanical Problems in the Corpus of Aristotle," DigitalCommons@University of Nebraska - Lincoln, 2007.
  10. ^ J. J. O'Connor and E. F. Robertson. Archytas of Tarentum. The MacTutor History of Mathematics archive. Visited 11 August 2011.
  11. ^ Eutocius, commentary on Archimedes' On the sphere and cylinder.
  12. ^ 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.
  13. ^ http://mathforum.org/dr.math/faq/davies/cubedbl.htm
  14. ^ Heath, Thomas Little, Sir (1921). A history of Greek mathematics. New York: Dover Publications. p. 214. ISBN 0486240738. OCLC 7703465.

References

Further reading

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

External links

Archytas (crater)

Archytas is a lunar impact crater that protrudes into the northern edge of Mare Frigoris. To the northwest is the comparably sized crater Timaeus, and the smaller Protagoras lies in the opposite direction to the southeast. Further to the southwest, beyond the opposite edge of the mare, is the dark-floored crater Plato.

The rim of Archytas is sharp-edged and shows little appearance of erosion due to subsequent impacts. The outer wall is nearly circular, with a slight outward bulge in the southeast. The interior is rough, with a ring of material deposited at the base of the inner wall. Just to the east of the crater midpoint is a pair of central peaks.

The surface surrounding the crater is relatively smooth to the south due to the lava flows that formed the mare. The surface is more rugged to the north and northeast. The satellite crater Archytas B, located to the northwest of Archytas, forms a lava-flooded bay along the edge of the Mare Frigoris.

Archytas (disambiguation)

Archytas (428–347 BC) was an Ancient Greek philosopher, mathematician, astronomer, statesman, and strategist

Archytas can also refer to:

Archytas of Amphissa, a Greek poet who lived around the 3rd or 4th century BCE

Archytas of Mytilene, an ancient Greek musician whose date is uncertain

Archytas (crater), a lunar impact crater

Archytas (fly), a genus of flies in the family Tachinidae

14995 Archytas, a main-belt asteroid

Archytas (fly)

Archytas is a genus of flies in the family Tachinidae.

Archytas analis

Archytas analis is a species of bristle fly in the family Tachinidae. It is found in North America.

Archytas apicifer

Archytas apicifer is a medium to large sized (approximately 10-15 mm) Nearctic tachinid fly. The species name was authored by the German entomologist Johann Friedrich Jaennicke (1867) and presumably named after the Greek classical philosopher and mathematician Archytas. The larvae are parasites of several caterpillar species.

Archytas metallicus

Archytas metallicus is a species of bristle fly in the family Tachinidae. It is found in North America.

Archytas of Amphissa

Archytas (Ancient Greek: Ἀρχύτας) of Amphissa was a Greek poet who was probably a contemporary of Euphorion of Chalcis, about 300 BCE, since it was a matter of doubt with the ancients themselves whether the epic poem Γέρανος (Geranos) was the work of Archytas or Euphorion.Plutarch quotes from him a hexameter verse concerning the country of the Ozolian Locrians. Two other lines, which he is said to have inserted in the poem Hermes of Eratosthenes, are preserved in the writings of Stobaeus. He seems to have been the same person whom Diogenes Laërtius calls an epigrammatist, and upon whom Bion of Smyrna wrote an epigram which he quotes.

Archytas of Mytilene

Archytas (Ancient Greek: Ἀρχύτας) of Mytilene was a musician of ancient Greece who may perhaps also have been the author of the work Περὶ Αὐλῶν, which is sometimes ascribed to Archytas of Tarentum.

Archytas rufiventris

Archytas rufiventris is a species of bristle fly in the family Tachinidae. It is found in North America.

Brachypeza archytas

Brachypeza archytas, commonly known as the sage orchid, is an epiphytic orchid that is endemic to Christmas Island, an Australian territory in the north-eastern Indian Ocean. It has many cord-like roots, four or five leaves arranged like a fan and a large number of small, crowded, short-lived, white flowers.

Eudoxus of Cnidus

Eudoxus of Cnidus (; Ancient Greek: Εὔδοξος ὁ Κνίδιος, Eúdoxos ho Knídios; c. 390? – c. 337 BC) was an ancient Greek astronomer, mathematician, scholar, and student of Archytas and Plato. All of his works are lost, though some fragments are preserved in Hipparchus' commentary on Aratus's poem on astronomy. Sphaerics by Theodosius of Bithynia may be based on a work by Eudoxus.

Mechanics (Aristotle)

Mechanics (or Mechanica or Mechanical Problems; Greek: Μηχανικά) is a text traditionally attributed to Aristotle, though his authorship of it is disputed. Thomas Winter has suggested that the author was Archytas. However, Coxhead says that it is only possible to conclude that the author was one of the Peripatetics.During the Renaissance, an edition of this work was published by Francesco Maurolico.

Musical system of ancient Greece

The musical system of ancient Greece evolved over a period of more than 500 years from simple scales of tetrachords, or divisions of the perfect fourth, to The Perfect Immutable System, encompassing a span of fifteen pitch keys (see tonoi below) (Chalmers 1993, chapt. 6, p. 99)

Any discussion of ancient Greek music, theoretical, philosophical or aesthetic, is fraught with two problems: there are few examples of written music, and there are many, sometimes fragmentary, theoretical and philosophical accounts. This article provides an overview that includes examples of different kinds of classification while also trying to show the broader form evolving from the simple tetrachord to the system as a whole.

Ninth Letter (Plato)

The Ninth Letter of Plato, also called Epistle IX or Letter IX, is an epistle that is traditionally ascribed to Plato. In the Stephanus pagination, it spans III. 357d–358b.

The letter is ostensibly written to Archytas of Tarentum, whom Plato met during his first trip to Sicily in 387 BC. Archytas had sent a letter with Archippus and Philonides, two Pythagoreans who had gone on to mention to Plato that Archytas was unhappy about not being able to get free of his public responsibilities. The Ninth Letter is sympathetic, noting that nothing is more pleasant than to attend to one's own business, especially when that business is the one that Archytas would engage in (viz. philosophy). Yet everyone has responsibilities to one's fatherland (πατρίς), parents, and friends, to say nothing of the need to provide for daily necessities. When the fatherland calls, it is improper not to answer, especially as a refusal will leave politics to the care of worthless men. The letter then declares that enough has been said of this subject, and concludes by noting that Plato will take care of Echecrates, who is still a youth (νεανίσκος), for Archytas' sake and that of Echecrates' father, as well as for the boy himself.

R. G. Bury describes the Ninth Letter as "a colourless and commonplace effusion which we would not willingly ascribe to Plato, and which no correspondent of his would be likely to preserve;" he also notes "certain peculiarities of diction which point to a later hand." A character by the name of Echecrates also appears in the Phaedo, though Bury suggests that he, if the same person mentioned here, could hardly have been called a youth by the time Plato met Archytas. Despite the fact that Cicero attests to its having been written by Plato, most scholars consider it a literary forgery.

Papilio menatius

Papilio menatius is a butterfly of the family Papilionidae.

Septimal comma

A septimal comma is a small musical interval in just intonation that contains the number seven in its prime factorization. There is more than one such interval, so the term septimal comma is ambiguous, but it most commonly refers to the interval 64/63 (27.26 cents). Play

Use of septimal commas introduces new intervals that extend tuning beyond common-practice, extending music to the 7-limit, including the 7/6 septimal minor third, the 7/5 septimal tritone and the 8/7 septimal major second. Composers who made extensive use of these intervals include Harry Partch and Ben Johnston. Johnston uses a "7" as an accidental to indicate a note is lowered 49 cents, or an upside down seven ("ㄥ" or "") to indicate a note is raised 49 cents (36/35).

Tachininae

Tachininae is a subfamily of flies in the family Tachinidae.

Twelfth Letter

The Twelfth Letter of Plato, also known as Epistle XII or Letter XII, is an epistle that tradition has ascribed to Plato, though it is almost certainly a literary forgery. Of all the Epistles, it is the only one that is followed by an explicit denial of its authenticity in the manuscripts. In the Stephanus pagination, it spans 359c–e of Vol. III.

Like the Ninth Letter, the Twelfth Letter is purportedly addressed to Archytas. It thanks him for sending Plato some treatises, which it then goes on to praise effusively, declaring its author worthy of his ancestors and including in their number Myrians, colonists from Troy during the reign of Laomedon. It then promises to send to Archytas some of Plato's unfinished treatises.

Diogenes Laërtius preserves this letter in his Lives and Opinions of Eminent Philosophers, as well as a letter from Archytas which presumably occasioned the Twelfth Letter; This letter points to the treatises having been those of Ocellos of Lucania, a Pythagorean. Because the writings which are attributes to Ocellos are forgeries from the First Century BC, the Twelfth Letter is probably also a forgery, and by the same forger, intended to stamp the treatises with Plato's authority. There is no other mention of a Trojan colony in Italy from the reign of Laomedon, let alone of Lucania or the Lucani having been descended from the otherwise unknown "Myrians." R. G. Bury also notes that the Twelfth Letter, along with the Ninth, spell Archytas with an α, whereas Plato spells it in more authoritative epistles with an η (Αρχύτης).

Vacerra bonfilius

Vacerra bonfilius is a butterfly in the family Hesperiidae. It is found in Panama, Brazil, Bolivia and Venezuela.

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