stanford - archytas

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Archytas First published Thu Jun 26, 2003; substantive revision Mon Jul 25, 2011

Copyright © 2011 byCarl Huffman

Archytas of Tarentum was a Greek mathematician, political leader and philosopher, active in the first half of the fourth century BC (i.e., during Plato'slifetime). He was the last prominent figure in the early Pythagorean tradition andthe dominant political figure in Tarentum, being elected general sevenconsecutive times. He sent a ship to rescue Plato from the clutches of the tyrantof Syracuse, Dionysius II, in 361, but his personal and philosophical connections

to Plato are complex, and there are many signs of disagreement between the two philosophers. A great number of works were forged in Archytas' name starting inthe first century BC, and only four fragments of his genuine work survive,although these are supplemented by a number of important testimonia. Archytaswas the first to solve one of the most celebrated mathematical problems inantiquity, the duplication of the cube. We also have his proof showing that ratiosof the form (n+1) : n, which are important in music theory, cannot be divided bya mean proportional. He was the most sophisticated of the Pythagorean harmonictheorists and provided mathematical accounts of musical scales used by the

practicing musicians of his day. Fr. 1 of Archytas may be the earliest text toidentify the group of four canonical sciences (logistic [arithmetic], geometry,astronomy and music), which would become known as thequadrivium in themiddle ages. There are also some indications that he contributed to thedevelopment of the science of optics and laid the mathematical foundations forthe science of mechanics. He saw the ultimate goal of the sciences as thedescription of individual things in the world in terms of ratio and proportion andaccordingly regarded logistic, the science of number and proportion, as themaster science. Rational calculation and an understanding of proportion werealso the bases of the just state and of the good life for an individual. He gavedefinitions of things that took account of both their matter and their form.Although we have little information about his cosmology, he developed the mostfamous argument for the infinity of the universe in antiquity.

1. Life and Works1.1 Family, Teachers, and Pupils; DateArchytas, son of Hestiaeus (see Aristoxenus in Diels-Kranz 1952, chap. 47,

passage A1; abbreviated as DK47 A1), lived in the Greek city of Tarentum, on

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1.2 SourcesApart from the surviving fragments of his writings, our knowledge of Archytas'life and work depends heavily on authors who wrote in the second half of thefourth century, in the fifty years after Archytas' death. Archytas' importance bothas an intellectual and as a political leader is reflected in the number of writingsabout him in this period, although only fragments of these works have been

preserved. Aristotle wrote a work in three volumes on the philosophy ofArchytas, more than on any other of his predecessors, as well as a second work,consisting of a summary of Plato's Timaeus and the writings of Archytas (A13).Unfortunately almost nothing of these works has survived. Aristotle's pupilEudemus discussed Archytas prominently in his history of geometry (A6 andA14) and in his work on physics (A23 and A24). Another pupil of Aristotle's,Aristoxenus, wrote a Life of Archytas, which is the basis for much of the

biographical tradition about him (A1, A7, A9). Aristoxenus (375-ca. 300) was ina good position to have accurate information about Archytas. He was born inTarentum and grew up during the height of Archytas' prominence in the city. Inaddition to whatever personal knowledge he had of Archytas, he draws on hisown father Spintharus, who was a younger contemporary of Archytas, as asource (e.g., A7). Aristoxenus began his philosophical career as a Pythagoreanand studied with the Pythagorean Xenophilus at Athens, so that it is notsurprising that his portrayal of Archytas is largely positive. Nonetheless,Archytas' opponents are given a fair hearing (e.g., Polyarchus in A9), andArchytas himself is represented as not without small flaws of character (A7).Other fourth-century sources such as the Seventh Letter in the Platonic corpusand Demosthenes' (?) Erotic Oration focus on the connection between Archytasand Plato (see below).

1.3 Archytas and TarentumArchytas is unique among Greek philosophers for the prominent role he playedin the politics of his native city. He was elected general ( stratêgos) seven years insuccession at one point in his career (A1), a record that reminds us of Pericles atAthens. His election was an exception to a law, which forbade election insuccessive years, and thus attests to his reputation in Tarentum. Aristoxenus

reports that Archytas was never defeated in battle and that, when at one point hewas forced to withdraw from his post by the envy of his enemies, the Tarentinesimmediately suffered defeat (A1). He probably served as part of a board ofgenerals (there was a board of ten at Athens). The analogy with Athens suggeststhat as a general he may also have had special privileges in addressing theassembly at Tarentum on issues of importance to the city, so that his position asgeneral gave him considerable political as well as military power. At some pointin his career, he may have been designated as ageneral autokratôr (“plenipotentiary”) (A2), which gave him special latitude indealing with diplomatic and military matters without consulting the assembly,

although this was not dictatorial power and all arrangements probably requiredthe eventual approval of the assembly. We do not know when Archytas served


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his seven successive years as general. Some have supposed that they mustcoincide with the seven year period which includes Plato's second and third visitsto Italy and Sicily, 367 – 361 (e.g., Wuilleumier 1939, 68 – 9), but Archytas neednot have been stratêgos to play the role assigned to him during these years inthe Seventh Letter . The evidence suggests that most of Archytas' militarycampaigns were directed not at other Greeks but at native Italic peoples such asthe Messapians and Lucanians, with whom Tarentum had been in constantconflict since its founding.

It is important to recognize that the Tarentum in which Archytas exercised suchinfluence was not some insignificant backwater. Spartan colonists founded it in706. It was initially overshadowed by other Greek colonies in southern Italy suchas Croton, although it had the best harbor on the south coast of Italy and was thenatural stopping point for any ships sailing west from mainland Greece. Archytaswill have grown up in a Tarentum that, in accord with its foundation by Sparta,

took the Peloponnesian and Syracusan side against Athens in the PeloponnesianWar (Thuc. VI 44; VI 104; VII 91). Athens allied with the Messapians (Thuc.VII 33), the long-standing enemy of the Tarentines, against whom Archytaswould later lead expeditions (A7). After the Peloponnesian War, Tarentumappears to have avoided direct involvement in the conflict between the tyrant ofSyracuse, Dionysius I, and a league of Greek cities in southern Italy headed byCroton. After Dionysius crushed the league, Tarentum emerged as the most

powerful Greek state in southern Italy and probably became the new head of theleague of Italiot Greek cities (A2). In the period from 380 – 350, when Archytaswas in his prime and old age, Tarentum was one of the most powerful cities in

the Greek world (Purcell 1994, 388). Strabo's description of its military might(VI 3.4) compares favorably with Thucydides' account of Athens at the beginning of the Peloponnesian war (II. 13).

Despite its ancestral connections to Sparta, which was an oligarchy, Tarentumappears to have been a democracy during Archytas' lifetime. According toAristotle ( Pol . 1303a), the democracy was founded after a large part of theTarentine aristocracy was killed in a battle with a native people, the Iapygians, in473. Herodotus confirms that this was the greatest slaughter of Greeks of whichhe was aware (VII 170). There is no evidence that Tarentum was anything but ademocracy between the founding of the democracy in 473 and Archytas' deathca. 350. Some scholars have argued that Tarentum's ties to Sparta and thesupposed predilection of the Pythagoreans for aristocracy will have insured thatTarentum did not remain a democracy long and that it was not a democracyunder Archytas (Minar 1942, 88 – 90; Ciaceri 1927 – 32, II 446 – 7). Strabo,however, explicitly describes Tarentum as a democracy at the time of itsflourishing under Archytas (A4), and the descriptions of Archytas' power inTarentum stress his popularity with the masses and his election as general by thecitizens (A1 and A2). Finally, Aristotle's account of the structure of the Tarentinegovernment in the fourth century ( Pol . 1291b14), while possibly consistent withother forms of government, makes most sense if Tarentum was a democracy. The


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same is true of fr. B3 of Archytas, with its emphasis on a more equal distributionof wealth.

1.4 Archytas and Plato

Archytas was most famous in antiquity and is most famous in the modern worldfor having sent a ship to rescue Plato from the tyrant of Syracuse, Dionysius II, in361. In both the surviving ancient lives of Archytas (by Diogenes Laertius, VIII79 – 83, and in the Suda) the first thing mentioned about him, after the name of hiscity-state and his father, is his rescue of Plato (A1 and A2). This story is told ingreatest detail in the Seventh Letter ascribed to Plato. It has accordingly beentypical to identify Archytas as “the friend of Plato” (Mathieu 1987). Archytas

first met Plato over twenty years earlier, when Plato visited southern Italy andSicily for the first time in 388/7, during his travels after the death of Socrates (Pl.[?], Ep. VII 324a, 326b-d; Cicero, Rep. I 10. 16; Philodemus, Acad. Ind. X 5 – 11;

cf. D.L. III 6). Some scholars have seen Archytas as the dominant figure in therelationship (Zhmud 2006, 93) and even as the “new model philosopher forPlato” (Vlastos 1991, 129) and the archetype of Plato's philosopher -king (Guthrie1962, 333). The actual situation appears to be considerably more complicated.The ancient evidence, apart from the Seventh Letter , presents the relationship

between Archytas and Plato in diametrically opposed ways. One tradition does present Archytas as the Pythagorean master at whose feet Plato sat, after Socrateshad died (e.g., Cicero, Rep. I 10.16), but another tradition makes Archytas thestudent of Plato, to whom he owed his fame and success in Tarentum(Demosthenes [?], Erotic Oration 44).

The Seventh Letter itself is of contested authenticity, although most scholarsregard it either as the work of Plato himself or of a student of Plato who hadconsiderable familiarity with Plato's involvement in events in Sicily (see e.g.,Brisson 1987; Lloyd 1990; Schofield 2000). The letter appears to serve as anapologia for Plato's involvement in events in Sicily. Lloyd has recently argued,however, that the letter also serves to distance Plato from Pythagoreanism andfrom Archytas (1990). Nothing in the letter suggests that Plato was ever the pupilof Archytas; instead the relationship is much closer to that presented in the EroticOration. Plato is presented as the dominant figure upon whom Archytas depends

both philosophically and politically. Archytas writes to Plato claiming thatDionysius II has made great progress in philosophy, in order to urge Plato tocome to Sicily a third time (339d-e). These claims are belied as soon as Platoarrives (340b). The letter thus suggests that, far from being the Pythagoreanmaster from whom Plato learned his philosophy, Archytas had a very imperfectunderstanding of what Plato considered philosophy to be. The letter makes clearthat Plato does have a relationship of xenia, “guest-friendship,” with Archytas andothers at Tarentum (339e, 350a). This relationship was probably established onPlato's first visit in 388/7, since Plato uses it as a basis to establish a similarrelationship between Archytas and Dionysius II during his second visit in 367(338c). It is also the relationship in terms of which Plato appeals to Archytas forhelp, when he is in danger after the third trip to Sicily goes badly (350a). Such a


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friendship need not imply any close personal intimacy, however. Aristotleclassifies xenia as a friendship for utility and points out that such friends do notnecessarily spend much time together or even find each other's company pleasant( EN 1156a26 ff.). Apart from Archytas' rescue of Plato in 361 (even this isdescribed as devised by Plato [350a]), Plato is clearly the dominant figure in therelationship. Archytas is portrayed as Plato's inferior in his understanding of

philosophy, and Plato is even presented as responsible for some of Archytas' political success, insofar as he establishes the relationship between Archytas andDionysius II, which is described as of considerable political importance (339d).

How are we to unravel the true nature of the relationship between Plato andArchytas in the light of this conflicting evidence? Apart from the Seventh Letter ,Plato never makes a direct reference to Archytas. He does, however, virtuallyquote a sentence from Archytas' book on harmonics in Book VII ofthe Republic (530d), and his discussions of the science of stereometry shortly

before this are likely to have some connections to Archytas' work in solidgeometry (528d). It is thus in the context of the discussion of the sciences thatPlato refers to Archytas, and the remains of Archytas' work focus precisely onthe sciences (e.g., fr. B1). Both strands of the tradition can be reconciled, if wesuppose that Plato's first visit to Italy and Sicily was at least in part motivated byhis desire to meet Archytas, as the first tradition claims, but that he soughtArchytas out not as a new “model philosopher” but rather as an expert in the

mathematical sciences, in which Plato had developed a deep interest.In Republic VII, Plato is critical of Pythagorean harmonics and of current workin solid geometry on philosophical grounds, so that, while he undoubtedly

learned a considerable amount of mathematics from Archytas, he clearlydisagreed with Archytas' understanding of the philosophical uses of the sciences.In 388 Tarentum had not yet reached the height of its power, and Archytas is notlikely to have achieved his political dominance yet, so that there may also besome truth to the claim of the second tradition that Archytas did not achieve hisgreat practical success until after his contact with Plato; whether or not thatsuccess had any direct relationship to his contact with Plato is more doubtful. Ontheir first meeting in 388/7, Plato and Archytas established a relationship ofguest-friendship, which obligated them to further each other's interests, whichthey did, as the events of 367 – 361 show. Plato and Archytas need not have been

in agreement on philosophical issues and are perhaps better seen as competitivecolleagues engaged in an ongoing debate as to the value of the sciences for philosophy (Huffman 2005, 32 – 42).

1.5 The Authenticity QuestionMore pages of text have been preserved in Archytas' name than in the name ofany other Pythagorean. Unfortunately the vast majority of this material is rightlyregarded as spurious. The same is true of the Pythagorean tradition in general;the vast majority of texts which purport to be by early Pythagoreans are, in fact,later forgeries. Some of these forgeries were produced for purely monetaryreasons; a text of a “rare” work by a famous Pythagorean could fetch a


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considerable sum from book collectors. There were characteristics unique to thePythagorean tradition, however, that led to a proliferation of forgeries. Starting asearly as the later fourth century BC, Pythagoras came to be regarded, in somecircles, as the philosopher par excellence, to whom all truth had been revealed.All later philosophy, insofar as it was true, was a restatement of this originalrevelation (see, e.g., O'Meara 1989). In order to support this view of Pythagoras,texts were forged in the name of Pythagoras and other early Pythagoreans, toshow that they had, in fact, anticipated the most important ideas of Plato andAristotle. These pseudo-Pythagorean texts are thus characterized by the use ofcentral Platonic and Aristotelian ideas, expressed in the technical terminologyused by Plato and Aristotle. Some of the forgeries even attempt to improve onPlato and Aristotle by adding refinements to their positions, which were firstadvanced several hundred years after their deaths. The date and place of origin ofthese pseudo-Pythagorean treatises is difficult to determine, but most seem tohave been composed between 150 BC and 100 AD (Burkert 1972b; Centrone1990; Moraux 1984); Rome (Burkert 1972b) and Alexandria (Centrone 1990) arethe most likely places of origin. Archytas is the dominant figure in this pseudo-Pythagorean tradition. In Thesleff 1965's collection of the pseudo-Pythagoreanwritings, forty-five of the two-hundred and forty-five pages (2-48), about 20%,comprising some 1,200 lines, are devoted to texts forged in Archytas' name. Onthe other hand, the fragments likely to be genuine, which are collected in DK, fillout only a hundred lines of text. Thus, over ten times more spurious than genuinematerial has been preserved in Archytas' name. It may well be that the style andDoric dialect of the pseudo-Pythagorean writings were also based on the modelof Archytas' genuine writings.

1.6 Spurious Works Ascribed to ArchytasThe treatises under Archytas' name collected in Thesleff 1965 have been almostuniversally regarded as spurious, except for On Law and Justice, where there has

been considerable controversy. Most are only preserved in fragments, althoughthere are two brief complete works. The most famous of these forgeriesis Concerning the Whole System [sc. of Categories] or Concerning the TenCategories (preserved complete, see Szlezak 1972). This work along with thetreatise On Opposites (Thesleff 1965, 15.3 – 19.2) and the much later Ten

Universal Assertions (preserved complete, first ascribed to Archytas in the15thcentury AD; see Szlezak 1972) represent the attempt to claim Aristotle'sdoctrine of categories for Archytas and the Pythagoreans. This attempt was tosome extent successful; both Simplicius and Iamblichus regarded the Archytanworks on categories as genuine anticipations of Aristotle (CAG VIII. 2, 9 – 25). Concerning the Ten Categories and On Opposites are very frequently citedin the ancient commentaries on Aristotle's Categories. Pseudo-Archytasidentifies ten categories with names that are virtually identical to those used byAristotle, and his language follows Aristotle closely in many places. The divisionof Archytas' work into two treatises, Concerning the Ten Categories and On

Opposites, reflects the work of Andronicus of Rhodes, who first separated thelast six chapters of Aristotle's Categories from the rest. Thus, the works in


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Archytas' name must have been forged after Andronicus' work in the first centuryBC. Other spurious works in metaphysics and epistemology include On

Principles (Thesleff 1965, 19.3 – 20.17), On Intelligence and Perception (Thesleff 1965, 36.12 – 39.25), which includes a paraphrase of thedivided line passage in Plato's Republic; On Being (Thesleff 1965, 40.1 – 16)and On Wisdom(Thesleff 1965, 43.24 – 45.4). The authenticity of this latter workhas recently been defended on the grounds that its admitted similarities to

passages in Aristotle are a result of Archytas' influence on Aristotle rather thanan indication that the work was forged on the basis of Aristotle (Johnson 2008,193 – 194). It is indeed true that Aristotle devoted several lost works to Archytasand must have been familiar with his thought. However, the issue of authenticitywithin the Pythagorean tradition has a different character than is the case withother ancient authors. In the case of an author such as Plato, where the vastmajority of surviving works are surely authentic, the onus of proof is on anyonewho wants to argue that a work is spurious. In the Pythagorean tradition, on theother hand, where surely spurious works far outnumber genuine ones, thesituation is reversed. The onus of proof rests on anyone who regards aPythagorean work as genuine to show that it does not fit the pattern of the forgedPythagorean treatises and that its contents can be corroborated by evidencedating before the third century, when the Pythagorean pseudepigrapha start to begenerated. Since On Wisdom does share with the pseudepigrapha thecharacteristic of using important Aristotelian distinctions (Huffman 2005, 598 – 599), even if it is not as blatant a copy of Aristotle's ideas as the works oncategories ascribed to Archtyas, it is much more likely that it was forged on the

basis of Aristotle than that Aristotle is using On Wisdom without attribution. Inorder for the latter situation to be probable there would need to be fourth-centuryevidence independent of On Wisdom which ascribed the ideas found in it toArchytas.

There are also fragments of two surely spurious treatises on ethics and politics,which have recent editions with commentary: On the Good and Happy

Man (Centrone 1990), which shows connections to Arius Didymus, an author ofthe first century BC, and On Moral Education (Centrone 1990), which has ties toCarneades (2nd c. BC). The status of one final treatise is less clear. Thefragments of On Law and Justice (Thesleff 1965, 33.1 – 36.11) were studied in

some detail by Delatte (1922), who showed that the treatise deals with the political conceptions of the fourth century and who came to the modestconclusion that the work might be by Archytas, since there were no positiveindications of late composition. Thesleff similarly concluded that the treatise“may be authentic or at least comparatively old” (1961, 112), while Minar

maintained that “it has an excellent claim to authenticity” (1942, 111). Its

authenticity has recently been supported by Johnson (2008, 194 – 198). On theother hand, DK did not include the fragments of On Law and Justice among thegenuine fragments, and most recent scholars have argued that the treatise isspurious. Aalders provides the most detailed treatment, although a number of his

arguments are inconclusive (1968, 13 – 20). Other opponents of authenticity areBurkert (1972a), Moraux (1984, 670 – 677) and Centrone (2000). The connections


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of On Law and Justice to the genuine fr. B2 of Archytas speak for itsauthenticity, but its similarities, sometimes word for word, to pseudo-Pythagorean treatises by “Diotogenes” (Thesleff 76.2– 3, 71. 21 –2), “Damippos”(Thesleff 68.26) and “Metopos” (Thesleff 119.28) argue for its spuriousness.

Moreover, the authentic Fr. 3 of Archytas shows that calculation (logismos) wasthe key concept in his political philosophy. Its total absence from On Law and

Justice, whose focus is political philosophy, along with the absence of other keyterms in Fr. 3 (e.g., pleonexia, homonoia and isotēs) is hard to explain, if On Lawand Justice is authentic (Huffman 2003, 599 – 606).

Some testimonia suggest that there were even more pseudo-Archytan treatises,which have not survived even in fragments (Thesleff 47.8 ff.). Two spuriousletters of Archytas survive. One is the letter to which the pseudo-Platonic Twelfth

Letter is responding (D.L. VIII 79 – 80), and the other is the purported letter ofArchytas to Dionysius II, which was sent along with the ship in order to secure

Plato's release in 361 (D.L. III 21 – 2). Archytas was a popular figure in theMiddle Ages and early Renaissance, when works continued to be forged in hisname, usually with the spelling Architas or Archita. The Ars geometriae, whichis ascribed to Boethius, but was in reality composed in the 12th century (Folkerts1970, 105), ascribes discoveries in mathematics to Architas which are clearlyspurious (Burkert 1972a, 406). Several alchemical recipes involving the wax ofthe left ear of a dog and the heart of a wolf are ascribed to Architas in ps.-Albertus Magnus, The Marvels of the World ( De mirabilibus mundi – 13th centuryAD). Numerous selections from a book entitled On Events in Nature (deeventibus in natura, also cited as de effectibus in natura and as de eventibus

futurorum) by Archita Tharentinus (or Tharentinus, or just Tharen) are preservedin the medieval texts known as The Light of the Soul ( Lumen Animae), whichwere composed in the fourteenth century and circulated widely in Europe in thefifteenth century as a manual for preachers (Rouse 1971; Thorndike 1934, III546 – 60). An apocryphal work, The Circular Theory of the Things in the Heaven,

by Archytas Maximus [!], which has never been published in full, is preserved inCodex Ambrosianus D 27 sup. (See Catalogus Codicum AstrologorumGraecorum, ed. F. Cumont et al., Vol. III, p. 11). For more on the issue ofauthenticity, see Huffman 2005, 91 – 100 and 595 – 618.

1.7 Genuine Works and Testimonia No list of Archytas' works has come down to us from antiquity, so that we don'tknow how many books he wrote. In the face of the large mass of spurious works,it is disappointing that only a few fragments of genuine works have survived.Most scholars accept as genuine the four fragments printed by Diels and Kranz(B1 – 4). Burkert (1972a, 220 n.14 and 379 n. 46) raised some concerns about theauthenticity of even some of these fragments, but see the responses of Bowen(1982) and Huffman (1985 and 2005). Our evidence for the titles of Archytas'genuine writings depends largely on the citations given by the authors who quotethe fragments. Fragments B1 and B2 are reported to come from a treatiseentitled Harmonics, and the major testimonia about Archytas' harmonic theory are


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likely to be ultimately based on this book (A16 – 19). This treatise began with adiscussion of the basic principles of acoustics (B1), defined the three types ofmean which are of importance in music theory (B2), and went on to presentArchytas' mathematical descriptions of the tetrachord (the fourth) in the threemain genera (chromatic, diatonic, and enharmonic – A16-A19). B3 probablycomes from a work On Sciences, which may have been a more generaldiscussion of the value of mathematics for human life in general and for theestablishment of a just state in particular. New support for its authenticity has

been provided by Schofield (2009). B4 comes from a workentitled Discourses ( Diatribai). The fragment itself asserts the priority of thescience of calculation (ha logistika, “logistic”) to the other sciences, such asgeometry, and thus suggests a technical work of mathematics. Thetitle Diatribai would more normally suggest a treatise of ethical content,however, so that in this work the sciences may have been evaluated in terms oftheir contribution to the wisdom that leads to a good life.

A relatively rich set of testimonia, many from authors of the fourth century BC,indicate that Archytas wrote other books as well. Archytas' famous argument forthe unlimited extent of the universe (A24), his theory of vision (A25), and hisaccount of motion (A23, A23a) all suggest that he may have written a work oncosmology. Aristotle's comments in the Metaphysics suggest that Archytas mayhave written a book on definition (A22), and A20 and A21 might suggest a workon arithmetic. Perhaps there was a treatise on geometry or solid geometry inwhich Archytas' solution to the problem of doubling the cube (A14 – 15) was

published. There is also a tradition of anecdotes about Archytas, which probably

ultimately derives from Aristoxenus' Life of Archytas (A7, A8, A9, A11). It is possible that even the testimonia for Archytas' argument for an unlimiteduniverse and his theory of vision were derived from anecdotes preserved byAristoxenus, and not at all from works of Archytas' own.

It is uncertain whether the treatises On Flutes (B6), On Machines (B1 and B7),and On Agriculture (B1 and B8), which were in circulation under the name ofArchytas, were in fact by Archytas of Tarentum or by other men of the samename. Diogenes Laertius lists three other writers with the name Archytas (VIII82). The treatise On the Decad mentioned by Theon (B5) might be by Archytas,

but the treatise by Philolaus with which it is paired is spurious (Huffman 1993,347 – 350), thus suggesting that the same may be true of the treatise underArchytas' name.

2. Archytas as Mathematician and HarmonicTheorist2.1 Doubling the CubeArchytas was the first person to arrive at a solution to one of the most famous

mathematical puzzles in antiquity, the duplication of the cube. The most romanticversion of the story, which occurs in many variations and ultimately goes back to


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Eratosthenes (3rd c. BC), reports that the inhabitants of the Greek island of Deloswere beset by a plague and, when they consulted an oracle for advice, were toldthat, if they doubled the size of a certain altar, which had the form of a cube, the

plague would stop (Eutocius, in Archim. sphaer. et cyl. II [III 88.3 – 96.27Heiberg/Stamatis]). The simple-minded response to the oracle, which is actuallyassigned to the Delians in some versions, is to build a second altar identical to thefirst one and set it on top of the first (Philoponus, In Anal. post., CAGXIII.3,102.12 – 22). The resulting altar does indeed have a volume twice that of the firstaltar, but it is no longer a cube. The next simple-minded response is to assumethat, since we want an altar that is double in volume, while still remaining a cube,we should build the new altar with a side that is double the length of the side ofthe original altar. This approach fails as well. Doubling the side of the altar

produces a new altar that is not twice the volume of the original altar but eighttimes the volume. If the original altar had a side of two, then its volume would be23 or 8, while an altar built on a side twice as long will have a volume of 4 3 or 64.What then is the length of the side which will produce a cube with twice thevolume of the original cube? The Delians were at a loss and presented their

problem to Plato in the Academy. Plato then posed the “Delian Problem,” as it

came to be known, to mathematicians associated with the Academy, and no lessthan three solutions were devised, those of Eudoxus, Menaechmus, and Archytas.

It is not clear whether or not the story about the Delians has any basis in fact.Even if it does, it should not be understood to suggest that the problem ofdoubling the cube first arose in the fourth century with the Delians. We are toldthat the mathematician, Hippocrates of Chios, who was active in the second half

of the fifth century, had already confronted the problem and had reduced it to aslightly different problem (Eutocius, in Archim. sphaer. et cyl. II [III 88.3 – 96.27Heiberg/Stamatis]). Hippocrates recognized that if we could find two mean

proportionals between the length of the side of the original cube G, and length D,where D = 2G, so that G : x :: x : y :: y : D, then the cube on length x will bedouble the cube on length G. Exactly how Hippocrates came to see this isconjectural and need not concern us here, but that he was right can be seenrelatively easily. Each of the values in the continued proportion G : x :: x : y :: y :D is equal to G : x, so we can set them all equal to G : x. If we do this andmultiply the three ratios together we get the value G3 : x3. On the other hand, if

we take the same continued proportion and carry out the multiplication in theoriginal terms, then G : x times x : y yields G : y, and G : y times the remainingterm gives G : D. Thus G : D = G3 : x3, but D is twice G so x3 is twice G3.Remember that G was the length of the side of the original cube, so the cube thatis twice the cube built on G, will be the cube built on x. The Greeks did not thinkof the problem as a problem in algebra but rather as a problem in geometry. AfterHippocrates the problem of doubling the cube was always seen as the problem offinding two lines such that they were mean proportionals between G, the lengthof the side of the original cube, and D, a length which is double G. It was to thisform of the problem that Archytas provided the first solution.


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Archytas' solution has been rightly hailed as “the most remarkable of all [thesolutions]” and as a “bold construction in three dimensions” (Heath 1921, 246);

Mueller calls it “atour de force of the spatial imagination” (1997, 312 n. 23). Weowe the preservation of Archytas' solution to Eutocius, who in the sixth centuryAD collected some eleven solutions to the problem as part of his commentary onthe second book of Archimedes' On the Sphere and Cylinder . Eutocius' sourcefor Archytas' solution was ultimately Aristotle's pupil Eudemus, who in the latefourth century BC wrote a history of geometry. The solution is complex and it isnot possible to go through it step by step here (see Huffman 2005,342 – 360 for adetailed treatment of the solution). Archytas proceeds by constructing a series offour similar triangles (see Figure 1 below) and then showing that the sides are

proportional so that AM : AI :: AI : AK :: AK : AD, where AM was equal to theside of the original cube (G) and AD was twice AM. Thus the cube double thevolume of the cube on AM should be built on AI. The real difficulty was inconstructing the four similar triangles, where the given length of the side of theoriginal cube and a length double that magnitude were two of the sides in thesimilar triangles. The key point for the construction of these triangles, point K,was determined as the intersection of two rotating plane figures. The first figureis a semicircle, which is perpendicular to the plane of the circle ABDZ and whichstarts on the diameter AED and, with point A remaining fixed, rotates to positionAKD. The second is the triangle APD, which rotates up out of the plane of thecircle ABDZ to position ALD. As each of these figures rotates, it traces a line onthe surface of a semicylinder, which is perpendicular to the plane of ABDZ andhas ABD as its base. The boldness and the imagination of the construction lies inenvisioning the intersection at point K of the line drawn by the rotatingsemicircle on the surface of the semicylinder with the line drawn by the rotatingtriangle on the same surface. We simply don't know what led Archytas to

produce this amazing feat of spatial imagination, in order to construct thetriangles with the sides in appropriate proportion.


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Figure 1

In the later tradition, Plato is reported to have criticized Archytas' solution forappealing to “constructions that use instruments and that are mechanical”

(Plutarch, Table Talk VIII 2.1 [718e]; Marc. XIV 5 – 6). Plato argued that thevalue of geometry and of the rest of mathematics resided in their ability to turnthe soul from the sensible to intelligible realm. The cube with which geometrydeals is not a physical cube or even a drawing of a cube but rather an intelligiblecube that fits the definition of the cube but is not a sense object. By employing

physical instruments, which “required much common handicraft,” and in effect

constructing machines to determine the two mean proportionals, Archytas wasfocusing not on the intelligible world but on the physical world and hencedestroying the value of geometry. Plato's quarrel with Archytas is a charmingstory, but it is hard to reconcile with Archytas' actual solution, which, as we have

seen, makes no appeal to any instruments or machines. The story of the quarrel,which is first reported in Plutarch in the first century AD, is also hard toreconcile with our earliest source for the story of the Delian problem,Eratosthenes. Eratosthenes had himself invented an instrument to determinemean proportionals, the mesolab (“mean-getter”), and he tells the story of theDelian problem precisely to emphasize that earlier solutions, including that ofArchytas, were in the form of geometrical demonstrations, which could not beemployed for practical purposes. He specifically labels Archytas' solutionas dysmêchana, “hardly mechanical.” Some scholars attempt to reconcilePlutarch's and Eratosthenes' versions by focusing on their different literary goals

(Knorr 1986, 22; van der Waerden 1963, 161; Wolfer 1954, 12 ff.; Sachs 1917,150); some suggest that the rotation of the semicircle and the triangle in


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Archytas' solution, might be regarded as mechanical, since motion is involved(Knorr 1986, 22). It may be, however, that Plutarch's story of a quarrel betweenPlato and Archytas over the use of mechanical devices in geometry is aninvention of the later tradition (Riginos 1976, 146; Zhmud 1998, 217) and

perhaps served as a sort of foundation myth for the science of mechanics, a mythwhich explained the separation of mechanics from philosophy as the result of aquarrel between two philosophers. In the Republic, Plato is critical of the solidgeometry of his day, but his criticism makes no mention of the use ofinstruments. His criticism instead focuses on the failure of solid geometry to bedeveloped into a coherent discipline alongside geometry and astronomy (528b-d). This neglect of solid geometry is ascribed to the failure of the Greek city-states to hold these difficult studies in honor, the lack of a director to organize thestudies, and the arrogance of the current experts in the field, who would notsubmit to such a director. Since Archytas' duplication of the cube shows him to

be one of the leading solid geometers of the time, it is hard to avoid theconclusion that Plato regarded him as one of the arrogant experts, who focusedon solving charming problems but failed to produce a coherent discipline of solidgeometry. Since Archytas was a leading political figure in Tarentum, it is also

possible that Plato was criticizing him for not making Tarentum a state whichheld solid geometry in esteem. For more on Archytas' solution to the duplicationof the cube, see Huffman 2005, 342 – 401.

2.2 Music and MathematicsOne of the most startling discoveries of early Greek science was that the

fundamental intervals of music, the octave, the fourth, and the fifth, correspondedto whole number ratios of string length. Thus, if we pluck a string of length x andthen a string of length 2 x, we will hear the interval of an octave between the twosounds. If the two string lengths are in the ratio 4 : 3, we will hear a fourth, and,if the ratio is 3 : 2, we will hear a fifth. This discovery that the phenomena ofmusical sound are governed by whole number ratios must have played a centralrole in the Pythagorean conception, first expressed by Philolaus, that all thingsare known through number (DK 44 B4). The next step in harmonic theory was todescribe an entire octave length scale in terms of mathematical ratios. Theearliest such description of a scale is found in Philolaus fr. B6. Philolaus

recognizes that, if we go up the interval of a fourth from any given note, and thenup the interval of a fifth, the final note will be an octave above the first note.Thus, the octave is made up of a fourth and a fifth. In mathematical terms, theratios that govern the fifth (3 : 2) and fourth (4 : 3) are added by multiplying theterms and thus produce an octave (3 : 2 × 4 : 3 = 12 : 6 = 2 : 1). The interval

between the note that is a fourth up from the starting note and the note that is afifth up was regarded as the basic unit of the scale, the whole tone, whichcorresponded to the ratio of 9 : 8 (subtraction of ratios is carried out by dividingthe terms, or cross multiplying: 3 : 2 / 4 : 3 = 9 : 8). The fifth was thus regardedas a fourth plus a whole tone, and the octave can be regarded as two fourths plus

a whole tone. The fourth consists of two whole tones with a remainder, whichhas the unlovely ratio of 256 : 243 (4 : 3 / 9 : 8 = 32 : 27 / 9 : 8 = 256 : 243).


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Philolaus' scale thus consisted of the following intervals: 9 : 8, 9 : 8, 256 : 243[these three intervals take us up a fourth], 9 : 8, 9 : 8, 9 : 8, 256 : 243 [these fourintervals make up a fifth and complete the octave from our staring note]. Thisscale is known as the Pythagorean diatonic and is the scale that Plato adopted inthe construction of the world soul in the Timaeus (36a-b).

Archytas took harmonic theory to a whole new level of theoretical andmathematical sophistication. Ptolemy, writing in the second century AD,identifies Archytas as having “engaged in the study of music most of all thePythagoreans” (A16). First, Archytas provided a general explanation of pitch,

arguing that the pitch of a sound depends on the speed with which the sound is propagated and travels (B1). Thus, if a stick is waved back and forth rapidly, itwill produce a sound that travels rapidly through the air, which will be perceivedas of a higher pitch than the sound produced by a stick waved more slowly.Archytas is correct to associate pitch with speed, but he misunderstood the role

of speed. The pitch does not depend on the speed with which a sound reaches us but rather on the frequency of impacts in a given period of time. A string thatvibrates more rapidly produces a sound of a higher pitch, but all sounds,regardless of pitch, travel at an equal velocity, if the medium is the same.Although Archytas' account of pitch was ultimately incorrect, it was veryinfluential. It was taken over and adapted by both Plato and Aristotle andremained the dominant theory throughout antiquity (Barker 1989, 41 n. 47).Second, Archytas introduced new mathematical rigor into Pythagoreanharmonics. One of the important results of the analysis of music in terms ofwhole number ratios is the recognition that it is not possible to divide the basic

musical intervals in half. The octave is not divided into two equal halves but intoa fourth and a fifth, the fourth is not divided into two equal halves but into twowhole tones and a remainder. The whole tone cannot be divided into two equalhalf tones. On the other hand, it is possible to divide a double octave in half.Mathematically this can be seen by recognizing that it is possible to insert a mean

proportional between the terms of the ratio corresponding to the double octave (4: 1) so that 4 : 2 :: 2 : 1. The double octave can thus be divided into two equal

parts each having a ratio of 2 : 1. The ratios which govern the basic musicalintervals (2 : 1, 4 : 3, 3 : 2, 9 : 8), all belong to a type of ratio known as asuperparticular ratio – roughly speaking, ratios of the form (n + 1) : n. Archytas

made a crucial contribution by providing a rigorous proof that there is no mean proportional between numbers in superparticular ratio (A19) and hence that the basic musical intervals cannot be divided in half. Archytas' proof was later takenover and modified slightly in the Sectio Canonis ascribed to Euclid (Prop. 3; seeBarker 1989, 195).

Archytas' final contribution to music theory has to do with the structure of thescale. The Greeks used a number of different scales, which were distinguished bythe way in which the fourth, or tetrachord, was constructed. These scales weregrouped into three main types or genera. One genus was called the diatonic; oneexample of this is the Pythagorean diatonic described above, which is built on thetetrachord with the intervals 9 : 8, 9 : 8 and 256 : 243 and was used by Philolaus


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and Plato. There is no doubt that Archytas knew of this diatonic scale, but hisown diatonic tetrachord was somewhat different, being composed of the intervals9 : 8, 8 : 7 and 28 : 27. Archytas also defined scales in the two other majorgenera, the enharmonic and chromatic. Archytas' enharmonic tetrachord iscomposed of the intervals 5 : 4, 36 : 35 and 28 : 27 and his chromatic tetrachordof the intervals 32 : 27, 243 : 224, and 28 : 27. There are several puzzles aboutthe tetrachords which Archytas adopts in each of the genera. First, why doesArchytas reject the Pythagorean diatonic used by Philolaus and Plato? Second,Ptolemy, who is our major source for Archytas' tetrachords (A16), argues thatArchytas adopted as a principle that all concordant intervals should correspond tosuperparticular ratios. The ratios in Archytas' diatonic and enharmonictetrachords are indeed superparticular, but two of the ratios in his chromatictetrachord are not superparticular (32 : 27 and 243 : 224). Why are these ratiosnot superparticular as well? Finally, Plato criticizes Pythagorean harmonics inthe Republic for seeking numbers in heard harmonies rather than ascending togeneralized problems (531c). Can any sense be made of this criticism in light ofArchytas' tetrachords? The basis for an answer to all of these questions iscontained in the work of Winnington-Ingram (1932) and Barker (1989, 46-52).The crucial point is that Archytas' account of the tetrachords in each of the threegenera can be shown to correspond to the musical practice of his day; Ptolemy'scriticisms miss the mark because of his ignorance of musical practice inArchytas' day, some 500 years before Ptolemy (Winnington-Ingram 1932, 207).Archytas is giving mathematical descriptions of scales actually in use; he arrivedat his numbers in part by observation of the way in which musicians tuned theirinstruments (Barker 1989, 50 – 51). He did not follow the Pythagorean diatonicscale because it did not correspond to any scale actually in use, although it doescorrespond to a method of tuning. The unusual numbers in Archytas' chromatictetrachord do correspond to a chromatic scale in use in Archytas' day. Barkertries to save Archytas' adherence to the principle that all concordant intervalsshould have superparticular ratios, but there is no direct evidence that he wasusing such a principle, and Ptolemy may be mistaken to apply it to him. Archytasthus provides a brilliant analysis of the music of his day, but it is precisely hisfocus on actual musical practice that draws Plato's ire. Plato does not want him tofocus on the music he hears about him (“heard harmonies”) but rather to ascend

to consider quite abstract questions about which numbers are harmonious withwhich. Plato might well have welcomed a principle of concordance based solelyon mathematical considerations, such as the principle that only superparticularratios are concordant, but Archytas wanted to explain the numbers of the musiche actually heard played. There is an important metaphysical issue at stake here.Plato is calling for the study of number in itself, apart from the sensible world,while Archytas, like Pythagoreans before him, envisages no split between asensible and an intelligible world and is looking for the numbers which governsensible things.

2.3 Evaluation of Archytas as Mathematician


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There have been tendencies both to overvalue and to undervalue Archytas'achievement as a mathematician. Van der Waerden went so far as to add toArchytas' accomplishments both Book VIII of Euclid's Elements and the treatiseon the mathematics of music known as theSectio Canonis, which is ascribed toEuclid in the ancient tradition (1962, 152 – 5). Although later scholars (e.g., Knorr1975: 244) repeat these assertions, they are based in part on a very subjectiveanalysis of Archytas' style. Archytas influenced the Sectio Canonis, sinceProposition 3 is based on a proof by Archytas (A19), but the treatise cannot be byArchytas, because its theory of pitch and its account of the diatonic andenharmonic tetrachords differ from those of Archytas. On the other hand, somescholars have cast doubt on Archytas' prowess as a mathematician, arguing thatsome of his work looks like “mere arithmology” and “mathematical

mystification” (Burkert 1972a, 386; Mueller 1997, 289). This judgment restslargely on a text that has been mistakenly interpreted as presenting Archytas' ownviews, whereas, in fact, it presents Archytas' report of his predecessors (A17).The duplication of the cube and Archytas' contributions to the mathematics ofmusic show that there can be no doubt that he was one of the leadingmathematicians of the first part of the fourth century BC. This was certainly the

judgment of antiquity. In his history of geometry, Eudemus identified Archytasalong with Leodamas and Theaetetus as the three most prominentmathematicians of Plato's generation (A6 = Proclus, in Eucl., prol. II 66, 14).

3. Archytas on the Sciences

3.1 The Value of the SciencesArchytas B1 is the beginning of his book on harmonics, and most of it is devotedto the basic principles of his theory of acoustics and, in particular, to his theory of

pitch described in section 2.2 above. In the first five lines, however, Archytas provides a proem on the value of the sciences (mathêmata) in general. There areseveral important features of this proem. First, Archytas identifies a set of foursciences: astronomy, geometry, “logistic” (arithmetic) and music. B1 is thus

probably the earliest text to identify the set of sciences that became known asthe quadrivium in the middle ages and that constitute four of the seven liberalarts. Second, Archytas does not present this classification of sciences as his owndiscovery but instead begins with praise of his predecessors who have worked inthese fields. Some scholars ar gue that, when he praises “those concerned with thesciences,” he is thinking only of the Pythagoreans (e.g., Zhmud 1997, 198 and

Lasserre 1954, 36), but this is wrongly to assume that all early Greekmathematics is Pythagorean. Archytas gives no hint that he is limiting hisremarks to Pythagoreans, and, in areas where we can identify those whoinfluenced him most, these figures are not limited to Pythagoreans (e.g.,Hippocrates of Chios in geometry, see section 2.1). He praises his predecessorsin the sciences, because, “having discerned well about the nature of wholes, theywere likely also to see well how things are in their parts” and to “have correctunderstanding about individual things as they are.” It is here that Archytas is


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putting forth his own understanding of the nature and value of the sciences; because of the brevity of the passage, much remains unclear. Archytas appears to be praising those concerned with the sciences for their discernment, their abilityto make distinctions (diagignôskein). He argues that they begin by distinguishingthe nature of wholes, the universal concepts of a science, and, because they dothis well, they are able to understand particular objects (the parts). Archytasappears to follow exactly this procedure in his Harmonics. He begins by definingthe most universal concept of the science, sound, and explains it in terms of otherconcepts such as impact, before going on to distinguish between audible andinaudible sounds and sounds of high and low pitch. The goal of the science is notthe making of these distinctions concerning universal concepts, however, butknowledge of the true nature of individual things. Thus, Archytas' harmonicsends with the mathematical description of the musical intervals that we hear

practicing musicians use (see section 2.2 above). Astronomy will end with amathematical description of the periods, risings and settings of the planets. Oneway to understand Archytas' project is to see him as working out the programsuggested by his predecessor in the Pythagorean tradition, Philolaus. One ofPhilolaus' central theses was that we only gain knowledge of things insofar as wecan give an account of them in terms of numbers (DK 44 B4). While Philolausonly took the first steps in this project, Archytas is much more successful ingiving an account of individual things in the phenomenal world in terms ofnumbers, as his description of the musical intervals shows.

Plato's account of the sciences in Book VII of the Republic can be seen as aresponse to Archytas' view of the sciences. First Plato identifies a group of five

rather than four sciences and decries the neglect of his proposed fifth science,stereometry (solid geometry), with a probable allusion to Archytas (see section2.1). Plato quotes with approval Archytas' assertion that “these sciences seem to

be akin” (B1), although he applies it just to harmonics and astronomy rather than

to Archytas' quadrivium and does not mention him by name. In the same passage,however, Plato pointedly rejects the Pythagorean attempt to search for numbersin “heard harmonies.” In doing so Plato is disagreeing with Archytas' attempt todetermine the numbers that govern things in the sensible world. For Plato, thevalue of the sciences is their ability to turn the eye of the soul from the sensibleto the intelligible realm. Book VII of the Republic with its elaborate argument for

the distinction between the intelligible and sensible realm, between the cave andthe intelligible world outside the cave, may be in large part directed at Archytas'attempt to use mathematics to explain the sensible world. As Aristotle repeatedlyemphasizes, the Pythagoreans differed from Plato precisely in their refusal toseparate numbers from things (e.g., Metaph. 987b27).

3.2 Logistic as the Master ScienceIn B4, Archytas asserts that “logistic seems to be far superior indeed to the other

arts in regard to wisdom.” What does Archytas mean by “logistic”? It appears to

be Archytas' term for the science of number, which was mentioned as one of thefour sister sciences in B1. There is simply not enough context in B4 or other texts


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of Archytas to determine the meaning of logistic from Archytas' usage alone. It isnecessary to rely to some extent on Plato, who is the only other early figure touse the term extensively. A later conception of logistic, as something that dealswith numbered things rather than numbers themselves, which is found in, e.g.,Geminus, should not be ascribed to Plato or Archytas (Klein 1968; Burkert1972a, 447 n. 119). In Plato, “logistic” can refer to everyday calculation, whatwe would call arithmetic (e.g. 3 × 700 = 2,100; see, Hp. Mi. 366c). In other

passages, however, Plato defines logistic in parallel with arithmêtikê, and treatsthe two of them as together constituting the science of number, on which

practical manipulation of number is based (Klein 1968, 23 – 24).Both arithmêtikê and logistic deal with the even and the odd. Arithmêtikê focusesnot on quantities but on kinds of numbers (Grg. 451b), beginning with the evenand the odd and presumably continuing with the types we find later in

Nicomachus ( Ar. 1.8 – 1.13), such as prime, composite and even-times even.Logistic, on the other hand, focuses on quantity, the “amount the odd and even

have both in themselves and in respect to one another” (Grg. 451c). An exampleof one part of logistic might be the study of various sorts of means and

proportions, which focus on the quantitative relations of numbers to one another(e.g., Nicomachus, Ar. II. 21 ff.). In B2, Archytas would probably considerhimself to be doing logistic, when he defines the three types of means which arerelevant to music (geometric, arithmetic, and harmonic). The geometric meanarises whenever three terms are so related that, as the first is to the second, so thesecond is to the third (e.g., 8 : 4 :: 4 : 2) and the arithmetic, when three terms areso related that the first exceeds the second by the same amount as the secondexceeds the third (e.g., 6 : 4 :: 4 : 2). Archytas, like Plato ( R. 525c), uses logisticnot just in this narrow sense of the study of relative quantity, but also todesignate the entire science of numbers including arithmêtikê.

Why does Archytas think that logistic is superior to the other sciences? In B4, he particular ly compares it to geometry, arguing that logistic (1) “deals with what itwishes more vividly than geometry” and (2) “completes demonstrations” where

geometry cannot, even “if there is any investigation concerning shapes.” This lastremark is surprising, since the study of shapes would appear to be the properdomain of geometry. The most common way of explaining Archytas' remark is tosuppose that he is arguing that logistic is mathematically superior to geometry, in

that certain proofs can only be completed by an appeal to logistic. Burkert seesthis as a reason for doubting the authenticity of the fragment, since the exactopposite seems to be true. Archytas could determine the cube root of twogeometrically, through his solution to the duplication of the cube, but could notdo so arithmetically, since the cube root of two is an irrational number (1972a,220 n. 14). Other scholars have pointed out, however, that certain proofs ingeometry do require an appeal to logistic (Knorr 1975, 311; Mueller 1992b, 90 n.12), e.g., logistic is required to recognize the incomensurability of the diagonalwith the side of the square, since incommensurability arises when twomagnitudes “have not to one another the ratio whichnumber has to number ”

(Euclid X 7). These suggestions show that logistic can be superior to geometry in


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certain cases, but they do not explain Archytas' more general assertion thatlogistic deals with whatever problems it wants more clearly than geometry.

However, it may be that B4 is not in fact comparing logistic to the other sciencesas sciences – in terms of their relative success in providing demonstrations. The

title of the work from which B4 is said to come, Discourses ( Diatribai), is mostcommonly used of ethical treatises. Moreover, it is specifically with regard towisdom ( sophia) that logistic is said to be superior, and, while sophia can refer totechnical expertise, it more commonly refers to the highest sort of intellectualexcellence, often the excellence that allows us to live a good life(Arist., EN 1141a12; Pl., R. 428d ff.). Is there any sense in which logistic makesus wiser than the other sciences? Since Archytas evidently agreed with Philolausthat we only understand individual things in the world insofar as we grasp thenumbers that govern them, it seems quite plausible that Archytas would regardlogistic as the science that makes us wise about the world. It is in this sense that

logistic will always be superior to geometry, even when dealing with shapes.Perhaps the most famous statue of the classical period is the Doryphoros by theArgive sculptor Polyclitus, which he also referred to as the Canon (i.e., thestandard). Although Polyclitus undoubtedly made use of geometry inconstructing this magnificent shape, in a famous sentence from his book, alsoentitled Canon, he asserts that his statue came to be not through many shapes but“through many numbers” (DK40 B2, see Huffman 2002a). Geometrical relationsalone will not determine the form of a given object, we have to assign specific

proportions, specific numbers. Archytas also thought that numbers and logisticwere the basis of the just state and hence the good life. In B3 he argues that it is

rational calculation (logismos) that produces the fairness on which the statedepends. Justice is a relation that needs to be stated numerically and it is throughsuch a statement that rich and poor can live together, each seeing that he haswhat is fair. Logistic will always be superior to the other sciences, because thosesciences will in the end rely on numbers to give us knowledge of the sounds wehear, the shapes we see and the movements of the heavenly bodies which weobserve.

3.3 Optics and MechanicsAristotle is the first Greek author to mention the sciences of optics andmechanics, describing optics as a subordinate science to geometry and mechanicsas a subordinate science to solid geometry ( APo. 78b34). Archytas does notmention either of these sciences in B1, when describing the work of his

predecessors in the sciences, nor does Plato mention them. This silence suggeststhat the two disciplines may have first developed in the first half of the fourthcentury, when Archytas was most active, and it is possible that he played animportant role in the development of both of them. In a recently identifiedfragment from his book on the Pythagoreans (Iamblichus, Comm. Math. XXV;see Burkert 1972a, 50 n. 112), Aristotle assigns a hitherto unrecognizedimportance to optics in Pythagoreanism. Just as the Pythagoreans were impressedwith the fact that musical intervals were based on whole number ratios, so they


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were impressed that the phenomena of optics could be explained in terms ofgeometrical diagrams. In addition to being an accomplished mathematician,Archytas had a theory of vision and evidently tried to explain some of the

phenomena involved in mirrors. In contrast to Plato, who argued that the visualray, which proceeded from the eye, requires the support of and coalesces withexternal light, Archytas explained vision in terms of the visual ray alone (A25). Itis tempting, then, to suppose that Archytas played a major role in thedevelopment of the mathematically based Pythagorean optics, to which Aristotlerefers. On the other hand, when Aristotle refers to Pythagoreans, he generallymeans Pythagoreans of the fifth century. Elsewhere he treats Archytasindependently of the Pythagorean tradition, writing works on Archytas whichwere distinct from his work on the Pythagoreans. It would thus be more naturalto read Aristotle's reference to Pythagorean optics as alluding to fifth-centuryPythagoreans such as Philolaus. Archytas will then have been responsible fordeveloping an already existing Pythagorean optical tradition into a science, ratherthan founding such a tradition.

Diogenes Laertius reports that Archytas was “the first to systematize mechanics

by using mathematical first principles” (VIII 83 = A1), and Archytas is

accordingly sometimes hailed by modern scholars as the founder of the scienceof mechanics. There is a puzzle, however, since, no ancient Greek author in thelater mechanical tradition (e.g., Heron, Pappus, Archimedes, Philon) everascribes any work in the field to Archytas. What did the ancients mean bymechanics? A rough definition would be “the description and explanation of theoperation of machines” (Knorr, Oxford Classical Dictionary, ed. 3, s.v.). The

earliest treatise in mechanics, the Mechanical Problems ascribed to Aristotle, begins with problems having to do with a simple machine, the lever. Pappus (AD320) refers to machines used to lift great weights, machines of war such as thecatapult, water lifting machines, amazing devices (automata), and machines thatserved as models of the heavens (1024.12 – 1025.4, on Pappus, see Cuomo2000). Pappus emphasizes, however, that, in addition to this practical part ofmechanics, there is a theoretical part that is heavily mathematical (1022. 13 – 15).Given his interest in describing physical phenomena in mathematical terms, itmight seem logical that Archytas would make important contributions tomechanics. The actual evidence is less conclusive. A great part of the tendency to

assign Archytas a role in the development of mechanics can be traced toPlutarch's story about the quarrel between Plato and Archytas over Archytas'supposed mechanical solution to the problem of doubling the cube. This story islikely to be false (see 2.1 above). Some scholars have argued that Archytasdevised machines of war (Diels 1965; Cambiano 1998), as Archimedes did later,

but this conclusion is based on questionable inferences and no ancient sourceascribes such machines to Archytas. The only mechanical device that can withsome probability be assigned to Archytas, apart from the children's toy known asa “clapper” (A10), is an automaton in the form of a wooden dove, which wasconnected to a pulley and counterweight and “flew” up from a lower perch to a

higher one, when set in motion by a puff of air (A10a). It has been suggestedthat, since ancient siege devices were called by the names of animals (e.g.,


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“tortoise” and “crow”), Archytas' “dove” might have been an early catapult of his

devising, or a projectile hurled by such a catapult, which was later misunderstoodto be a mechanical dove (Berryman 2003, 355). However, no ancient sourceexplains the dove in this way. A complicating factor here is that DiogenesLaertius reports (A1) that there was a book on mechanics in circulation, whichsome thought to be by a different Archytas, so that it is possible that the flyingdove is, in fact, the work of a separate Archytas. Archytas' solution to theduplication of the cube, although it was not mechanical itself, was of enormousimportance for mechanics, since the solution to the problem allows one not justto double a cube but also to construct bodies that are larger or smaller than agiven body in any given ratio. Thus, the solution permits the construction of afull-scale machine on the basis of a working model. Pappus cites the solution tothe duplication of the cube as one of the three most crucial geometrical theoremsfor practical mechanics ( Math. Coll. 1028. 18 – 21). It may then be that Archytas'

primary contribution to mechanics was precisely his solution to the duplication ofthe cube and that it is this solution which constituted the mathematical first

principles which Archytas provided for mechanics. It is more doubtful thatArchytas wrote a treatise on mechanics.

4. DefinitionsIn the Metaphysics, Aristotle praises Archytas for having offered definitionswhich took account of both form and matter (1043a14 – 26 = A22). The examplesgiven are “windlessness” (nênemia), which is defined as “stillness [the form] in aquantity of air [the matter],” and “calm-on-the-ocean” ( galênê), which is definedas “levelness [the form] of sea [the matter].” The terms form and matter areAristotle's, and we cannot be sure how Archytas conceptualized the two parts ofhis definitions. A plausible suggestion is that he followed his predecessorPhilolaus in adopting limiters and unlimiteds as his basic metaphysical principlesand that he saw his definitions as combinations of limiters, such as levelness andstillness, with unlimiteds, such as air and sea. The oddity of “windlessness” and

“calm-on-the-sea” as examples suggests that they were not the by-products ofsome other sort of investigation, e.g. cosmology, but were chosen precisely toillustrate principles of definition. Archytas may thus have devoted a treatise tothe topic. Aristotle elsewhere comments on the use of proportion in developing

definitions and uses these same examples (Top. 108a7). The ability to recognizelikeness in things of different genera is said to be the key. “Windlessness” and

“calm-on-the-ocean” are recognized as alike, and this likeness can be expressedin the following proportion: as nênemia is to the air so galênê is to the sea. It istempting to suppose that Archytas, who saw the world as explicable in terms ofnumber and proportion, also saw proportion as the key in developing definitions.This would explain another reference to Archytas in Aristotle.At Rhetoric1412a9-17 (= A12) Aristotle praises Archytas precisely for his abilityto see similarity, even in things which differ greatly, and gives as an exampleArchytas' assertion that an arbitrator and an altar are the same. DK oddly includethis text among the testimonia for Archytas' life, but it clearly is part of Archytas'work on definition. The definitions of both an altar and an arbitrator will appeal


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to their common functions as a refuge, while recognizing the different contextand way in which this function is carried out (for doubts about this reconstructionof Archytas' theory of definition, see Barker 2006, 314 – 318).

5. Cosmology and PhysicsWe have very little evidence for Archytas' cosmology, yet he was responsible forone of the most famous cosmological arguments in antiquity, an argument whichhas been hailed as “the most compelling argument ever produced for the infinity

of space” (Sorabji 1988, 125). The argument is ascribed to Archytas in a

fragment of Eudemus preserved by Simplicius (= A24), and it is probably toArchytas that Aristotle is referring when he describes the fifth and “most

important” reason that people believe in the existence of the unlimited

( Ph. 203b22 ff.). Archytas asks anyone who argues that the universe is limited toengage in a thought experiment: “If I arrived at the outermost edge of the heaven,

could I extend my hand or staff into what is outside or not? It would be paradoxical [given our normal assumptions about the nature of space] not to beable to extend it.” The end of the staff, once extended will mark a new limit.Archytas can advance to the new limit and ask the same question again, so thatthere will always be something, into which his staff can be extended, beyond thesupposed limit, and hence that something is clearly unlimited. Neither Plato norAristotle accepted this argument, and both believed that the universe was limited.

Nonetheless, Archytas' argument had great influence and was taken over andadapted by the Stoics, Epicureans (Lucretius I 968 – 983), Locke and Newton,among others, while eliciting responses from Alexander and Simplicius (Sorabji

1988, 125 – 141). Not all scholars have been impressed by the argument (seeBarnes 1982, 362), and modern notions of space allow for it to be finite withouthaving an edge, and without an edge Archytas' argument cannot get started (butsee Sorabji 1988, 160 – 163). Beyond this argument, there is only exiguousevidence for Archytas' system of the physical world. Eudemus praises Archytasfor recognizing that the unequal and uneven are not identical with motion asPlato supposed (see Ti. 52e and 57e) but rather the causes of motion (A23).Another testimonium suggests that Archytas thought that all things are moved inaccordance with proportion (Arist., Prob. 915a25 – 32 = A23a). The sametestimonium indicates that different sorts of proportion defined different sorts ofmotion. Archytas asserted that “the proportion of equality” (arithmetic

proportion?) defined natural motion, which he regarded as curved motion. Thisexplanation of natural motion is supposed to explain why certain parts of plantsand animals (e.g. the stem, thighs, arms and trunk) are rounded rather thantriangular or polygonal. Some scholars argue that it was the influence ofArchytas that led Plato and Eudemus to emphasize uniform circular movement inexplaining the heavens (Zhmud 2006: 97). An explanation of motion in terms of

proportion fits well with the rest of evidence for Archytas, but the details remainobscure.

6. Ethics and Political Philosophy


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Archytas' search for the numbers in things was not limited to the natural world.Political relationships and the moral action of individuals were also explained interms of number and proportion. In B3, rational calculation is identified as the

basis of the stable state:

Once calculation (logismos) was discovered, it stopped discord and increasedconcord. For people do not want more than their share, and equality exists, oncethis has come into being. For by means of calculation we will seek reconciliationin our dealings with others. Through this, then, the poor receive from the

powerful, and the wealthy give to the needy, both in the confidence that they willhave what is fair on account of this.

The emphasis on equality (isotas) and fairness (to ison) suggests that Archytasenvisages rational calculation (logismos) as heavily mathematical. On the otherhand, logismos is not identical to the technical science of number (logistic – see3.2 above) but is rather a practical ability to understand numerical calculations,

including basic proportions, an ability that is shared by most human beings. It isthe clarity of calculation and proportion that does away with the constant strivingfor more ( pleonexia), which produces discord in the state. Since the state is basedon a widely shared human ability to calculate, an ability that the rich and poorshare, Archytas was led to support a more democratic constitution (see 1.3above) than Plato, who emphasizes the expert mathematical knowledge of a few( R. 546a ff.). Zhmud (2006: 60 – 76) points out connections between B3 andIsocrates and argues that Isocrates is referring to Archytas, when he says thatsome praise the sciences for their utility and others try to demonstrate that theycontribute greatly to virtue ( Busiris23). However, Archytas seems to accept both

of these views about the sciences, while Isocrates refers to two different groupsof people. Isocrates' reference is also very general and makes no allusion to thecentral terms of B3 so that it is doubtful that he has Archytas in mind. For furtherdiscussion of the argument of B3 see Huffman 2005: 182 – 224 and Schofield2008.

Most of our evidence for Archytas' ethical views is, unfortunately, not based onfragments of his writings but rather on anecdotes, which probably ultimatelyderive from Aristoxenus' Life of Archytas. The good life of the individual, no lessthan the stability of the state, appears to have been founded on rational

calculation. Aristoxenus presented a confrontation between the Syracusanhedonist, Polyarchus, and Archytas. Polyarchus' long speech is preserved byAthenaeus and Archytas' response by Cicero (A9 = Deip. 545a andSen. XII 39 – 41 respectively). Polyarchus' defense of always striving for more ( pleonexia) andof the pursuit of pleasure is reminiscent of Plato's presentations of Callicles andThrasymachus, but is not derived from those presentations and is better seen asan important parallel development (Huffman 2002). Archytas bases his responseon the premise that reason (= rational calculation) is the best part of us and the

part that should govern our actions. Polyarchus might grant such a premise, sincehis is a rational hedonism. Archytas responds once again with a thought

experiment. We are to imagine someone in the throes of the greatest possible bodily pleasure (sexual orgasm?). Surely we must agree that a person in such a


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state is not able to engage in rational calculation. It thus appears that bodily pleasure is in itself antithetical to reason and that, the more we succeed inobtaining it, the less we are able to reason. Aristotle appears to refer to thisargument in the Nicomachean Ethics (1152b16 – 18). Archytas' argument isspecifically directed against bodily pleasure and he did not think that all pleasurewas disruptive; he enjoyed playing with children (A8) and recognized that the

pleasures of friendship were part of a good life (Cicero, Amic. XXIII 88). Otheranecdotes emphasize that our actions must be governed by reason rather than theemotions: Archytas refused to punish the serious misdeeds of his slaves, becausehe had become angry and did not want to act out of anger (A7); he restrainedhimself from swearing aloud by writing his curses on a wall instead (A11).

7. Importance and InfluenceArchytas fits the common stereotype of a Pythagorean better than anyone else

does. He is by far the most accomplished Pythagorean mathematician, makingimportant contributions to geometry, logistic/arithmetic and harmonics. He wasmore successful as a political leader than any other ancient philosopher, and thereis a rich anecdotal tradition about his personal self-control. It is striking,however, that there are essentially no testimonia connecting Archytas tometempsychosis or the religious aspect of Pythagoreanism. Archytas is a

prominent figure in the rebirth of interest in Pythagoreanism in first century BCRome: Horace, Propertius and Cicero all highlight him. As the last prominentmember of the early Pythagorean tradition, more pseudo-Pythagorean workscame to be forged in his name than any other Pythagorean, including Pythagoras

himself. His name, with the spelling Architas, continued to exert power inMedieval and Renaissance texts, although the accomplishments assigned to himin those texts are fanciful.

Scholars have typically emphasized the continuities between Plato and Archytas(e.g., Kahn 2001, 56), but the evidence suggests that Archytas and Plato were inserious disagreement on a number of issues. Plato's only certain reference toArchytas is part of a criticism of his approach to harmonics in Book VII ofthe Republic, where there is probably also a criticism of his work in solidgeometry. Plato's attempt to argue for the split between the intelligible and

sensible world in Books VI and VII of the Republic may well be a protrepticdirected at Archytas, who refused to separate numbers from things. It issometimes thought that the eponymous primary speaker in Plato's Timaeus, whois described as a leading political figure and philosopher from southern Italy(20a), must be a stand-in for Archytas. The Timaeus, however, is a most un-Archytan document. It is based on the split between the sensible and intelligibleworld, which Archytas did not accept. Plato argues that the universe is limited,while Archytas is famous for this argument to show that it is unlimited. Platoconstructs the world soul according to ratios that are important in harmonictheory, but he uses Philolaus' ratios rather than Archytas'. Plato does adopt

Archytas' theory of pitch with some modification, but Archytas and Platodisagree on the explanation of sight. Archytas' refusal to split the intelligible


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from the sensible may have made him a more attractive figure to Aristotle, whodevoted four books to him (Huffman 2005: 583 – 594) and praised his definitionsfor treating the composite of matter and form, not of form separate from matter( Metaph. 1043a14 – 26). Archytas' vision of the role of mathematics in the state iscloser to Aristotle's mathematical account of distributive and redistributive

justice ( EN 1130b30 ff.) than to Plato's emphasis on the expert mathematicalknowledge of the guardians. Clearly Archytas was an important influence on

both Plato and Aristotle, but the exact nature of those philosophical relationshipsis complex.

BibliographyTexts and Commentaries

Diels, H. and W. Kranz, 1952, Die Fragmente der Vorsokratiker (in threevolumes), 6th edition, Dublin and Zürich: Weidmann. The Archytasmaterial is in Volume 1, Chapter 47, pp. 421 – 439 (Greek texts of thefragments and testimonia with translations in German). Referred to as DK.

Huffman, C. A., 2005, Archytas of Tarentum: Pythagorean, Philosopherand Mathematician King , Cambridge: Cambridge University Press (Themost complete and up-to-date collection of fragments and testimonia withtranslations and commentary in English).

Timpanaro Cardini, M., 1958 – 64, Pitagorici, Testimonianze e frammenti,

3 vols., Firenze: La Nuova Italia, Vol. 2, 262 – 385 (Greek texts of thefragments and testimonia with translations and commentary in Italian).

General Bibliography

Barker, A. D., 1989, Greek Musical Writings, Vol. II: Harmonic and Acoustic Theory, Cambridge: Cambridge University Press. (A goodaccount of Archytas' harmonic theory.)

–––, 1994, „Ptolemy's Pythagoreans, Archytas, and Plato's conception ofmathematics‟, Phronesis, 39(2): 113 – 135.

–––, 2006, „Archytas Unbound: A Discussion of Carl A.

Huffman, Archytas of Tarentum‟, Oxford Studies in Ancient Philosophy,31: 297 – 321.

Barnes, J., 1982, The Presocratic Philosophers, London: Routledge.

Berryman, Sylvia, 2003, „Ancient Automata and MechanicalExplanation‟, Phronesis, 48(4): 344 – 369.

Blass, F., 1884, „De Archytae Tarentini fragmentis mathematicis‟,

in Mélanges Graux, Paris: Ernest Thorin.

Bowen, A. C., 1982, „The foundations of early Pythagorean harmonicscience: Archytas, fragment 1‟, Ancient Philosophy, 2: 79 – 104.


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Brisson, Luc, 1987, Platon: Lettres, Paris: Flammarion.

Burkert, W., 1961, „Hellenistische Pseudopythagorica‟, Philologus, 105:16 – 43, 226 – 246.

––– , 1972a, Lore and Science in Ancient Pythagoreanism, E. Minar (tr.),

Cambridge, Mass.: Harvard University Press; 1st German edn., 1962. –––, 1972b, „Zur geistesgeschichtlichen Einordnung einiger

Pseudopythagorica‟, in Pseudepigrapha I , Fondation Hardt EntretiensXVIII, Vandoeuvres-Genève, 25 – 55.

Cambiano, Giuse ppe, 1998, „Archimede Meccanico et La Meccanica diArchita‟, Elenchos, 19(2): 291 – 324.

Cassio, Albio Cesare, 1988, „Nicomachus of Gerasa and the Dialect of

Archytas, Fr. 1‟, Classical Quarterly (ns), 38: 135 – 139.

Centrone, Bruno, 1990, Pseudopythagorica Ethica, Naples: Bibliopolis. –––, 1994a, „Archytas de Tarente‟, in Dictionnaire des Philosophes Antiques, vol. 1, Richard Goulet (ed.), Paris: CNRS Éditions, 339 – 342.

–––, 1994b, „Pseudo-Archytas‟, in Dictionnaire des Philosophes Antiques,vol. 1, Richard Goulet (ed.), Paris: CNRS Éditions, 342 – 345.

–––, 2000, „Platonism and Pythagoreanism in the early empire‟, in TheCambridge History of Greek and Roman Political Thought , ChristopherRowe and Malcolm Schofield (eds.), Cambridge: Cambridge UniversityPress, 559 – 584.

Ciaceri, E., 1927 – 32, Storia della Magna Grecia, I – III, Milan-Rome:Albrighi, Segati & C.

Cuomo, S., 2000, Pappus of Alexandria and the Mathematics of Late Antiquity, Cambridge: Cambridge University Press.

Delatte, A., 1922, Essai sur la politique pythagoricienne, Liège and Paris:H. Vaillant-Carmanne and Édouard Champion.

Diels, H., 1965, Antike Technik , 3rd edn., Osnabrück: Otto Zeller.

Diogenes Laertius, Lives of Eminent Philosophers, R. D. Hicks (tr.),

Cambridge, Mass.: Harvard University Press, 1925.Eutocius, 1915, Archimedis opera omnia cum commentariis Eutocii, vol.3, J. L. Heiberg and E. Stamatis (eds.), Leipzig: Teubner (repr. Stuttgart:1972).

Folkerts, M.,1970, “Boethius” Geometrie II: Ein Mathematisches Lehrbuch des Mittelalters, Wiesbaden: Franz Steiner.

Frank, E., 1923, Plato und die sogenannten Pythagoreer: Ein Kapitel ausder Geschichte des griechischen Geistes, Halle: Max Niemeyer.


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Frischer, Bernard, 1984, „Horace and the Monuments: A NewInterpretation of the Archytas Ode (C.1.28)‟, Harvard Studies in Classical

Philology, 88: 71 – 102.

Gruppe, O. F., 1840, Über die Fragmente des Archytas und der ältern

Pythagoreer , Berlin: O. Sichler.Guthrie, W. K. C., 1962, A History of Greek Philosophy, Vol. 1,Cambridge: Cambridge University Press.

Harvey, F. D., 1965 –66, „Two Kinds of Equality‟, Classica et Mediaevalia, 26: 101 – 146.

Heath, T. L., 1921, A History of Greek Mathematics, 2 vols., Oxford:Clarendon Press.

Huff man, C. A., 1985, „The Authenticity of Archytas Fr. 1‟, ClassicalQuarterly, 35(2): 344 – 348.

––– , 1993, Philolaus of Croton: Pythagorean and Presocratic,Cambridge: Cambridge University Press.

–––, 1999, „Limite et illimité chez les premiers philosophes grecs‟, in La Fêlure du Plaisir : Études sur le Philèbe de Platon, Vol. II: Contextes, M.Dixsaut (ed.), Paris: Vrin, 11 – 31.

–––, 2001, „The Philolaic Method: The Pythagoreanism behind

the Philebus‟, in Essays in Ancient Greek Philosophy VI: Before Plato, A.Preus (ed.), Albany: State University of New York Press, 67 – 85.

–––, 2002a, „Polyclète et les Présocratiques‟, in Qu' est-ce que la Philosophie Présocratique?, André Laks and Claire Louguet (eds.),Villeneuve d'Ascq: Septentrion, 303 – 327.

–––, 2002b, „Archytas and the Sophists‟, in Presocratic Philosophy: Essays in Honour of Alexander Mourelatos, Victor Caston and Daniel W.Graham (eds.), Aldershot: Ashgate, 251 – 270.

Johnson, Monte Ransome, 2008, „Sources for the Philosophy of

Archytas‟, Ancient Philosophy, 28: 173 – 199.

Kahn, C., 2001, Pythagoras and the Pythagoreans, Indianapolis: Hackett.

Knorr, W. R., 1975, The Evolution of the Euclidean Elements, Dordrecht:D. Reidel.

––– , 1986, The Ancient Tradition of Geometric Problems, Boston:Birkhäuser.

––– , 1989, Textual Studies in Ancient and Medieval Geometry, Boston:Birkhäuser.

Krafft, Fritz, 1970, Dynamishce und Statische Betrachtungsweise in der Antiken Mechanik , Wiesbaden: Franz Steiner.

Krischer, Tilman (1995) „Die Rolle der Magna Graecia in der Geschichte

der Mechanik‟, Antike und Abendland , 41: 60 – 71.


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Lloyd, G. E. R., 1979, Magic, Reason, and Experience, Cambridge:Cambridge University Press.

––– , 1987, The Revolutions of Wisdom, Cambridge: Cambridge UniversityPress.

–––, 1990, „Plato and Archytas in the Seventh Letter‟, Phronesis, 35(2):159 – 174.

Mathieu, Bernard, 1987, „Archytas de Tarente pythagoricien et ami de

Platon‟, Bulletin de l'Association G. Budé, 239 – 255.

Minar, Edwin L., 1942, Early Pythagorean Politics in Practice andTheory, Baltimore: Waverly Press.

Moraux, P., 1984, Der Aristotelismus bei den Griechen von Andronikosbis Alexander von Aphrodisias, Zweiter Band: Der Aristotelismus im I.und II. Jh. n. Chr., Berlin: Walter De Gruyter.

Mueller, I., 1997, „Greek arithmetic, geometry and harmonics: Thales toPlato‟, in Routledge History of Philosophy Vol. I: From the Beginning to

Plato, C. C. W. Taylor (ed.), London: Routledge, 271 – 322.

Netz, R., 1999, The Shaping of Deduction in Greek Mathematics,Cambridge: Cambridge University Press.

Nisbet, R.G.M. and Hubbard, M.A., 1970, A Commentary on Horace:Odes, Book 1, Oxford: Clarendon Press.

O'Meara, D. J., 1989, Pythagoras Revived. Mathematics and Philosophy

in Late Antiquity, Oxford: Clarendon Press. ––– , 2003, Platonopolis: Platon

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