Archytas (Stanford Encyclopedia of Philosophy) Cite this entry Search the SEP • Advanced Search • Tools • RSS FeedTable of Contents• What's New• Archives• Projected ContentsEditorial Information• About the SEP• Editorial Board• How to Cite the SEP• Special CharactersSupport the SEPContact the SEP ©Metaphysics Research Lab,CSLI,Stanford University  Open access to the SEP is made possible by a world-wide funding initiative. Please Read How You Can Help Keep the Encyclopedia FreeArchytasFirst published Thu Jun 26, 2003; substantive revision Wed Jul 25, 2007Archytas of Tarentum was a Greek mathematician, political leader andphilosopher, active in the first half of the fourth century BC (i.e.,during Plato's lifetime). He was the last prominent figure in theearly Pythagorean tradition and the dominant political figure inTarentum, being elected general seven consecutive times. He sent aship to rescue Plato from the clutches of the tyrant of Syracuse,Dionysius II, in 361, but his personal and philosophical connectionsto Plato are complex, and there are many signs of disagreement betweenthe two philosophers. A great number of works were forged in Archytas'name starting in the first century BC, and only four fragments of hisgenuine work survive, although these are supplemented by a number ofimportant testimonia. Archytas was the first to solve one of the mostcelebrated mathematical problems in antiquity, the duplication of thecube. We also have his proof showing that ratios of the form(n+1) : n, which are important in music theory,cannot be divided by a mean proportional. He was the mostsophisticated of the Pythagorean harmonic theorists and providedmathematical accounts of musical scales used by the practicingmusicians of his day. He was the first to identify the group of fourcanonical sciences (logistic [arithmetic], geometry, astronomy andmusic), which would become known as the quadrivium in themiddle ages. There are also some indications that he contributed tothe development of the science of optics and laid the mathematicalfoundations for the science of mechanics. He saw the ultimate goal ofthe sciences as the description of individual things in the world interms of ratio and proportion and accordingly regarded logistic, thescience of number and proportion, as the master science. Rationalcalculation and an understanding of proportion were also the bases ofthe just state and of the good life for an individual. He gavedefinitions of things that took account of both their matter and theirform. Although we have little information about his cosmology, hedeveloped the most famous argument for the infinity of the universe inantiquity.1. Life and Works 1.1 Family, Teachers, and Pupils; Date 1.2 Sources 1.3 Archytas and Tarentum 1.4 Archytas and Plato 1.5 The Authenticity Question 1.6 Spurious Works Ascribed to Archytas 1.7 Genuine Works and Testimonia 2. Archytas as Mathematician and Harmonic Theorist 2.1 Doubling the Cube 2.2 Music and Mathematics 2.3 Evaluation of Archytas as Mathematician 3. Archytas and the Sciences 3.1 The Value of the Sciences 3.2 “Logistic” as the Master Science 3.3 Optics and Mechanics 4. Definitions5. Cosmology and Physics6. Ethics and Political Philosophy7. Importance and InfluenceBibliographyOther Internet ResourcesRelated Entries1. Life and Works1.1 Family, Teachers, and Pupils; Date Archytas, son of Hestiaeus (see Aristoxenus in Diels-Kranz 1952,chap. 47, passage A1; abbreviated as DK47 A1), lived in the Greek cityof Tarentum, on the heel of the boot of Italy. The later traditionalmost universally identifies him as a Pythagorean (e.g., A1, A2, A7,A16). Aristotle and his pupil Eudemus do not explicitly call Archytasa Pythagorean and appear to treat him as an important independentthinker. Plato never refers to Archytas by name except in theSeventh Letter, if that is by Plato, and he is not called aPythagorean there. In the Republic, however, when Platoquotes a sentence which appears in Fr. 1 of Archytas (DK47 B1), heexplicitly labels it as part of Pythagorean harmonics (530d). Cicero(de Orat. III 34. 139) reports that Archytas was the pupil ofPhilolaus, and this is not improbable. Philolaus was the mostprominent Pythagorean of the preceding generation (ca. 470-390) andmay have taught in Tarentum (Huffman 1993, 6). Archytas'achievements in mathematics depend on the work of Hippocrates ofChios, but we have no evidence that he studied with Hippocrates. Theonly pupil of Archytas who is more than a name, is Eudoxus(ca. 390-340), the prominent mathematician. Eudoxus presumably did notlearn his famous hedonism from Archytas (see DK47 A9), and it isspecifically geometry that he is said to have studied with Archytas(Diogenes Laertius VIII 86). Archytas was, roughly speaking, a contemporary of Plato, but it isdifficult to be more precise about his dates. Aristotle's pupil,Eudemus, presents him as the contemporary of Plato (born 428/7) andLeodamas (born ca. 430), on the one hand, and of Theaetetus (born ca.415), on the other (A6). Since it would be difficult to call him thecontemporary of Theaetetus, if he were born much earlier than 435,this is the earliest he was likely to have been born. On the otherhand, he could have been born as late as 410 and still be considered acontemporary of Plato. Strabo associates Archytas with the flourishingof Tarentum, before a period of decline, in which Tarentum hiredmercenary generals (A4). Since the mercenaries appear ca. 340, itseems likely that Archytas was dead by 350 at the latest. Such a dateis in accord with other evidence (A5 = [Demosthenes],Erot. Or. 61.46), which connects Archytas to Timotheus, whodied ca. 355, and with Plato's (?) Seventh Letter(350a), which presents Archytas as still active in Tarentum in361. Thus Archytas was born between 435 and 410 and died between 360and 350. Some scholars (e.g., Ciaceri 1927-32: III 4) have supposed that thespeaker of the Roman poet, Horace's, Archytas Ode (I 28 = A3) isArchytas himself and hence have concluded that Archytas died in ashipwreck. The standard interpretation, however, rightly recognizesthat the speaker is not Archytas but a shipwrecked sailor whoapostrophizes Archytas (Nisbet and Hubbard 1970, 317ff.). The odetells us nothing about Archytas' death, but it is one of manypieces of evidence for the fascination with Archytas by Roman authorsof the first century BC (Propertius IV 1b.77; Varro in B8; Cicero,Rep. I 38.59, I 10.16; Fin. V 29.87;Tusc. IV 36.78, V 23.64, de Orat. III34.139; Amic. XXIII 88; Sen. XII 39-41),perhaps because Pythagoreanism had come to be seen as a native Italianphilosophy, and not a Greek import (Burkert 1961; Powell 1995, 11ff.).1.2 Sources Apart from the surviving fragments of his writings, our knowledge ofArchytas' life and work depends heavily on authors who wrote inthe second half of the fourth century, in the fifty years afterArchytas' death. Archytas' importance both as anintellectual and as a political leader is reflected in the number ofwritings about him in this period, although only fragments of theseworks have been preserved. Aristotle wrote a work in three volumes onthe philosophy of Archytas, more than on any other of hispredecessors, as well as a second work, consisting of a summary ofPlato's Timaeus and the writings of Archytas (A13).Unfortunately almost nothing of these works hassurvived. Aristotle's pupil Eudemus discussed Archytasprominently in his history of geometry (A6 and A14) and in his work onphysics (A23 and A24). Another pupil of Aristotle's,Aristoxenus, wrote a Life of Archytas, which is the basis formuch of the biographical tradition about him (A1, A7, A9). Aristoxenus(375-ca. 300) was in a good position to have accurate informationabout Archytas. He was born in Tarentum and grew up during the heightof Archytas' prominence in the city. In addition to whateverpersonal knowledge he had of Archytas, he draws on his own fatherSpintharus, who was a younger contemporary of Archytas, as a source(e.g., A7). Aristoxenus began his philosophical career as aPythagorean and studied with the Pythagorean Xenophilus at Athens, sothat it is not surprising that his portrayal of Archytas is largelypositive. Nonetheless, Archytas' opponents are given a fairhearing (e.g., Polyarchus in A9), and Archytas himself is representedas not without small flaws of character (A7). Other fourth-centurysources such as the Seventh Letter in the Platonic corpus andDemosthenes' (?) Erotic Oration focus on the connectionbetween Archytas and Plato (see below).1.3 Archytas and Tarentum Archytas is unique among Greek philosophers for the prominent role heplayed in the politics of his native city. He was elected general(stratêgos) seven years in succession at one point inhis career (A1), a record that reminds us of Pericles at Athens. Hiselection 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 andthat, when at one point he was forced to withdraw from his post by theenvy of his enemies, the Tarentines immediately suffered defeat (A1).He probably served as part of a board of generals (there was a boardof ten at Athens). The analogy with Athens suggests that as a generalhe may also have had special privileges in addressing the assembly atTarentum on issues of importance to the city, so that his position asgeneral gave him considerable political as well as military power. Atsome point in his career, he may have been designated as a generalautokratôr (“plenipotentiary”) (A2), whichgave him special latitude in dealing with diplomatic and militarymatters without consulting the assembly, although this was notdictatorial power and all arrangements probably required the eventualapproval of the assembly. We do not know when Archytas served hisseven successive years as general. Some have supposed that they mustcoincide with the seven year period which includes Plato's second andthird visits to Italy and Sicily, 367-361 (e.g., Wuilleumier 1939,68-9), but Archytas need not have been stratêgos toplay the role assigned to him during these years in the SeventhLetter. The evidence suggests that most of Archytas'military campaigns were directed not at other Greeks but at nativeItalic peoples such as the Messapians and Lucanians, with whomTarentum had been in constant conflict since its founding. It is important to recognize that the Tarentum in which Archytasexercised such influence was not some insignificant backwater. Spartancolonists founded it in 706. It was initially overshadowed by otherGreek colonies in southern Italy such as Croton, although it had thebest harbor on the south coast of Italy and was the natural stoppingpoint for any ships sailing west from mainland Greece. Archytas willhave grown up in a Tarentum that, in accord with its foundation bySparta, took the Peloponnesian and Syracusan side against Athens inthe Peloponnesian War (Thuc. VI 44; VI 104; VII 91). Athens alliedwith the Messapians (Thuc. VII 33), the long-standing enemy of theTarentines, against whom Archytas would later lead expeditions (A7).After the Peloponnesian War, Tarentum appears to have avoided directinvolvement in the conflict between the tyrant of Syracuse, DionysiusI, and a league of Greek cities in southern Italy headed by Croton.After Dionysius crushed the league, Tarentum emerged as the mostpowerful Greek state in southern Italy and probably became the newhead of the league of Italiot Greek cities (A2). In the period from380-350, when Archytas was in his prime and old age, Tarentum was oneof the most powerful cities in the Greek world (Purcell 1994,388). Strabo's description of its military might (VI 3.4) comparesfavorably with Thucydides' account of Athens at the beginning ofthe Peloponnesian war (II. 13). Despite its ancestral connections to Sparta, which was an oligarchy,Tarentum appears to have been a democracy during Archytas'lifetime. According to Aristotle (Pol. 1303a), the democracywas founded after a large part of the Tarentine aristocracy was killedin a battle with a native people, the Iapygians, in 473. Herodotusconfirms that this was the greatest slaughter of Greeks of which hewas aware (VII 170). There is no evidence that Tarentum was anythingbut a democracy between the founding of the democracy in 473 andArchytas' death ca. 350. Some scholars have argued thatTarentum's ties to Sparta and the supposed predilection of thePythagoreans for aristocracy will have insured that Tarentum did notremain a democracy long and that it was not a democracy under 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 ofArchytas' power in Tarentum stress his popularity with themasses and his election as general by the citizens (A1 and A2).Finally, Aristotle's account of the structure of the Tarentinegovernment in the fourth century (Pol. 1291b14), whilepossibly consistent with other forms of government, makes most senseif Tarentum was a democracy. The same is true of fr. B3 of Archytas,with its emphasis on a more equal distribution of wealth.1.4 Archytas and Plato Archytas was most famous in antiquity and is most famous in themodern world for having sent a ship to rescue Plato from the tyrant ofSyracuse, Dionysius II, in 361. In both the surviving ancient lives ofArchytas (by Diogenes Laertius, VIII 79-83, and in the Suda)the first thing mentioned about him, after the name of his city-stateand his father, is his rescue of Plato (A1 and A2). This story is toldin greatest detail in the Seventh Letter ascribed toPlato. It has accordingly been typical to identify Archytas as“the friend of Plato” (Mathieu 1987). Archytas first metPlato over twenty years earlier, when Plato visited southern Italy andSicily for the first time in 388/7, during his travels after the deathof 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 “newmodel philosopher for Plato” (Vlastos 1991, 129), and he hasbeen regarded as the archetype of Plato's philosopher-king(Guthrie 1962, 333). The actual situation appears to be considerablymore complicated. The ancient evidence, apart from the SeventhLetter, presents the relationship between Archytas and Plato indiametrically opposed ways. One tradition does present Archytas as thePythagorean master at whose feet Plato sat, after Socrates had died(e.g., Cicero, Rep. I 10.16), but another tradition makesArchytas the student of Plato, to whom he owed his fame and success inTarentum (Demosthenes [?], Erotic Oration 44). The Seventh Letter itself is of contested authenticity,although most scholars regard it either as the work of Plato himselfor of a student of Plato who had considerable familiarity withPlato'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 hasrecently argued, however, that the letter also serves to distancePlato from Pythagoreanism and from Archytas (1990). Nothing in theletter suggests that Plato was ever the pupil of Archytas; instead therelationship is much closer to that presented in the EroticOration. Plato is presented as the dominant figure upon whomArchytas depends both philosophically and politically. Archytas writesto Plato claiming that Dionysius II has made great progress inphilosophy, in order to urge Plato to come to Sicily a third time(339d-e). These claims are belied as soon as Plato arrives (340b). Theletter thus suggests that, far from being the Pythagorean master fromwhom Plato learned his philosophy, Archytas had a very imperfectunderstanding of what Plato considered philosophy to be. The lettermakes clear that Plato does have a relationship of xenia,“guest-friendship,” with Archytas and others at Tarentum(339e, 350a). This relationship was probably established onPlato's first visit in 388/7, since Plato uses it as a basis toestablish a similar relationship between Archytas and Dionysius IIduring his second visit in 367 (338c). It is also the relationship interms of which Plato appeals to Archytas for help, when he is indanger after the third trip to Sicily goes badly (350a). Such afriendship need not imply any close personal intimacy,however. Aristotle classifies xenia as a friendship forutility and points out that such friends do not necessarily spend muchtime together or even find each other's company pleasant(EN 1156a26 ff.). Apart from Archytas' rescue of Platoin 361 (even this is described as devised by Plato [350a]), Plato isclearly the dominant figure in the relationship. Archytas is portrayedas Plato's inferior in his understanding of philosophy, andPlato is even presented as responsible for some of Archytas'political success, insofar as he establishes the relationship betweenArchytas and Dionysius II, which is described as of considerablepolitical importance (339d). How are we to unravel the true nature of the relationship betweenPlato and Archytas in the light of this conflicting evidence? Apartfrom the Seventh Letter, Plato never makes a direct referenceto Archytas. He does, however, virtually quote a sentence fromArchytas' book on harmonics in Book VII of the Republic(530d), and his discussions of the science of stereometry shortlybefore this are likely to have some connections to Archytas'work in solid geometry (528d). It is thus in the context of thediscussion of the sciences that Plato refers to Archytas, and theremains of Archytas' work focus precisely on the 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 inpart motivated by his desire to meet Archytas, as the first traditionclaims, but that he sought Archytas out not as a new “modelphilosopher” but rather as an expert in the mathematicalsciences, in which Plato had developed a deep interest. InRepublic VII, Plato is critical of Pythagorean harmonics andof current work in solid geometry on philosophical grounds, so that,while he undoubtedly learned a considerable amount of mathematics fromArchytas, he clearly disagreed with Archytas' understanding of thephilosophical uses of the sciences. In 388 Tarentum had not yetreached the height of its power, and Archytas is not likely to haveachieved his political dominance yet, so that there may also be sometruth to the claim of the second tradition that Archytas did notachieve his great practical success until after his contact withPlato; whether or not that success had any direct relationship to hiscontact with Plato is more doubtful. On their first meeting in 388/7,Plato and Archytas established a relationship of guest-friendship,which obligated them to further each other's interests, whichthey did, as the events of 367-361 show. Plato and Archytas need nothave been in agreement on philosophical issues and are perhaps betterseen as competitive colleagues engaged in an ongoing debate as to thevalue of the sciences for philosophy.1.5 The Authenticity Question More pages of text have been preserved in Archytas' name thanin the name of any other Pythagorean. Unfortunately the vast majorityof this material is rightly regarded as spurious. The same is true ofthe Pythagorean tradition in general; the vast majority of texts whichpurport to be by early Pythagoreans are, in fact, laterforgeries. Some of these forgeries were produced for purely monetaryreasons; a text of a “rare” work by a famous Pythagoreancould fetch a considerable sum from book collectors. There werecharacteristics unique to the Pythagorean tradition, however, that ledto a proliferation of forgeries. Starting as early as the later fourthcentury BC, Pythagoras came to be regarded, in some circles, as thephilosopher par excellence, to whom all truth had been revealed. Alllater philosophy, insofar as it was true, was a restatement of thisoriginal revelation (see, e.g., O'Meara 1989). In order tosupport this view of Pythagoras, texts were forged in the name ofPythagoras and other early Pythagoreans, to show that they had, infact, anticipated the most important ideas of Plato andAristotle. These pseudo-Pythagorean texts are thus characterized bythe use of central Platonic and Aristotelian ideas, expressed in thetechnical terminology used by Plato and Aristotle. Some of theforgeries even attempt to improve on Plato and Aristotle by addingrefinements to their positions, which were first advanced severalhundred years after their deaths. The date and place of origin ofthese pseudo-Pythagorean treatises is difficult to determine, but mostseem to have been composed between 150 BC and 100 AD (Burkert 1972b;Centrone 1990; Moraux 1984); Rome (Burkert 1972b) and Alexandria(Centrone 1990) are the most likely places of origin. Archytas is thedominant figure in this pseudo-Pythagorean tradition. In Thesleff1965's collection of the pseudo-Pythagorean writings, forty-fiveof the two-hundred and forty-five pages (2-48), about 20%, comprisingsome 1,200 lines, are devoted to texts forged in Archytas'name. On the other hand, the fragments likely to be genuine, which arecollected in DK, fill out only a hundred lines of text. Thus, over tentimes more spurious than genuine material has been preserved inArchytas' name. It may well be that the style and Doric dialectof the pseudo-Pythagorean writings were also based on the model ofArchytas' genuine writings.1.6 Spurious Works Ascribed to Archytas All of the treatises under Archytas' name collected in Thesleff1965 are universally regarded as spurious, except for On Law andJustice, where some controversy remains. Most are only preservedin fragments, although there are two brief complete works. The mostfamous of these forgeries is Concerning the Whole System [sc. ofCategories] or Concerning the Ten Categories (preservedcomplete, see Szlezak 1972). This work along with the treatise OnOpposites (Thesleff 1965, 15.3-19.2) and the much later TenUniversal Assertions (preserved complete, first ascribed toArchytas in the 15th century AD; see Szlezak 1972)represent the attempt to claim Aristotle's doctrine ofcategories for Archytas and the Pythagoreans. This attempt was to someextent successful; both Simplicius and Iamblichus regarded theArchytan works on categories as genuine anticipations of Aristotle(CAG VIII. 2, 9-25). Concerning the Ten Categoriesand On Opposites are very frequently cited in the ancientcommentaries on Aristotle's Categories. Pseudo-Archytasidentifies ten categories with names that are virtually identical tothose used by Aristotle, and his language follows Aristotle closely inmany places. The division of 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 therest. Thus, the works in Archytas' name must have been forgedafter Andronicus' work in the first century BC. Other spuriousworks 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; OnBeing (Thesleff 1965, 40.1-16) and On Wisdom (Thesleff1965, 43.24-45.4). There are also fragments of two surely spurious treatises on ethicsand politics, which have recent editions with commentary: On theGood and Happy Man (Centrone 1990), which shows connections toArius Didymus, an author of the first century BC, and On MoralEducation (Centrone 1990), which has ties to Carneades (2ndc. BC). The status of one final treatise is less clear. The fragmentsof On Law and Justice (Thesleff 1965, 33.1-36.11) werestudied in some detail by Delatte (1922), who showed that the treatisedeals with the political conceptions of the fourth century and whocame to the modest conclusion that the work might be by Archytas,since there were no positive indications of late composition. Thesleffsimilarly concluded that the treatise “may be authentic or atleast comparatively old” (1961, 112), while Minar maintainedthat “it has an excellent claim to authenticity” (1942,111). On the other hand, DK did not include the fragments of OnLaw and Justice among the genuine fragments, and most recentscholars have argued that the treatise is spurious. Aalders providesthe most detailed treatment, although a number of his arguments areinconclusive (1968, 13-20). Other opponents of authenticity areBurkert (1972a), Moraux (1984, 670-677) and Centrone (2000). Theconnections of On Law and Justice to the genuine fr. B2 ofArchytas speak for its authenticity, but its ties topseudo-Pythagorean treatises by “Diotogenes” (Thesleff76.2-3, 71. 21-2), “Damippos” (Thesleff 68.26) and“Metopos” (Thesleff 119.28) argue for itsspuriousness. Some testimonia suggest that there were even more pseudo-Archytantreatises, which have not survived even in fragments (Thesleff 47.8ff.). Two spurious letters of Archytas survive. One is the letter towhich the pseudo-Platonic Twelfth Letter is responding (D.L.VIII 79-80), and the other is the purported letter of Archytas toDionysius II, which was sent along with the ship in order to securePlato's release in 361 (D.L. III 21-2). Archytas was a popularfigure in the Middle Ages and early Renaissance, when works continuedto be forged in his name, usually with the spelling Architas orArchita. The Ars geometriae, which is ascribed to Boethius,but was in reality composed in the 12th century (Folkerts1970, 105), ascribes discoveries in mathematics to Architas which areclearly spurious (Burkert 1972a, 406). Several alchemical recipesinvolving the wax of the left ear of a dog and the heart of a wolf areascribed to Architas in ps.-Albertus Magnus, The Marvels of theWorld (De mirabilibus mundi - 13th centuryAD). Numerous selections from a book entitled On Events inNature (de eventibus in natura, also cited as deeffectibus in natura and as de eventibus futurorum) byArchita Tharentinus (or Tharentinus, or just Tharen) are preserved inthe medieval texts known as The Light of the Soul (LumenAnimae), which were composed in the fourteenth century andcirculated widely in Europe in the fifteenth century as a manual forpreachers (Rouse 1971; Thorndike 1934, III 546-60). An apocryphalwork, The Circular Theory of the Things in the Heaven, byArchytas Maximus [!], which has never been published in full, ispreserved in Codex Ambrosianus D 27 sup. (See Catalogus CodicumAstrologorum Graecorum, ed. F. Cumont et al., Vol. III, p.11).1.7 Genuine Works and Testimonia No list of Archytas' works has come down to us from antiquity,so that we don't know how many books he wrote. In the face ofthe large mass of spurious works, it is disappointing that only a fewfragments of genuine works have survived. Most scholars accept asgenuine 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 ofBowen (1982) and Huffman (1985). Our evidence for the titles ofArchytas' genuine writings depends largely on the citationsgiven by the authors who quote the fragments. Fragments B1 and B2 arereported to come from a treatise entitled Harmonics, and themajor testimonia about Archytas' harmonic theory are likely tobe ultimately based on this book (A16-19). This treatise began with adiscussion of the basic principles of acoustics (B1), defined thethree types of mean which are of importance in music theory (B2), andwent on to present Archytas' mathematical descriptions of thetetrachord (the fourth) in the three main genera (chromatic, diatonic,and enharmonic - A16-A19). B3 probably comes from a work OnSciences, which may have been a more general discussion of thevalue of mathematics for human life in general and for theestablishment of a just state in particular. B4 comes from a workentitled Discourses (Diatribai). The fragment itselfasserts the priority of the science of calculation (halogistika, “logistic”) to the other sciences, such asgeometry, and thus suggests a technical work of mathematics. The titleDiatribai would more normally suggest a treatise of ethicalcontent, however, so that in this work the sciences may have beenevaluated in terms of their contribution to the wisdom that leads to agood life. A relatively rich set of testimonia, many from authors of the fourthcentury BC, indicate that Archytas wrote other books aswell. Archytas' famous argument for the unlimited extent of theuniverse (A24), his theory of vision (A25), and his account of motion(A23, A23a) all suggest that he may have written a work oncosmology. Aristotle's comments in the Metaphysics suggestthat Archytas wrote a book on definition (A22), and A20 and A21 mightsuggest a work on arithmetic. Perhaps there was a treatise on geometryor solid geometry in which Archytas' solution to the problem ofdoubling the cube (A14-15) was published. There is also a tradition ofanecdotes about Archytas, which probably ultimately derives fromAristoxenus' Life of Archytas (A7, A8, A9, A11). It ispossible that even the testimonia for Archytas' argument for anunlimited universe and his theory of vision were derived fromanecdotes preserved by Aristoxenus, and not at all from works ofArchytas' own. It is uncertain whether the treatises On Flutes (B6), OnMachines (B1 and B7), and On Agriculture (B1 and B8),which were in circulation under the name of Archytas, were in fact byArchytas of Tarentum or by other men of the same name. DiogenesLaertius lists three other writers with the name Archytas (VIII 82).The treatise On the Decad mentioned by Theon (B5) might be byArchytas, but the treatise by Philolaus with which it is paired isspurious (Huffman 1993, 347-350), thus suggesting that the same may betrue of the treatise under Archytas' name as well.2. Archytas as Mathematician and Harmonic Theorist2.1 Doubling the Cube Archytas was the first person to arrive at a solution to one of themost famous mathematical puzzles in antiquity, the duplication of thecube. The most romantic version of the story, which occurs in manyvariations and ultimately goes back to Eratosthenes (3rd c. BC),reports that the inhabitants of the Greek island of Delos were besetby a plague and, when they consulted an oracle for advice, were toldthat, if they doubled the size of a certain altar, which had the formof a cube, the plague would stop (Eutocius, in Archim. sphaer. etcyl. II [III 88.3-96.27 Heiberg/Stamatis]). The simple-mindedresponse to the oracle, which is actually assigned to the Delians insome versions, is to build a second altar identical to the first oneand set it on top of the first (Philoponus, In Anal. post.,CAG XIII.3, 102.12-22). The resulting altar does indeed have avolume twice that of the first altar, but it is no longer a cube. Thenext simple-minded response is to assume that, since we want an altarthat is double in volume, while still remaining a cube, we shouldbuild the new altar with a side that is double the length of the sideof the original altar. This approach fails as well. Doubling the sideof the altar produces a new altar that is not twice the volume of theoriginal altar but eight times the volume. If the original altar had aside of two, then its volume would be 23 or 8, while analtar built on a side twice as long will have a volume of43 or 64. What then is the length of the side which willproduce a cube with twice the volume of the original cube? The Delianswere at a loss and presented their problem to Plato in theAcademy. Plato then posed the “Delian Problem,” as it cameto be known, to mathematicians associated with the Academy, and noless than three solutions were devised, those of Eudoxus, Menaechmus,and Archytas. It is not clear whether or not the story about the Delians has anybasis in fact. Even if it does, it should not be understood to suggestthat the problem of doubling the cube first arose in the fourth centurywith the Delians. We are told that the mathematician, Hippocrates ofChios, who was active in the second half of the fifth century, hadalready confronted the problem and had reduced it to a slightlydifferent problem (Eutocius, in Archim. sphaer. et cyl. II[III 88.3-96.27 Heiberg/Stamatis]). Hippocrates recognized that if wecould find two mean proportionals between the length of the side of theoriginal cube G, and length D, where D = 2G, so that G : x :: x : y ::y : D, then the cube on length x will be double the cube on length G.Exactly how Hippocrates came to see this is conjectural and need notconcern us here, but that he was right can be seen relatively easily.Each of the values in the continued proportion G : x :: x : y :: y : Dis equal to G : x, so we can set them all equal to G : x. If we do thisand multiply the three ratios together we get the value G3 :x3. On the other hand, if we take the same continuedproportion and carry out the multiplication in the original terms, thenG : x times x : y yields G : y, and G : y times the remaining termgives G : D. Thus G : D = G3 : x3, but D is twiceG so x3 is twice G3. Remember that G was thelength of the side of the original cube, so the cube that is twice thecube 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 ingeometry. After Hippocrates the problem of doubling the cube was alwaysseen as the problem of finding two lines such that they were meanproportionals between G, the length of the side of the original cube,and D, a length which is double G. It was to this form of the problemthat Archytas provided the first solution. Archytas' solution has been rightly hailed as “the mostremarkable of all [the solutions]” and as a “boldconstruction in three dimensions” (Heath 1921, 246); Muellercalls it “a tour de force of the spatialimagination” (1997, 312 n. 23). We owe the preservation ofArchytas' solution to Eutocius, who in the sixth century AD collectedsome eleven solutions to the problem as part of his commentary on thesecond book of Archimedes' On the Sphere andCylinder. Eutocius' source for Archytas' solution was ultimatelyAristotle's pupil Eudemus, who in the late fourth century BC wrote ahistory of geometry. The solution is complex and it is not possible togo through it step by step here (see Huffman 2005,342-360 for adetailed treatment of the solution). Archytas proceeds by constructinga series of four similar triangles (see Figure 1 below) and thenshowing that the sides are proportional so that AM : AI :: AI : AK ::AK : AD, where AM was equal to the side of the original cube (G) andAD was twice AM. Thus the cube double the volume of the cube on AMshould be built on AI. The real difficulty was in constructing thefour similar triangles, where the given length of the side of theoriginal cube and a length double that magnitude were two of the sidesin the similar triangles. The key point for the construction of thesetriangles, point K, was determined as the intersection of two rotatingplane figures. The first figure is a semicircle, which isperpendicular to the plane of the circle ABDZ and which starts on thediameter AED and, with point A remaining fixed, rotates to positionAKD. The second is the triangle APD, which rotates up out of the planeof the circle ABDZ to position ALD. As each of these figures rotates,it traces a line on the surface of a semicylinder, which isperpendicular to the plane of ABDZ and has ABD as its base. Theboldness and the imagination of the construction lies in envisioningthe intersection at point K of the line drawn by the rotatingsemicircle on the surface of the semicylinder with the line drawn bythe rotating triangle on the same surface. We simply don't know whatled Archytas to produce this amazing feat of spatial imagination, inorder to construct the triangles with the sides in appropriateproportion.   Figure 1 In the later tradition, Plato is reported to have criticizedArchytas' solution for appealing to “constructions thatuse instruments and that are mechanical” (Plutarch, TableTalk VIII 2.1 [718e]; Marc. XIV 5-6). Plato argued thatthe value of geometry and of the rest of mathematics resided in theirability to turn the soul from the sensible to intelligible realm. Thecube with which geometry deals is not a physical cube or even adrawing of a cube but rather an intelligible cube that fits thedefinition of the cube but is not a sense object. By employingphysical instruments, which “required much commonhandicraft,” and in effect constructing machines to determinethe two mean proportionals, Archytas was focusing not on theintelligible world but on the physical world and hence destroying thevalue of geometry. Plato's quarrel with Archytas is a charmingstory, but it is hard to reconcile with Archytas' actualsolution, which, as we have seen, makes no appeal to any instrumentsor machines. The story of the quarrel, which is first reported inPlutarch in the first century AD, is also hard to reconcile with ourearliest source for the story of the Delian problem, Eratosthenes.Eratosthenes had himself invented an instrument to determine meanproportionals, the mesolab (“mean-getter”), andhe tells the story of the Delian problem precisely to emphasize thatearlier solutions, including that of Archytas, were in the form ofgeometrical demonstrations, which could not be employed for practicalpurposes. He specifically labels Archytas' solution asdysmêchana, “hardly mechanical.” Somescholars attempt to reconcile Plutarch'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 inArchytas' solution, might be regarded as mechanical, since motion isinvolved (Knorr 1986, 22). It may be, however, that Plutarch'sstory of a quarrel between Plato and Archytas over the use ofmechanical devices in geometry is an invention of the later tradition(Riginos 1976, 146; Zhmud 1998, 217) and perhaps served as a sort offoundation myth for the science of mechanics, a myth which explainedthe separation of mechanics from philosophy as the result of a quarrelbetween two philosophers. In the Republic, Plato is criticalof the solid geometry of his day, but his criticism makes no mentionof the use of instruments. His criticism instead focuses on thefailure of solid geometry to be developed into a coherent disciplinealongside geometry and astronomy (528b-d). This neglect of solidgeometry is ascribed to the failure of the Greek city-states to holdthese difficult studies in honor, the lack of a director to organizethe studies, and the arrogance of the current experts in the field,who would not submit to such a director. Since Archytas'duplication of the cube shows him to be one of the leading solidgeometers of the time, it is hard to avoid the conclusion that Platoregarded him as one of the arrogant experts, who focused on solvingcharming 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 makingTarentum a state which held solid geometry in esteem.2.2 Music and Mathematics One of the most startling discoveries of early Greek science was thatthe fundamental intervals of music, the octave, the fourth, and thefifth, corresponded to whole number ratios of string length. Thus, ifwe pluck a string of length x and then a string of length 2x, we willhear the interval of an octave between the two sounds. If the twostring lengths are in the ratio 4 : 3, we will hear a fourth, and, ifthe ratio is 3 : 2, we will hear a fifth. This discovery that thephenomena of musical sound are governed by whole number ratios musthave played a central role in the Pythagorean conception, firstexpressed by Philolaus, that all things are known through number (DK44 B4). The next step in harmonic theory was to describe an entireoctave length scale in terms of mathematical ratios. The earliest suchdescription of a scale is found in Philolaus fr. B6. Philolausrecognizes that, if we go up the interval of a fourth from any givennote, and then up the interval of a fifth, the final note will be anoctave above the first note. Thus, the octave is made up of a fourthand a fifth. In mathematical terms, the ratios that govern the fifth(3 : 2) and fourth (4 : 3) are added by multiplying the terms and thusproduce an octave (3 : 2 x 4 : 3 = 12 : 6 = 2 : 1). The intervalbetween the note that is a fourth up from the starting note and thenote that is a fifth up was regarded as the basic unit of the scale,the whole tone, which corresponded to the ratio of 9 : 8 (subtractionof ratios is carried out by dividing the terms, or cross multiplying:3 : 2 / 4 : 3 = 9 : 8). The fifth was thus regarded as a fourth plus awhole tone, and the octave can be regarded as two fourths plus a wholetone. 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). Philolaus' scale thus consisted of the followingintervals: 9 : 8, 9 : 8, 256 : 243 [these three intervals take us up afourth], 9 : 8, 9 : 8, 9 : 8, 256 : 243 [these four intervals make upa fifth and complete the octave from our staring note]. This scale isknown as the Pythagorean diatonic and is the scale that Plato adoptedin the 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 centuryAD, identifies Archytas as having “engaged in the study of musicmost of all the Pythagoreans” (A16). First, Archytas provided ageneral explanation of pitch, arguing that the pitch of a sounddepends on the speed with which the sound is propagated and travels(B1). Thus, if a stick is waved back and forth rapidly, it willproduce a sound that travels rapidly through the air, which will beperceived as of a higher pitch than the sound produced by a stickwaved more slowly. Archytas is correct to associate pitch with speed,but he misunderstood the role of speed. The pitch does not depend onthe speed with which a sound reaches us but rather on the frequency ofimpacts in a given period of time. A string that vibrates more rapidlyproduces a sound of a higher pitch, but all sounds, regardless ofpitch, travel at an equal velocity, if the medium is thesame. Although Archytas' account of pitch was ultimatelyincorrect, it was very influential. It was taken over and adapted byboth Plato and Aristotle and remained the dominant theory throughoutantiquity (Barker 1989, 41 n. 47). Second, Archytas introduced newmathematical rigor into Pythagorean harmonics. One of the importantresults of the analysis of music in terms of whole number ratios isthe recognition that it is not possible to divide the basic musicalintervals in half. The octave is not divided into two equal halves butinto a fourth and a fifth, the fourth is not divided into two equalhalves but into two whole tones and a remainder. The whole tone cannotbe divided into two equal half tones. On the other hand, it ispossible to divide a double octave in half. Mathematically this can beseen by recognizing that it is possible to insert a mean proportionalbetween 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 intotwo equal parts each having a ratio of 2 : 1. The ratios which governthe basic musical intervals (2 : 1, 4 : 3, 3 : 2, 9 : 8), all belongto a type of ratio known as a superparticular ratio -- roughlyspeaking, ratios of the form (n + 1) : n. Archytas made a crucialcontribution by providing a rigorous proof that there is no meanproportional between numbers in superparticular ratio (A19) and hencethat the basic musical intervals cannot be divided inhalf. Archytas' proof was later taken over and modified slightlyin the Sectio Canonis ascribed to Euclid (Prop. 3; see Barker1989, 195). Archytas' final contribution to music theory has to do with thestructure of the scale. The Greeks used a number of different scales,which were distinguished by the way in which the fourth, ortetrachord, was constructed. These scales were grouped into three maintypes or genera. One genus was called the diatonic; one example ofthis is the Pythagorean diatonic described above, which is built onthe tetrachord with the intervals 9 : 8, 9 : 8 and 256 : 243 and wasused by Philolaus and Plato. There is no doubt that Archytas knew ofthis diatonic scale, but his own diatonic tetrachord was somewhatdifferent, being composed of the intervals 9 : 8, 8 : 7 and 28 :27. Archytas also defined scales in the two other major genera, theenharmonic and chromatic. Archytas' enharmonic tetrachord iscomposed of the intervals 5 : 4, 36 : 35 and 28 : 27 and his chromatictetrachord of the intervals 32 : 27, 243 : 224, and 28 : 27. There areseveral puzzles about the tetrachords which Archytas adopts in each ofthe genera. First, why does Archytas reject the Pythagorean diatonicused by Philolaus and Plato? Second, Ptolemy, who is our major sourcefor Archytas' tetrachords (A16), argues that Archytas adopted asa principle that all concordant intervals should correspond tosuperparticular ratios. The ratios in Archytas' diatonic andenharmonic tetrachords are indeed superparticular, but two of theratios in his chromatic tetrachord are not superparticular (32 : 27and 243 : 224). Why are these ratios not superparticular as well?Finally, Plato criticizes Pythagorean harmonics in theRepublic for seeking numbers in heard harmonies rather thanascending to generalized problems (531c). Can any sense be made ofthis criticism in light of Archytas' tetrachords? The basis foran answer to all of these questions is contained in the work ofWinnington-Ingram (1932) and Barker (1989, 46-52). The crucial pointis that Archytas' account of the tetrachords in each of thethree genera can be shown to correspond to the musical practice of hisday; Ptolemy's criticisms miss the mark because of his ignoranceof musical practice in Archytas' day, some 500 years beforePtolemy (Winnington-Ingram 1932, 207). Archytas is giving mathematicaldescriptions of scales actually in use; he arrived at his numbers inpart by observation of the way in which musicians tuned theirinstruments (Barker 1989, 50-51). He did not follow the Pythagoreandiatonic scale because it did not correspond to any scale actually inuse, although it does correspond to a method of tuning. The unusualnumbers in Archytas' chromatic tetrachord do correspond to achromatic scale in use in Archytas' day. Barker tries to saveArchytas' adherence to the principle that all concordantintervals should have superparticular ratios, but there is no directevidence that he was using such a principle, and Ptolemy may bemistaken to apply it to him. Archytas thus provides a brilliantanalysis of the music of his day, but it is precisely his focus onactual musical practice that draws Plato's ire. Plato does notwant him to focus on the music he hears about him (“heardharmonies”) but rather to ascend to consider quite abstractquestions about which numbers are harmonious with which. Plato mightwell have welcomed a principle of concordance based solely onmathematical considerations, such as the principle that onlysuperparticular ratios are concordant, but Archytas wanted to explainthe numbers of the music he actually heard played. There is animportant metaphysical issue at stake here. Plato is calling for thestudy of number in itself, apart from the sensible world, whileArchytas, like Pythagoreans before him, envisages no split between asensible and an intelligible world and is looking for the numberswhich govern sensible things.2.3 Evaluation of Archytas as Mathematician There have been tendencies both to overvalue and to undervalueArchytas' achievement as a mathematician. Van der Waerden wentso far as to add to Archytas' accomplishments both Book VIII ofEuclid's Elements and the treatise on the mathematicsof music known as the Sectio Canonis, which is ascribed toEuclid in the ancient tradition (1962, 152-5). Although later scholars(e.g., Knorr 1975: 244) repeat these assertions, they are based inpart on a very subjective analysis of Archytas' style. Archytasinfluenced the Sectio Canonis, since Proposition 3 is basedon a proof by Archytas (A19), but the treatise cannot be by Archytas,because its theory of pitch and its account of the diatonic andenharmonic tetrachords differ from those of Archytas. On the otherhand, some scholars have cast doubt on Archytas' prowess as amathematician, arguing that some of his work looks like “merearithmology” and “mathematical mystification”(Burkert 1972a, 386; Mueller 1997, 289). This judgment rests largelyon a text that has been mistakenly interpreted as presentingArchytas' own views, whereas, in fact, it presentsArchytas' report of his predecessors (A17). The duplication ofthe cube and Archytas' contributions to the mathematics of musicshow that there can be no doubt that he was one of the leadingmathematicians of the first part of the fourth century BC. This wascertainly the judgment of antiquity. In his history of geometry,Eudemus identified Archytas along with Leodamas and Theaetetus as thethree most prominent mathematicians of Plato's generation (A6 =Proclus, in Eucl., prol. II 66, 14).3. Archytas on the Sciences3.1 The Value of the Sciences Archytas B1 is the beginning of his book on harmonics, and most of itis devoted to the basic principles of his theory of acoustics and, inparticular, to his theory of pitch described in section 2.2 above. Inthe first five lines, however, Archytas provides a proem on the valueof the sciences (mathêmata) in general. There areseveral important features of this proem. First, Archytas identifies aset of four sciences: astronomy, geometry, “logistic”(arithmetic) and music. B1 is thus the earliest text to identify theset of sciences that became known as the quadrivium in themiddle ages and that constitute four of the seven liberalarts. Second, Archytas does not present this classification ofsciences as his own discovery but instead begins with praise of hispredecessors who have worked in these fields. Some scholars arguethat, when he praises “those concerned with the sciences,”he is thinking only of the Pythagoreans (e.g., Zhmud 1997, 198 andLasserre 1954, 36), but this is wrongly to assume that all early Greekmathematics is Pythagorean. Archytas gives no hint that he is limitinghis remarks to Pythagoreans, and, in areas where we can identify thosewho influenced him most, these figures are not limited to Pythagoreans(e.g., Hippocrates of Chios in geometry, see section 2.1). He praiseshis predecessors in the sciences, because, “having discernedwell about the nature of wholes, they were likely also to see well howthings are in their parts” and to “have correctunderstanding about individual things as they are.” It is herethat Archytas is putting forth his own understanding of the nature andvalue of the sciences; because of the brevity of the passage, muchremains unclear. Archytas appears to be praising those concerned withthe sciences for their discernment, their ability to make distinctions(diagignôskein). He argues that they begin bydistinguishing the nature of wholes, the universal concepts of ascience, and, because they do this well, they are able to understandparticular objects (the parts). Archytas appears to follow exactlythis procedure in his Harmonics. He begins by defining themost universal concept of the science, sound, and explains it in termsof other concepts such as impact, before going on to distinguishbetween audible and inaudible sounds and sounds of high and lowpitch. The goal of the science is not the making of these distinctionsconcerning universal concepts, however, but knowledge of the truenature of individual things. Thus, Archytas' harmonics ends withthe mathematical description of the musical intervals that we hearpracticing musicians use (see section 2.2 above). Astronomy will endwith a mathematical description of the periods, risings and settingsof the planets. One way to understand Archytas' project is tosee him as working out the program suggested by his predecessor in thePythagorean tradition, Philolaus. One of Philolaus' centraltheses was that we only gain knowledge of things insofar as we cangive an account of them in terms of numbers (DK 44 B4). WhilePhilolaus only took the first steps in this project, Archytas is muchmore successful in giving an account of individual things in thephenomenal world in terms of numbers, as his description of themusical intervals shows. Plato's account of the sciences in Book VII of theRepublic can be seen as a response to Archytas' view ofthe sciences. First Plato identifies a group of five rather than foursciences and decries the neglect of his proposed fifth science,stereometry (solid geometry), with a probable allusion to Archytas(see section 2.1). 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 toArchytas' quadrivium and does not mention him byname. In the same passage, however, Plato pointedly rejects thePythagorean attempt to search for numbers in “heardharmonies.” In doing so Plato is disagreeing withArchytas' attempt to determine the numbers that govern things inthe sensible world. For Plato, the value of the sciences is theirability to turn the eye of the soul from the sensible to theintelligible realm. Book VII of the Republic with itselaborate argument for the distinction between the intelligible andsensible realm, between the cave and the intelligible world outsidethe cave, may be in large part directed at Archytas' attempt touse mathematics to explain the sensible world. As Aristotle repeatedlyemphasizes, the Pythagoreans differed from Plato precisely in theirrefusal to separate numbers from things (e.g.,Metaph. 987b27).3.2 Logistic as the Master Science In B4, Archytas asserts that “logistic seems to be far superiorindeed to the other arts in regard to wisdom.” What doesArchytas mean by “logistic”? It appears to beArchytas' term for the science of number, which was mentioned asone of the four sister sciences in B1. There is simply not enoughcontext in B4 or other texts of Archytas to determine the meaning oflogistic from Archytas' usage alone. It is necessary to rely tosome extent on Plato, who is the only other early figure to use theterm extensively. A later conception of logistic, as something thatdeals with numbered things rather than numbers themselves, which isfound in, e.g., Geminus, should not be ascribed to Plato or Archytas(Klein 1968; Burkert 1972a, 447 n. 119). In Plato,“logistic” can refer to everyday calculation, what wewould call arithmetic (e.g. 3 x 700 = 2,100; see, Hp. Mi.366c). In other passages, however, Plato defines logistic in parallelwith arithmêtikê, and treats the two of them astogether constituting the science of number, on which practicalmanipulation of number is based (Klein 1968, 23-24). Botharithmêtikê and logistic deal with the even andthe odd. Arithmêtikê focuses not on quantitiesbut on kinds of numbers (Grg. 451b), beginning with the evenand the odd and presumably continuing with the types we find later inNicomachus (Ar. 1.8 - 1.13), such as prime, composite andeven-times even. Logistic, on the other hand, focuses on quantity, the“amount the odd and even have both in themselves and in respectto one another” (Grg. 451c). An example of one part oflogistic 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 wouldprobably consider himself to be doing logistic, when he defines thethree types of means which are relevant to music (geometric,arithmetic, and harmonic). The geometric mean arises whenever threeterms 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, whenthree terms are so related that the first exceeds the second by thesame amount as the second exceeds the third (e.g. 6 : 4 :: 4 : 2).Archytas, like Plato (R. 525c), uses logistic not just inthis narrow sense of the study of relative quantity, but also todesignate the entire science of numbers includingarithmêtikê. Why does Archytas think that logistic is superior to the othersciences? In B4, he particularly compares it to geometry, arguing thatlogistic 1) “deals with what it wishes more vividly thangeometry” and 2) “completes demonstrations” wheregeometry cannot, even “if there is any investigation concerningshapes.” This last remark is surprising, since the study ofshapes would appear to be the proper domain of geometry. The mostcommon way of explaining Archytas' remark is to suppose that heis arguing that logistic is mathematically superior to geometry, inthat certain proofs can only be completed by an appeal to logistic.Burkert sees this as a reason for doubting the authenticity of thefragment, since the exact opposite seems to be true. Archytas coulddetermine the cube root of two geometrically, through his solution tothe duplication of the cube, but could not do so arithmetically, sincethe cube root of two is an irrational number (1972a, 220 n. 14). Otherscholars have pointed out, however, that certain proofs in geometry dorequire an appeal to logistic (Knorr 1975, 311; Mueller 1992b, 90n. 12), e.g., logistic is required to recognize the incomensurabilityof the diagonal with the side of the square, since incommensurabilityarises when two magnitudes “have not to one another the ratiowhich number has to number” (Euclid X 7). Thesesuggestions show that logistic can be superior to geometry in certaincases, but they do not explain Archytas' more general assertionthat logistic deals with whatever problems it wants more clearly thangeometry. However, it may be that B4 is not in fact comparing logistic to theother sciences as sciences -- in terms of their relative success inproviding demonstrations. The title of the work from which B4 is saidto come, Discourses (Diatribai), is most commonlyused of ethical treatises. Moreover, it is specifically with regard towisdom (sophia) that logistic is said to be superior, and,while sophia can refer to technical expertise, it morecommonly refers to the highest sort of intellectual excellence, oftenthe excellence that allows us to live a good life (Arist., EN1141a12; Pl., R. 428d ff.). Is there any sense in whichlogistic makes us wiser than the other sciences? Since Archytasevidently agreed with Philolaus that we only understand individualthings in the world insofar as we grasp the numbers that govern them,it seems quite plausible that Archytas would regard logistic as thescience that makes us wise about the world. It is in this sense thatlogistic will always be superior to geometry, even when dealing withshapes. Perhaps the most famous statue of the classical period is theDoryphoros by the Argive sculptor Polyclitus, which he also referredto as the Canon (i.e., the standard). Although Polyclitus undoubtedlymade use of geometry in constructing this magnificent shape, in afamous sentence from his book, also entitled Canon, heasserts that his statue came to be not through many shapes but“through many numbers” (DK40 B2, see Huffman2002a). Geometrical relations alone will not determine the form of agiven object, we have to assign specific proportions, specificnumbers. Archytas also thought that numbers and logistic were thebasis of the just state and hence the good life. In B3 he argues thatit is rational calculation (logismos) that produces thefairness on which the state depends. Justice is a relation that needsto be stated numerically and it is through such a statement that richand poor can live together, each seeing that he has what isfair. Logistic will always be superior to the other sciences, becausethose sciences will in the end rely on numbers to give us knowledge ofthe sounds we hear, the shapes we see and the movements of theheavenly bodies which we observe.3.3 Optics and Mechanics Aristotle is the first Greek author to mention the sciences of opticsand mechanics, describing optics as a subordinate science to geometryand mechanics as a subordinate science to solid geometry(APo. 78b34). Archytas does not mention either of thesesciences in B1, when describing the work of his predecessors in thesciences, nor does Plato mention them. This silence suggests that thetwo disciplines may have first developed in the first half of thefourth century, when Archytas was most active, and it is possible thathe played an important role in the development of both of them. In arecently identified fragment from his book on the Pythagoreans(Iamblichus, Comm. Math. XXV; see Burkert 1972a, 50 n. 112),Aristotle assigns a hitherto unrecognized importance to optics inPythagoreanism. Just as the Pythagoreans were impressed with the factthat musical intervals were based on whole number ratios, so they wereimpressed that the phenomena of optics could be explained in terms ofgeometrical diagrams. In addition to being an accomplishedmathematician, Archytas had a theory of vision and evidently tried toexplain some of the phenomena involved in mirrors. In contrast toPlato, who argued that the visual ray, which proceeded from the eye,requires the support of and coalesces with external light, Archytasexplained vision in terms of the visual ray alone (A25). It istempting, then, to suppose that Archytas played a major role in thedevelopment of the mathematically based Pythagorean optics, to whichAristotle refers. On the other hand, when Aristotle refers toPythagoreans, he generally means Pythagoreans of the fifth century.Elsewhere he treats Archytas independently of the Pythagoreantradition, writing works on Archytas which were distinct from his workon the Pythagoreans. It would thus be more natural to readAristotle's reference to Pythagorean optics as alluding tofifth-century Pythagoreans such as Philolaus. Archytas will then havebeen responsible for developing an already existing Pythagoreanoptical tradition into a science, rather than founding such atradition. Diogenes Laertius reports that Archytas was “the first tosystematize mechanics by using mathematical first principles”(VIII 83 = A1), and Archytas is accordingly sometimes hailed by modernscholars as the founder of the science of mechanics. There is apuzzle, however, since, no ancient Greek author in the latermechanical tradition (e.g., Heron, Pappus, Archimedes, Philon) everascribes any work in the field to Archytas. What did the ancients meanby mechanics? A rough definition would be “the description andexplanation of the operation of machines” (Knorr, OxfordClassical Dictionary, ed. 3, s.v.). The earliest treatise inmechanics, the Mechanical Problems ascribed to Aristotle,begins with problems having to do with a simple machine, thelever. Pappus (AD 320) refers to machines used to lift great weights,machines of war such as the catapult, water lifting machines, amazingdevices (automata), and machines that served as models of the heavens(1024.12 - 1025.4, on Pappus, see Cuomo 2000). Pappus emphasizes,however, that, in addition to this practical part of mechanics, thereis a theoretical part that is heavily mathematical(1022. 13-15). Given his interest in describing physical phenomena inmathematical terms, it might seem logical that Archytas would makeimportant contributions to mechanics. The actual evidence is lessconclusive. A great part of the tendency to assign Archytas a role inthe development of mechanics can be traced to Plutarch's storyabout the quarrel between Plato and Archytas over Archytas'supposed mechanical solution to the problem of doubling the cube. Thisstory is likely to be false (see 2.1 above). Some scholars haveargued that Archytas devised machines of war (Diels 1965; Cambiano1998), as Archimedes did later, but this conclusion is based onquestionable inferences and no ancient source ascribes such machinesto Archytas. The only mechanical device that can with some probabilitybe assigned to Archytas, apart from the children's toy known asa “clapper” (A10), is an automaton in the form of a woodenbird connected to a pulley and counterweight, which “flew”up from a lower perch to a higher one, when set in motion by a puff ofair (A10a). The complicating factor here is that Diogenes Laertiusreports (A1) that there was a book on mechanics in circulation, whichsome thought to be by a different Archytas, so that it is possiblethat the flying dove is, in fact, the work of a separateArchytas. Archytas' solution to the duplication of the cube,although it was not mechanical itself, was of enormous importance formechanics, since the solution to the problem allows one not just todouble a cube but also to construct bodies that are larger or smallerthan a given body in any given ratio. Thus, the solution permits theconstruction of a full-scale machine on the basis of a workingmodel. Pappus cites the solution to the duplication of the cube as oneof the three most crucial geometrical theorems for practical mechanics(Math. Coll. 1028. 18-21). It may then be thatArchytas' primary contribution to mechanics was precisely hissolution to the duplication of the cube and that it is this solutionwhich constituted the mathematical first principles which Archytasprovided for mechanics. It is more doubtful that Archytas wrote atreatise on mechanics.4. Definitions In the Metaphysics, Aristotle praises Archytas for havingoffered definitions which took account of both form and matter(1043a14-26 = A22). The examples given are “windlessness”(nênemia), which is defined as “stillness [theform] in a quantity of air [the matter],” and“calm-on-the-ocean” (galênê), whichis defined as “levelness [the form] of sea [the matter].”The terms form and matter are Aristotle's, and we cannot be sure howArchytas conceptualized the two parts of his definitions. A plausiblesuggestion is that he followed his predecessor Philolaus in adoptinglimiters and unlimiteds as his basic metaphysical principles and thathe saw his definitions as combinations of limiters, such as levelnessand stillness, with unlimiteds, such as air and sea. The oddity of“windlessness” and “calm-on-the sea” asexamples suggests that they were not the by-products of some othersort of investigation, e.g. cosmology, but were chosen precisely toillustrate principles of definition. Archytas may thus have devoted atreatise to the topic. Aristotle elsewhere comments on the use ofproportion in developing definitions and uses these same examples(Top. 108a7). The ability to recognize likeness in things ofdifferent genera is said to be the key. “Windlessness” and“calm-on -the-ocean” are recognized as alike, and thislikeness can be expressed in the following proportion: asnênemia is to the air so galênê isto the sea. It is tempting to suppose that Archytas, who saw the worldas explicable in terms of number and proportion, also saw proportionas the key in developing definitions. This would explain anotherreference to Archytas in Aristotle. At Rhetoric 1412a9-17 (=A12) Aristotle praises Archytas precisely for his ability to seesimilarity, even in things which differ greatly, and gives as anexample Archytas' assertion that an arbitrator and an altar are thesame. DK oddly include this text among the testimonia for Archytas'life, but it clearly is part of Archytas' work on definition. Thedefinitions of both an altar and an arbitrator will appeal to theircommon functions as a refuge, while recognizing the different contextand way in which this function is carried out (for doubts about thisreconstruction of Archytas' theory of definition, see Barker 2006,314-318).5. Cosmology and Physics We have very little evidence for Archytas' cosmology, yet hewas responsible for one of the most famous cosmological arguments inantiquity, an argument which has been hailed as “the mostcompelling argument ever produced for the infinity of space”(Sorabji 1988, 125). The argument is ascribed to Archytas in afragment of Eudemus preserved by Simplicius (= A24), and it isprobably to Archytas that Aristotle is referring when he describes thefifth and “most important” reason that people believe inthe existence of the unlimited (Ph. 203b22 ff.). Archytasasks anyone who argues that the universe is limited to engage in athought experiment: “If I arrived at the outermost edge of theheaven, could I extend my hand or staff into what is outside or not?It would be paradoxical [given our normal assumptions about the natureof space] not to be able to extend it.” The end of the staff,once extended will mark a new limit. Archytas can advance to the newlimit and ask the same question again, so that there will always besomething, into which his staff can be extended, beyond the supposedlimit, and hence that something is clearly unlimited. Neither Platonor Aristotle accepted this argument, and both believed that theuniverse was limited. Nonetheless, Archytas' argument had greatinfluence and was taken over and adapted by the Stoics, Epicureans(Lucretius I 968-983), Locke and Newton, among others, while elicitingresponses from Alexander and Simplicius (Sorabji 1988, 125-141). Notall scholars have been impressed by the argument (see Barnes 1982,362), and modern notions of space allow for it to be finite withouthaving an edge, and without an edge Archytas' argument cannotget started (but see Sorabji 1988, 160-163). Beyond this argument,there is only exiguous evidence for Archytas' system of thephysical world. Eudemus praises Archytas for recognizing that theunequal and uneven are not identical with motion as Plato supposed(see Ti. 52e and 57e) but rather the causes of motion(A23). Another testimonium suggests that Archytas thought that allthings are moved in accordance with proportion (Arist.,Prob. 915a25-32 = A23a). The same testimonium indicates thatdifferent sorts of proportion defined different sorts ofmotion. Archytas asserted that “the proportion ofequality” (arithmetic proportion?) defined natural motion, whichhe regarded as curved motion. This explanation of natural motion issupposed to explain why certain parts of plants and animals (e.g. thestem, thighs, arms and trunk) are rounded rather than triangular orpolygonal. An explanation of motion in terms of proportion fits wellwith the rest of evidence for Archytas, but the details remainobscure.6. Ethics and Political Philosophy Archytas' search for the numbers in things was not limited tothe natural world. Political relationships and the moral action ofindividuals were also explained in terms of number and proportion. InB3, rational calculation is identified as the basis of the stablestate: Once calculation (logismos) was discovered, it stoppeddiscord and increased concord. For people do not want more than theirshare, and equality exists, once this has come into being. For by meansof calculation we will seek reconciliation in our dealings with others.Through this, then, the poor receive from the powerful, and the wealthygive to the needy, both in the confidence that they will have what isfair on account of this. The emphasis on equality (isotas) and fairness (toison) suggests that Archytas envisages rational calculation(logismos) as heavily mathematical. On the other hand,logismos is not identical to the technical science of number(logistic - see 3.2 above) but is rather a practical ability tounderstand numerical calculations, including basic proportions, anability that is shared by most human beings. It is the clarity ofcalculation and proportion that does away with the constant strivingfor more (pleonexia), which produces discord in the state.Since the state is based on a widely shared human ability tocalculate, an ability that the rich and poor share, Archytas was ledto support a more democratic constitution (see 1.3 above) than Plato,who emphasizes the expert mathematical knowledge of a few(R. 546a ff.). Most of our evidence for Archytas' ethical views is,unfortunately, not based on fragments of his writings but rather onanecdotes, which probably ultimately derive from Aristoxenus'Life of Archytas. The good life of the individual, no lessthan the stability of the state, appears to have been founded onrational calculation. Aristoxenus presented a confrontation betweenthe Syracusan hedonist, Polyarchus, and Archytas. Polyarchus'long speech is preserved by Athenaeus and Archytas' response byCicero (A9 = Deip. 545a and Sen. XII 39-41respectively). Polyarchus' defense of always striving for more(pleonexia) and of the pursuit of pleasure is reminiscent ofPlato's presentations of Callicles and Thrasymachus, but is notderived from those presentations and is better seen as an importantparallel development (Huffman 2002). Archytas bases his response onthe premise that reason (= rational calculation) is the best part ofus and the part that should govern our actions. Polyarchus mightgrant such a premise, since his is a rational hedonism. Archytasresponds once again with a thought experiment. We are to imaginesomeone in the throes of the greatest possible bodily pleasure (sexualorgasm?). Surely we must agree that a person in such a state is notable to engage in rational calculation. It thus appears that bodilypleasure is in itself antithetical to reason and that, the more wesucceed in obtaining it, the less we are able to reason. Aristotleappears to refer to this argument in the Nicomachean Ethics(1152b16-18). Archytas' argument is specifically directedagainst bodily pleasure and he did not think that all pleasure wasdisruptive; he enjoyed playing with children (A8) and recognized thatthe pleasures of friendship were part of a good life (Cicero,Amic. XXIII 88). Other anecdotes emphasize that our actionsmust be governed by reason rather than the emotions: Archytas refusedto punish the serious misdeeds of his slaves, because he had becomeangry and did not want to act out of anger (A7); he restrained himselffrom swearing aloud by writing his curses on a wall instead (A11).7. Importance and Influence Archytas fits the common stereotype of a Pythagorean better thananyone else does. He is by far the most accomplished Pythagoreanmathematician, making important contributions to geometry,logistic/arithmetic and harmonics. He was more successful as apolitical leader than any other ancient philosopher, and there is arich anecdotal tradition about his personal self-control. It isstriking, however, that there are essentially no testimonia connectingArchytas to metempsychosis or the religious aspect of Pythagoreanism.Archytas is a prominent figure in the rebirth of interest inPythagoreanism in first century BC Rome: Horace, Propertius and Ciceroall highlight him. As the last prominent member of the earlyPythagorean tradition, more pseudo-Pythagorean works came to be forgedin 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 assignedto him in those texts are fanciful. Scholars have typically emphasized the continuities between Plato andArchytas (e.g., Kahn 2001, 56), but the evidence suggests thatArchytas and Plato were in serious disagreement on a number of issues.Plato's only certain reference to Archytas is part of a criticism ofhis approach to harmonics in Book VII of the Republic, wherethere is probably also a criticism of his work in solid geometry.Plato's attempt to argue for the split between the intelligibleand sensible world in Books VI and VII of the Republic maywell be a protreptic directed at Archytas, who refused to separatenumbers from things. It is sometimes thought that the eponymousprimary speaker in Plato's Timaeus, who is described asa leading political figure and philosopher from southern Italy (20a),must be a stand-in for Archytas. The Timaeus, however, is amost un-Archytan document. It is based on the split between thesensible and intelligible world, which Archytas did not accept. Platoargues that the universe is limited, while Archytas is famous for thisargument to show that it is unlimited. Plato constructs the world soulaccording to ratios that are important in harmonic theory, but he usesPhilolaus' ratios rather than Archytas'. Plato does adoptArchytas' theory of pitch with some modification, but Archytasand Plato disagree on the explanation of sight. Archytas'refusal to split the intelligible from the sensible may have made hima more attractive figure to Aristotle, who devoted four books to himand praised his definitions for treating the composite of matter andform, not of form separate from matter. Archytas' vision of therole of mathematics in the state is closer to Aristotle'smathematical account of distributive and redistributive justice(EN 1130b30 ff.) than to Plato's emphasis on the expertmathematical knowledge of the guardians. 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An Alternative Theory of Date and Purpose’,Pseudepigrapha I, Fondation Hardt Entretiens XVIII,Vandoeuvres-Genève, 59-87.Thorndike, Lynn, 1934, A History of Magic and ExperimentalScience, New York: Columbia University Press.Vlastos, G., 1991, Socrates: Ironist and MoralPhilosopher, Ithaca: Cornell University Press.de Vogel, C. J., 1966, Pythagoras and EarlyPythagoreanism, Assen: Van Gorcum.van der Waerden, B. L., 1943, ‘Die Harmonielehre derPythagoreer’, Hermes 78: 163-199.–––, 1947-9, ‘Die Arithmetik derPythagoreer’, Mathematische Annalen 120: 127-153,676-700.–––, 1963, Science Awakening,A. Dresden (tr.), New York: John Wiley & Sons.Winnington-Ingram, R. P., 1932, ‘Aristoxenus and the Intervalsof Greek Music’, Classical Quarterly 26: 195- 208.Wolfer, E. P., 1954, Eratosthenes von Kyrene als Mathematikerund Philosoph, Groningen: P. Noordhoff.Wuilleumier, Pierre, 1939, Tarente des Origines à laConquête Romaine, Paris: E. De Boccard.Zhmud, L., 1997, Wissenschaft, Philosophie und Religion imfrühen Pythagoreismus, Berlin: Akademie Verlag.–––, 1998, ‘Plato as Architect ofScience’, Phronesis 43.3: 210-244.–––, 2002, ‘Eudemus' History ofMathematics’, in Eudemus of Rhodes, W. W. Fortenbaughand I. Bodnar (eds.), New Brunswick: Transaction, 263-306. Other Internet ResourcesArchytas (The MacTutor History of Mathematics Archive, School ofMathematics and Statistics, University of St Andrews, Scotland)Related Entries Aristotle | Philolaus | Plato | Pythagoras | Pythagoreanism Copyright © 2007 byCarl Huffman<cahuff@depauw.edu> |
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