## 1. Life and Works


### 1.1 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 city of
Tarentum, on the heel of the boot of Italy. The later tradition almost
universally identifies him as a Pythagorean (e.g., A1, A2, A7, A16).
Aristotle and his pupil Eudemus do not explicitly call Archytas a
Pythagorean and appear to treat him as an important independent
thinker. Plato never refers to Archytas by name except in the
*Seventh Letter*, if that is by Plato, and he is not called a
Pythagorean there. In the *Republic*, however, when Plato
quotes a sentence which appears in Fr. 1 of Archytas (DK47 B1), he
explicitly labels it as part of Pythagorean harmonics (530d). Cicero
(*de Orat.* III 34. 139) reports that Archytas was the pupil of
Philolaus, and this is not improbable. Philolaus was the most
prominent Pythagorean of the preceding generation (ca. 470-390)
and may have taught in Tarentum. Archytas' achievements in
mathematics depend on the work of Hippocrates of Chios, but we have no
evidence that he studied with Hippocrates. The only pupil of Archytas
who is more than a name, is Eudoxus (ca. 390-340), the prominent
mathematician. Eudoxus presumably did not learn his famous hedonism
from Archytas (see DK47 A9), and it is specifically geometry that he
is said to have studied with Archytas (Diogenes Laertius VIII 86).



Archytas was, roughly speaking, a contemporary of Plato, but it is
difficult to be more precise about his dates. Aristotle's pupil,
Eudemus, presents him as the contemporary of Plato (born 428/7) and
Leodamas (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 the
contemporary of Theaetetus, if he were born much earlier than 435,
this is the earliest he was likely to have been born. On the other
hand, he could have been born as late as 410 and still be considered a
contemporary of Plato. Strabo associates Archytas with the flourishing
of Tarentum, before a period of decline, in which Tarentum hired
mercenary generals (A4). Since the mercenaries appear ca. 340, it
seems likely that Archytas was dead by 350 at the latest. Such a date
is in accord with other evidence (A5 = [Demosthenes], *Erot.
Or.* 61.46), which connects Archytas to Timotheus, who died ca.
355, and with Plato's (?) *Seventh Letter* (350a), which
presents Archytas as still active in Tarentum in 361. Thus Archytas
was born between 435 and 410 and died between 360 and 350.



Some scholars (e.g., Ciaceri 1927-32: III 4) have supposed that
the speaker of the Roman poet, Horace's, Archytas Ode (I 28 =
A3) is Archytas himself and hence have concluded that Archytas died in
a shipwreck. The standard interpretation, however, rightly recognizes
that the speaker is not Archytas but a shipwrecked sailor who
apostrophizes Archytas (Nisbet and Hubbard 1970, 317ff.). The ode
tells us nothing about Archytas' death, but it is one of many
pieces of evidence for the fascination with Archytas by Roman authors
of 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.* III 34.139; *Amic.* XXIII
88; *Sen.* XII 39-41), perhaps because Pythagoreanism had
come to be seen as a native Italian philosophy, and not a Greek import
(Burkert 1961; Powell 1995, 11 ff.).


### 1.2 Sources



Apart 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 the fourth century, in the fifty years after
Archytas' death. Archytas' importance both as an
intellectual and as a political leader is reflected in the number of
writings about him in this period, although only fragments of these
works have been preserved. Aristotle wrote a work in three volumes on
the philosophy of Archytas, 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. Scholars
have generally regarded them as authentic (Huffman 2005: 583-94)
but Schofield has recently raised some doubts (2014: 81-2).
Aristotle's pupil Eudemus discussed Archytas prominently in his
history of geometry (A6 and A14) 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 in a good position to have accurate information about
Archytas. He was born in Tarentum and grew up during the height of
Archytas' prominence in the city. In addition to whatever
personal knowledge he had of Archytas, he draws on his own father
Spintharus, who was a younger contemporary of Archytas, as a source
(e.g., A7). Aristoxenus began his philosophical career as a
Pythagorean and studied with the Pythagorean Xenophilus at Athens, so
that it is not surprising that his portrayal of Archytas is largely
positive. Nonetheless, Archytas' opponents are given a fair
hearing (e.g., Polyarchus in A9), and Archytas himself is represented
as not without small flaws of character (A7). Other fourth-century
sources such as the *Seventh Letter* in the Platonic corpus and
Demosthenes' (?) *Erotic Oration* focus on the connection
between Archytas and Plato (see below).


### 1.3 Archytas and Tarentum



Archytas is unique among Greek philosophers for the prominent role he
played in the politics of his native city. He was elected general
(*strategos*) seven years in succession at one point in
his career (A1), a record that reminds us of Pericles at Athens. His
election was an exception to a law, which forbade election in
successive years, and thus attests to his reputation in Tarentum.
Aristoxenus reports that Archytas was never defeated in battle and
that, when at one point he was forced to withdraw from his post by the
envy of his enemies, the Tarentines immediately suffered defeat (A1).
He probably served as part of a board of generals (there was a board
of ten at Athens). The analogy with Athens suggests that as a general
he may also have had special privileges in addressing the assembly at
Tarentum on issues of importance to the city, so that his position as
general gave him considerable political as well as military power. At
some point in his career, he may have been designated as a general
*autokrator* ("plenipotentiary") (A2), which
gave him special latitude in dealing with diplomatic and military
matters without consulting the assembly, although this was not
dictatorial power and all arrangements probably required the eventual
approval of the assembly. We do not know when Archytas served his
seven successive years as general. Some have supposed that they must
coincide with the seven year period which includes Plato's
second and third visits to Italy and Sicily, 367-361 (e.g.,
Wuilleumier 1939, 68-9), but Archytas need not have been
*strategos* to play the role assigned to him during these
years in the *Seventh Letter*. The evidence suggests that most
of Archytas' military campaigns were directed not at other
Greeks but at native Italic peoples such as the Messapians and
Lucanians, with whom Tarentum had been in constant conflict since its
founding.



It is important to recognize that the Tarentum in which Archytas
exercised such influence was not some insignificant backwater. Spartan
colonists founded it in 706. It was initially overshadowed by other
Greek colonies in southern Italy such as Croton, although it had the
best harbor on the south coast of Italy and was the natural stopping
point for any ships sailing west from mainland Greece. Archytas will
have grown up in a Tarentum that, in accord with its foundation by
Sparta, took the Peloponnesian and Syracusan side against Athens in
the Peloponnesian War (Thuc. VI 44; VI 104; VII 91). Athens allied
with the Messapians (Thuc. VII 33), the long-standing enemy of the
Tarentines, against whom Archytas would later lead expeditions (A7).
After the Peloponnesian War, Tarentum appears to have avoided direct
involvement in the conflict between the tyrant of Syracuse, Dionysius
I, and a league of Greek cities in southern Italy headed by Croton.
After Dionysius crushed the league, Tarentum emerged as the most
powerful Greek state in southern Italy and probably became the new
head of the league of Italiot Greek cities (A2). In the period from
380-350, when Archytas was 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,
Tarentum appears to have been a democracy during Archytas'
lifetime. According to Aristotle (*Pol*. 1303a), the democracy
was founded after a large part of the Tarentine aristocracy was killed
in a battle with a native people, the Iapygians, in 473. Herodotus
confirms that this was the greatest slaughter of Greeks of which he
was aware (VII 170). There is no evidence that Tarentum was anything
but a democracy between the founding of the democracy in 473 and
Archytas' death ca. 350. Some scholars have argued that
Tarentum's ties to Sparta and the supposed predilection of the
Pythagoreans for aristocracy will have insured that Tarentum did not
remain 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 its flourishing under Archytas (A4), and the descriptions of
Archytas' power in Tarentum stress his popularity with the
masses and his election as general by the citizens (A1 and A2).
Finally, Aristotle's account of the structure of the Tarentine
government in the fourth century (*Pol*. 1291b14), while
possibly consistent with other forms of government, makes most sense
if 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 the modern
world for having sent a ship to rescue Plato from the tyrant of
Syracuse, Dionysius II, in 361. In both the surviving ancient lives of
Archytas (by Diogenes Laertius, VIII 79-83, and in the
*Suda*) the first thing mentioned about him, after the name of
his city-state and his father, is his rescue of Plato (A1 and A2).
This story is told in greatest detail in the *Seventh Letter*
ascribed to Plato. It has accordingly been typical to identify
Archytas as "the friend of Plato" (Mathieu 1987). Archytas
first met Plato over twenty years earlier, when Plato visited southern
Italy and Sicily 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 the relationship (Zhmud 2006, 93) and even as the "new
model philosopher for Plato" (Vlastos 1991, 129) and the
archetype of Plato's philosopher-king (Guthrie 1962, 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 Socrates had died (e.g., Cicero, *Rep*. I
10.16), but another tradition makes Archytas the student of Plato, to
whom he owed his fame and success in Tarentum (Demosthenes [?],
*Erotic Oration* 44).



The *Seventh Letter* itself is of contested authenticity. Many
scholars regard it either as the work of Plato himself or of a student
of Plato who had considerable familiarity with Plato's
involvement in events in Sicily (see e.g., Brisson 1987; Lloyd 1990;
Schofield 2000), but grave doubts about this view have been raised by
Burnyeat and Frede (2015). The letter appears to serve as an apologia
for Plato's involvement in events in Sicily. Lloyd has argued,
however, that the letter also serves to distance Plato from
Pythagoreanism and from Archytas (1990). Nothing in the letter
suggests that Plato was ever the pupil of Archytas; instead the
relationship is much closer to that presented in the *Erotic
Oration*. Plato is presented as the dominant figure upon whom
Archytas depends both philosophically and politically. Archytas writes
to Plato claiming that Dionysius II has made great progress in
philosophy, in order to urge Plato to come to Sicily a third time
(339d-e). These claims are belied as soon as Plato arrives (340b). The
letter thus suggests that, far from being the Pythagorean master from
whom Plato learned his philosophy, Archytas had a very imperfect
understanding of what Plato considered philosophy to be. The letter
makes clear that Plato does have a relationship of *xenia*,
"guest-friendship," with Archytas and others at Tarentum
(339e, 350a). This relationship could have been established on
Plato's first visit in 388/7, since Plato uses it as a basis to
establish a similar relationship 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 for help, when he is in
danger after the third trip to Sicily goes badly (350a). Such a
friendship need not imply any close personal intimacy, however.
Aristotle classifies *xenia* as a friendship for utility and
points out that such friends do not necessarily 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 is described as devised by Plato [350a]), Plato is clearly the
dominant figure in the relationship. 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
and Dionysius II, which is described as of considerable political
importance (339d).



How are we to unravel the true nature of the relationship between
Plato and Archytas in the light of this conflicting evidence? Apart
from the *Seventh Letter*, Plato never makes a direct reference
to Archytas. He does, however, virtually quote a sentence from
Archytas' book on harmonics in Book VII of the *Republic*
(530d), and his discussions of the science of stereometry shortly
before this are likely to have some connections to Archytas'
work in solid geometry (528d). It is thus in the context of the
discussion of the sciences that Plato refers to Archytas, and the
remains of Archytas' work focus precisely on the sciences (e.g.,
fr. B1). Both strands of the tradition can be reconciled, if we
suppose that Plato's first visit to Italy and Sicily was at
least in part motivated by his desire to meet Archytas, as the first
tradition claims, but that he sought Archytas 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 work in solid geometry on philosophical grounds, so
that, while he undoubtedly learned a considerable amount of
mathematics from Archytas, he clearly disagreed 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
not likely to have achieved his political dominance yet, so that there
may also be some truth to the claim of the second tradition that
Archytas did not achieve his great practical success until after his
contact with Plato; whether or not that success had any direct
relationship to his contact 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, which they 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 competitive
colleagues engaged in an ongoing debate as to the value of the
sciences for philosophy (Huffman 2005, 32-42).


### 1.5 The Authenticity Question



More pages of text have been preserved in Archytas' name than in
the name of any other Pythagorean. Unfortunately the vast majority of
this material is rightly regarded 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 monetary reasons; a
text of a "rare" work by a famous Pythagorean could fetch
a considerable sum from book collectors. There were characteristics
unique to the Pythagorean tradition, however, that led to a
proliferation of forgeries. Starting as early as the later fourth
century BC, Pythagoras came to be regarded, in some circles, 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
original revelation (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, to show that they had, in
fact, anticipated the most important ideas of Plato and Aristotle.
These pseudo-Pythagorean texts are thus characterized by the use of
central Platonic and Aristotelian ideas, expressed in the technical
terminology used by Plato and Aristotle. Some of the forgeries even
attempt to improve on Plato and Aristotle by adding refinements to
their positions, which were first advanced several hundred years after
their deaths. The date and place of origin of these pseudo-Pythagorean
treatises is difficult to determine, but most seem 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 the dominant figure in
this pseudo-Pythagorean tradition, probably because of his connection
to Plato (Zhmud 2019). In Thesleff 1965's collection of the
pseudo-Pythagorean writings, 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. On the other hand,
the fragments likely to be genuine, which are collected in DK, fill
out only a hundred lines of text. Thus, over ten times more spurious
than genuine material has been preserved in Archytas' name. It
may well be that the style and Doric dialect of the pseudo-Pythagorean
writings were also based on the model of Archytas' genuine
writings.


### 1.6 Pseudepigrapha Ascribed to Archytas



The treatises under Archytas' name collected in Thesleff 1965
have been almost universally regarded as not by the historical
Archytas, except for *On Law and Justice*, where there has been
considerable controversy. Most are only preserved in fragments,
although there are two brief complete works. The most famous of these
pseudepigrapha is *Concerning the Whole System [sc. of
Categories]* or *Concerning the Ten Categories* (preserved
complete, see Szlezak 1972). This work along with the treatise *On
Opposites* (Thesleff 1965, 15.3-19.2) and the much later
*Ten Universal Assertions* (preserved complete, first ascribed
to Archytas in the 15th century AD; see Szlezak 1972)
represent the attempt to claim Aristotle's doctrine of
categories for Archytas and the Pythagoreans (see also Griffin 2015).
This attempt was to some extent successful; both Simplicius and
Iamblichus regarded the Archytan works on categories as genuine
anticipations of Aristotle (*CAG* VIII. 2, 9-25).
*Concerning the Ten Categories* and *On Opposites* are
very frequently cited in the ancient commentaries on Aristotle's
*Categories*. Pseudo-Archytas identifies ten categories with
names that are virtually identical to those used by Aristotle, and his
language follows Aristotle closely in many 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 the last six chapters of
Aristotle's *Categories* from the rest. Thus, the works
in Archytas' name must have been composed after
Andronicus' work in the first century BC. Other pseudepigrapha
in metaphysics and epistemology include *On Principles*
(Thesleff 1965, 19.3 - 20.17) and *On Intelligence and
Perception* (Thesleff 1965, 36.12-39.25), which includes a
paraphrase of the divided line passage in Plato's
*Republic*. Mansfeld has recently shown that Fragment 1 of the
latter work in Thesleff's collection belongs to the former work
(Mansfeld 2019). De Cesaris and Horky (2018) provide commentary on
*On Intelligence and Perception*, but much remains obscure in
this difficult work. Mansfeld (2019) shows that *On Intelligence
and Perception* is not likely to have been an influence on the
account of Pythagorean principles in Aetius *Placita* 1.3.8 as
has been suggested by De Cesaris and Horky (2018). Ulacco (2017)
establishes new texts (with commentary) for *On Principles*,
*On Intelligence and Perception* and *On Opposites*.
Further metaphysical and epistemological works include *On
Being* (Thesleff 1965, 40.1-16) and *On Wisdom*
(Thesleff 1965, 43.24-45.4). The authenticity of this latter
work has recently been defended on the grounds that its admitted
similarities to passages in Aristotle are a result of Archytas'
influence on Aristotle rather than an 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 Archytas and
must have been familiar with his thought. However, the issue of
authenticity within the Pythagorean tradition has a different
character than is the case with other ancient authors. In the case of
an author such as Plato, where the vast majority of surviving works
are surely authentic, the onus of proof is on anyone who wants to
argue that a work is spurious. In the Pythagorean tradition, on the
other hand, where surely spurious works far outnumber genuine ones,
the situation is reversed. The onus of proof rests on anyone who
regards a Pythagorean work as genuine to show that it does not fit the
pattern of the forged Pythagorean treatises and that its contents can
be corroborated by evidence dating before the third century, when the
Pythagorean pseudepigrapha start to be generated. Since *On
Wisdom* does share with the pseudepigrapha the characteristic 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 on categories ascribed to
Archytas, it is much more likely that it was forged on the basis of
Aristotle than that Aristotle is using *On Wisdom* without
attribution. In order for the latter situation to be probable there
would need to be fourth-century evidence independent of *On
Wisdom* which ascribed the ideas found in it to Archytas. Horky
2015 includes *On Wisdom* among the pseudepigrapha and provides
analysis of it. For recent discussions of the nature of the
pseudepigrapha ascribed to Archytas see Bonazzi 2013 and Centrone
2014. For the nature of the appropriation of Aristotle in
pseudepigrapha ascribed to Archytas and other early Pythagoreans see
Ulacco 2016.



There are also fragments of two pseudepigrapha 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 of the first century BC, and *On Moral Education*
(Centrone 1990), which has ties to Carneades (2nd c. BC). The status
of one final treatise is less clear. The fragments 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 modest
conclusion that the work might be by Archytas, since there were no
positive indications 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
was supported by Johnson (2008, 194-198) but more recently Horky
and Johnson argue that it was not written by Archytas himself and
propose the somewhat Byzantine theory that it was written by an author
who based it on a speech which they hypothesize that Aristoxenus
assigned to Archytas in his *Life of Archytas* (2020:
459-460). On the other hand, DK did not include the fragments of
*On Law and Justice* among the genuine fragments, and most
recent scholars have argued that the treatise is spurious. Aalders
provides the most detailed treatment, although a number of his
arguments are inconclusive (1968, 13-20). Other opponents of
authenticity are Burkert (1972a), Moraux (1984, 670-677),
Centrone (2000) and most recently Schofield (2014). The connections of
*On Law and Justice* to the genuine Fr. B2 of Archytas speak
for its authenticity, 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*) was the 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 key
terms in Fr. 3 (e.g., *pleonexia*,  *homonoia* and
*isotes*) is hard to explain, if *On Law and
Justice* is authentic (Huffman 2005, 599-606). For a recent
argument against the authenticity of *On Law and Justice*,
which further contrasts it with the authentic Fr. 3, see Schofield
2014: 82-5. Even if the work belongs among the pseudepigrapha
Horky and Johnson are surely right that the controversy about its
authenticity has led scholars to neglect the philosophic content of
the treatise (Horky and Johnson 2020: 487) and they provide a
commentary that attempts to elucidate this difficult text. They regard
the text as closely tied to genuine fragments of Archytas and suggest
that it can help us understand those fragments, but the similarities
are of a very general sort. As Zhmud has recently argued there is very
little that is genuinely Pythagorean in the pseudepigrapha (Zhmud
2019) and that also appears to be true of *On Law and Justice*.




Some testimonia suggest that there were even more pseudo-Archytan
treatises, which have not survived even in fragments (Thesleff 47.8
ff.). Two spurious letters 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 of Archytas
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 the Middle Ages and early Renaissance, when works
continued to be written in his name, usually with the spelling
Architas or Archita. The *Ars geometriae*, which is ascribed to
Boethius, but was in reality composed in the 12th century
(Folkerts 1970, 105), ascribes discoveries in mathematics to Architas
which are clearly spurious (Burkert 1972a, 406). Several alchemical
recipes involving the wax of the 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 century AD). Numerous selections from a book entitled
*On Events in Nature* (*de eventibus in natura*, also
cited as *de effectibus in natura* and as *de eventibus
futurorum*) by Archita Tharentinus (or Tharentinus, or just
Tharen) are preserved in the medieval texts known as *The Light of
the Soul* (*Lumen Animae*), which were composed in the
fourteenth century and circulated widely in Europe in the fifteenth
century as a manual for preachers (Rouse 1971; Thorndike 1934, III
546-60). An apocryphal work, *The Circular Theory of the
Things in the Heaven*, by Archytas Maximus [!], which has never
been published in full, is preserved in Codex Ambrosianus D 27 sup.
(See *Catalogus Codicum Astrologorum 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 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 the
authenticity 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 quote the fragments. Fragments B1 and B2 are
reported to come from a treatise entitled *Harmonics*, and the
major testimonia about Archytas' harmonic theory are likely to
be ultimately based on this book (A16-19). This treatise began
with a discussion of the basic principles of acoustics (B1), defined
the three types of mean which are of importance in music theory (B2),
and went on to present Archytas' mathematical descriptions of
the tetrachord (the fourth) in the three main genera (chromatic,
diatonic, and enharmonic - A16-A19). B3 probably comes from a
work *On Sciences*, which may have been a more general
discussion of the value of mathematics for human life in general and
for the establishment of a just state in particular. New support for
its authenticity has been provided by Schofield (2009). B4 comes from
a work entitled *Discourses* (*Diatribai*). The fragment
itself asserts the priority of the science of calculation (*ha
logistika*, "logistic") to the other sciences, such as
geometry, and thus suggests a technical work of mathematics. The title
*Diatribai* would more normally suggest a treatise of ethical
content, however, so that in this work the sciences may have been
evaluated in terms of their 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 for the unlimited extent of the
universe (A24), his theory of vision (A25), and his account of motion
(A23, A23a) all suggest that he may have written a work on cosmology.
Aristotle's comments in the *Metaphysics* suggest that
Archytas may have written a book on definition (A22), and A20 and A21
might suggest a work on arithmetic. Perhaps there was a treatise on
geometry or solid geometry in which 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 unlimited universe and his theory of
vision were derived from anecdotes preserved by Aristoxenus, 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 of Archytas, were in fact by
Archytas of Tarentum or by other men of the same name. Diogenes
Laertius lists three other writers with the name Archytas (VIII 82).
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 under Archytas' name.


## 2. Archytas as Mathematician and Harmonic Theorist


### 2.1 Doubling the Cube



Archytas 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 romantic version of the story, which occurs in many
variations and ultimately goes back to Eratosthenes (3rd c. BC),
reports that the inhabitants of the Greek island of Delos were beset
by a plague and, when they consulted an oracle for advice, were told
that, 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 88.3-96.27 Heiberg/Stamatis]). The
simple-minded response to the oracle, which is actually assigned to
the Delians in some versions, is to build a second altar identical to
the first one and set it on top of the first (Philoponus, *In Anal.
post., CAG* XIII.3, 102.12-22). The resulting altar does
indeed have a volume twice that of the first altar, but it is no
longer a cube. The next simple-minded response is to assume that,
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 of the 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 eight times the volume. If the
original altar had a side of two, then its volume would be
23 or 8, while an altar built on a side twice as long will
have a volume of 43 or 64. What then is the length of the
side which will produce a cube with twice the volume 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 less than 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 of doubling the cube first arose in the fourth
century with the Delians. We are told that 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 a
slightly different problem (Eutocius, *in Archim. sphaer. et
cyl.* II [III 88.3-96.27 Heiberg/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 be
double the cube on length G. Exactly how Hippocrates came to see this
is conjectural and need not concern 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 : D is equal to G : x, so we can set
them all equal to G : x. If we do this and multiply 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 the original terms, then G : x times x : y yields G
: y, and G : y times the remaining term 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 that is twice the cube built on G, will be
the cube built on x. The Greeks did not think of the problem as a
problem in algebra but rather as a problem in geometry. After
Hippocrates the problem of doubling the cube was always seen as the
problem of finding two lines such that they were mean proportionals
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 problem that
Archytas provided the first solution.



Archytas' solution has been rightly hailed as "the most
remarkable of all [the solutions]" and as a "bold
construction in three dimensions" (Heath 1921, 246); Mueller
calls it "a *tour de force* of the spatial
imagination" (1997, 312 n. 23). We owe the preservation of
Archytas' solution to Eutocius, who in the sixth century AD
collected some eleven solutions to the problem as part of his
commentary on the second book of Archimedes' *On the Sphere
and Cylinder*. Eutocius' source for Archytas' solution
was ultimately Aristotle's pupil Eudemus, who in the late fourth
century BC wrote a history of geometry. The solution is complex and it
is not possible to go through it step by step here (see Huffman 2005,
342-401 for a detailed treatment of the solution). Archytas
proceeds by constructing a series of four 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 the side of
the original cube (G) and AD was twice AM. Thus the cube double the
volume of the cube on AM should be built on AI. The real difficulty
was in constructing the four similar triangles, where the given length
of the side of the original cube and a length double that magnitude
were two of the sides in the similar triangles. The key point for the
construction of these triangles, point K, was determined as the
intersection of two rotating plane figures. The first figure is a
semicircle, which is perpendicular to the plane of the circle ABDZ and
which starts on the diameter AED and, with point A remaining fixed,
rotates to position AKD. The second is the triangle APD, which rotates
up out of the plane of the circle ABDZ to position ALD. As each of
these figures rotates, it traces a line on the surface of a
semicylinder, which is perpendicular to the plane of ABDZ and has ABD
as its base. The boldness and the imagination of the construction lies
in envisioning the intersection at point K of the line drawn by the
rotating semicircle on the surface of the semicylinder with the line
drawn by the rotating triangle on the same surface. We simply
don't know what led Archytas to produce this amazing feat of
spatial imagination, in order to construct the triangles with the
sides in appropriate proportion. For a recent attempt to situate
Archytas' solution in the mathematics of his time and make it
less "miraculous," see Menn 2015. 



> 
> ![figure](cube1.jpg)
>   
> 
> ![figure](cube2.jpg)
>   
> 
> **Figure 1**
> 



In the later tradition, Plato is reported to have criticized
Archytas' solution for appealing to "constructions that
use instruments and that are mechanical" (Plutarch, *Table
Talk* VIII 2.1 [718e]; *Marc.* XIV 5-6). Plato argued
that the value of geometry and of the rest of mathematics resided in
their ability to turn the soul from the sensible to intelligible
realm. The cube with which geometry deals is not a physical cube or
even a drawing of a cube but rather an intelligible cube 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 was focusing not on the
intelligible world but on the physical world and hence destroying the
value of geometry. Plato's quarrel with Archytas is a charming
story, 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 to reconcile with our
earliest source for the story of the Delian problem, Eratosthenes.
Eratosthenes had himself invented an instrument to determine mean
proportionals, the *mesolab* ("mean-getter"), and
he tells the story of the Delian problem precisely to emphasize that
earlier solutions, including that of Archytas, were in the form of
geometrical demonstrations, which could not be employed for practical
purposes. He specifically labels Archytas' solution as
*dysmechana*, "hardly mechanical." Some
scholars 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 in
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 between Plato and Archytas over
the use of mechanical devices in geometry is an invention 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 myth which
explained the separation of mechanics from philosophy as the result of
a quarrel between two philosophers. In the *Republic*, Plato is
critical of the solid geometry of his day, but his criticism makes no
mention of the use of instruments. His criticism instead focuses on
the failure of solid geometry to be developed 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 the 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 solid
geometers of the time, it is hard to avoid the conclusion that Plato
regarded him as one of the arrogant experts, who focused on solving
charming problems but failed to produce a coherent discipline of solid
geometry. Since Archytas was a leading political figure in Tarentum,
it is also possible that Plato was criticizing him for not making
Tarentum a state which held solid geometry in esteem. 



Brisson (2013) takes a sceptical stance towards the evidence and
concludes that Archytas never solved the problem of the duplication of
the cube. He argues that Plato would have mentioned a solution to the
problem if one existed in his day and that the mathematics of the
solution ascribed to Archytas is impossible for someone of his time,
since it employs conic sections, which were not developed until the
third century (2013: 220-1). However, while Plato does criticize
the state of stereometry (solid geometry) in his day, he also affirms
that there are people studying it and that some of their results have
charm and beauty (*Rep.* 528c-d). There is no reason that
Archytas' duplication could not be included among these results.
Moreover, the mathematics used in Archytas' solution in no way
relies on conic sections but relies on mathematics found in
Euclid's *Elements* Books 1, 3, 4, 6, and 11, which rely
on geometry of the fourth century when Archytas was active (Heath
1921, Knorr 1986, Mueller 1997 and Menn 2015 all regard the
mathematics as appropriate for Archytas). Menaechmus, who lived two
generations after Archytas, is the first person to solve the problem
using conic sections (see Menn 2015: 415-6). Brisson has to
discount the explicit tradition that already in the fourth century
Eudemus knew of Archytas' solution and assume that the solution
was developed by a compiler in the latter tradition, but it is
implausible that such a compiler could have developed the
sophisticated mathematics of the solution and that, if he were such an
accomplished mathematician, he would ascribe it to Archytas.


### 2.2 Music and Mathematics



One of the most startling discoveries of early Greek science was that
the fundamental intervals of music, the octave, the fourth, and the
fifth, corresponded to whole number ratios of string length. Thus, if
we pluck a string of length *x* and then a string of length
2*x*, we will hear the interval of an octave between the two
sounds. 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 of musical sound are governed by whole
number ratios must have played a central role in the Pythagorean
conception, first expressed by Philolaus, that all things are known
through number (DK 44 B4). The next step in harmonic theory was to
describe an entire octave length scale in terms of mathematical
ratios. The earliest 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 then up 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, the ratios
that govern the fifth (3 : 2) and fourth (4 : 3) are added by
multiplying the terms and thus produce an octave (3 : 2 x 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 a fifth up was regarded as
the basic unit of the scale, the whole tone, which corresponded to the
ratio of 9 : 8 (subtraction of 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 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, which has the unlovely ratio of 256 :
243 (4 : 3 / 9 : 8 = 32 : 27 / 9 : 8 = 256 : 243). 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 four intervals make up a fifth and complete the
octave from our staring note]. This scale is known as the Pythagorean
diatonic and is the scale that Plato adopted in the construction of
the world soul in the *Timaeus* (36a-b).



Archytas took harmonic theory to a whole new level of theoretical and
mathematical sophistication. Ptolemy, writing in the second century
AD, identifies Archytas as having "engaged in the study of music
most of all the Pythagoreans" (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, it will
produce a sound that travels rapidly through the air, which will be
perceived as 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 that vibrates 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 very influential. It was taken over and adapted by both Plato and
Aristotle and remained the dominant theory throughout antiquity
(Barker 1989, 41 n. 47; Barker 2014: 187). Second, Archytas introduced
new mathematical rigor into Pythagorean harmonics. One of the
important results of the analysis of music in terms of whole 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 into a fourth and a fifth, the fourth is not divided into
two equal halves but into two whole tones and a remainder. The whole
tone cannot be divided into two equal half 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 musical intervals (2 : 1, 4 : 3, 3 : 2,
9 : 8), all belong to a type of ratio known as a superparticular 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 taken over
and modified slightly in the *Sectio Canonis* ascribed to
Euclid (Prop. 3; see Barker 1989, 195). On Archytas' proof see
Huffman 2005: 451-70 and Barker 2007: 303-5.



Archytas' final contribution to music theory has to do with the
structure of the scale (for a more detailed account than what follows
see Huffman 2005: 402-25 and Barker 2007: 292-302). The
Greeks used a number of different scales, which were distinguished by
the way in which the fourth, or tetrachord, was constructed. These
scales were grouped into three main types or genera. One genus was
called the diatonic; one example of this is the Pythagorean diatonic
described above, which is built on the tetrachord with the intervals 9
: 8, 9 : 8 and 256 : 243 and was used by Philolaus and Plato. There is
no doubt that Archytas knew of this diatonic scale, but his own
diatonic tetrachord was somewhat different, being composed of the
intervals 9 : 8, 8 : 7 and 28 : 27. Archytas also defined scales in
the two other major genera, the enharmonic and chromatic.
Archytas' enharmonic tetrachord is composed of the intervals 5 :
4, 36 : 35 and 28 : 27 and his chromatic tetrachord of the intervals
32 : 27, 243 : 224, and 28 : 27. There are several puzzles about the
tetrachords which Archytas adopts in each of the genera. First, why
does Archytas reject the Pythagorean diatonic used by Philolaus and
Plato? Second, Ptolemy, who is our major source for Archytas'
tetrachords (A16), argues that Archytas adopted as a principle that
all concordant intervals should correspond to superparticular ratios.
The ratios in Archytas' diatonic and enharmonic tetrachords are
indeed superparticular, but two of the ratios in his chromatic
tetrachord are not superparticular (32 : 27 and 243 : 224). Why are
these ratios not superparticular as well? Finally, Plato criticizes
Pythagorean harmonics in the *Republic* for seeking numbers in
heard harmonies rather than ascending to generalized problems (531c).
Can any sense be made of this criticism in light of Archytas'
tetrachords? The basis for an answer to all of these questions is
contained 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 three genera can be shown to correspond to
the musical practice of his day; Ptolemy's criticisms miss the
mark because of his ignorance of musical practice in Archytas'
day, some 500 years before Ptolemy (Winnington-Ingram 1932, 207).
Archytas is giving mathematical descriptions of scales actually in
use; although mathematical considerations did play a role (Barker
2007: 295-302), he arrived at his numbers in part by observation
of the way in which musicians tuned their instruments (Barker 1989,
50-51). He did not follow the Pythagorean diatonic scale because
it did not correspond to any scale actually in use, although it does
correspond to a method of tuning. The unusual numbers in
Archytas' chromatic tetrachord do correspond to a chromatic
scale in use in Archytas' day. Ptolemy was wrong to suggest that
Archytas adhered to the principle that all concordant intervals should
have superparticular ratios (Huffman 2005: 422-3), although
Barker suggests that he may have been following a different but
related principle (2007: 301). Archytas thus provides a brilliant
analysis of the music of his day, but it is precisely his focus on
actual musical practice that draws Plato's ire. Plato does not
want him to focus on the music he hears about him ("heard
harmonies") but rather to ascend to consider quite abstract
questions about which numbers are harmonious with which. Plato might
well have welcomed a principle of concordance based solely on
mathematical considerations, such as the principle that only
superparticular ratios are concordant, but Archytas wanted to explain
the numbers of the music he 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 a
sensible and an intelligible world and is looking for the numbers
which govern sensible things. For discussions of Archytas as the
target of Plato's complaints in the  *Republic* see
Huffman 2005: 423-5 and Barker 2014:192-3.


### 2.3 Evaluation of Archytas as Mathematician



There have been tendencies both to overvalue and to undervalue
Archytas' achievement as a mathematician. Van der Waerden went
so far as to add to Archytas' accomplishments both Book VIII of
Euclid's *Elements* and the treatise on the mathematics
of music known as the *Sectio Canonis*, which is ascribed to
Euclid in the ancient tradition (1962, 152-5). Although later
scholars (e.g., Knorr 1975: 244) repeat these assertions, they are
based in part on a very subjective analysis of Archytas' style.
Archytas influenced the *Sectio Canonis*, since Proposition 3
is based on a proof by Archytas (A19), but the treatise cannot be by
Archytas, because its theory of pitch and its account of the diatonic
and enharmonic tetrachords differ from those of Archytas. On the other
hand, some scholars have cast doubt on Archytas' prowess as a
mathematician, arguing that some of his work looks like "mere
arithmology" and "mathematical mystification"
(Burkert 1972a, 386; Mueller 1997, 289). This judgment rests largely
on a text (A17) that has been mistakenly interpreted as presenting
Archytas' own views, whereas, in fact, it presents
Archytas' report of his predecessors (Huffman 2005:
428-37; Barker 2007: 193-5). The duplication of the cube
and Archytas' contributions to the mathematics of music (Barker
2007: 287 calls him "a heroic figure in the early history of
mathematical harmonics") show that there can be no doubt that he
was one of the leading mathematicians of the first part of the fourth
century BC. This was certainly the judgment of antiquity. In his
history of geometry, Eudemus identified Archytas along with Leodamas
and Theaetetus as the three most prominent mathematicians of
Plato's generation (A6 = Proclus, *in Eucl.*, prol. II
66, 14). Netz (2014) has recently argued that there were two networks
that accounted for most of the progress in ancient Greek mathematics.
The later used Archimedes as the paradigmatic figure, while the
earlier used Archytas. Netz suggests that we should apply Bertrand
Russell's description of Pythagoras as "one of the most
important men that ever lived" to Archytas instead both because
of his mathematical genius and also because of his pivotal position in
three different groups: (1) South Italian Pythagoreans, (2) Greek
mathematicians, and (3) philosophers whose work was in dialogue with
Plato (2014: 181-2). 


## 3. Archytas on the Sciences


### 3.1 The Value of the Sciences



Archytas B1 is the beginning of his book on harmonics, and most of it
is devoted to 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 (*mathemata*) in general. There are
several important features of this proem. First, Archytas identifies a
set of four sciences: astronomy, geometry, "logistic"
(arithmetic) and music. B1 is thus probably the earliest text to
identify the set of sciences that became known as the
*quadrivium* in the middle ages and that constitute four of the
seven liberal arts. Second, Archytas does not present this
classification of sciences as his own discovery but instead begins
with praise of his predecessors who have worked in these fields. Some
scholars argue that, when he praises "those concerned with the
sciences," 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 Greek mathematics is Pythagorean. Archytas gives no hint
that he is limiting his remarks to Pythagoreans, and, in areas where
we can identify those who influenced him most, these figures are not
limited to Pythagoreans (e.g., Hippocrates of Chios in geometry, see
section 2.1). He praises his predecessors in the sciences, because,
"having discerned well about the nature of wholes, they were
likely also to see well how things are in their parts" and to
"have correct understanding about individual things as they
are." It is here that Archytas is 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 ability to make distinctions (*diagignoskein*). He
argues that they begin by distinguishing the nature of wholes, the
universal concepts of a science, and, because they do this well, they
are able to understand particular objects (the parts). Archytas
appears to follow exactly this procedure in his *Harmonics*. He
begins by defining the most universal concept of the science, sound,
and explains it in terms of other concepts such as impact, before
going on to distinguish between audible and inaudible sounds and
sounds of high and low pitch. The goal of the science is not the
making of these distinctions concerning universal concepts, however,
but knowledge of the true nature of individual things. Thus,
Archytas' harmonics ends with the mathematical description of
the musical intervals that we hear practicing musicians use (see
section 2.2 above). Astronomy will end with a mathematical description
of the periods, risings and settings of the planets. One way to
understand Archytas' project is to see him as working out the
program suggested by his predecessor in the Pythagorean tradition,
Philolaus. One of Philolaus' central theses was that we only
gain knowledge of things insofar as we can give an account of them in
terms of numbers (DK 44 B4). While Philolaus only took the first steps
in this project, Archytas is much more successful in giving an account
of individual things in the phenomenal world in terms of numbers, as
his description of the musical intervals shows.



Plato's account of the sciences in Book VII of the
*Republic* can be seen as a response 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 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 to
Archytas' *quadrivium* and does not mention him by name.
In the same passage, however, Plato pointedly rejects the Pythagorean
attempt to search for numbers in "heard harmonies." In
doing so Plato is disagreeing with Archytas' attempt to
determine the numbers that govern things in the sensible world. For
Plato, the value of the sciences is their ability to turn the eye of
the soul from the sensible to the intelligible realm. Book VII of the
*Republic* with its elaborate argument for the distinction
between the intelligible and sensible realm, between the cave and the
intelligible world outside the cave, may be in large part directed at
Archytas' attempt to use mathematics to explain the sensible
world. As Aristotle repeatedly emphasizes, the Pythagoreans differed
from Plato precisely in their refusal 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 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 the four sister sciences in B1. There is simply not enough
context in B4 or other texts of Archytas to determine the meaning of
logistic from Archytas' usage alone. It is necessary to rely to
some extent on Plato, who is the only other early figure to use the
term extensively. A later conception of logistic, as something that
deals with numbered things rather than numbers themselves, which is
found 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 we
would call arithmetic (e.g. 3 x 700 = 2,100; see, *Hp.
Mi.* 366c). In other passages, however, Plato defines logistic in
parallel with *arithmetike*, and treats the two of
them as together constituting the science of number, on which
practical manipulation of number is based (Klein 1968, 23-24).
Both *arithmetike* and logistic deal with the even
and the odd. *Arithmetike* focuses not on
quantities but on kinds of numbers (*Grg.* 451b), beginning
with the even and 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 example of 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 consider himself to be doing logistic,
when he defines the three types of means which are relevant to music
(geometric, arithmetic, and harmonic). The geometric mean arises
whenever three terms are so related that, as the first is to the
second, so the second is to the third (e.g., 8 : 4 :: 4 : 2) and the
arithmetic, when three terms are so related that the first exceeds the
second by the same amount as the second exceeds the third (e.g., 6 : 4
:: 4 : 2). Archytas, like Plato (*R.* 525c), uses logistic not
just in this narrow sense of the study of relative quantity, but also
to designate the entire science of numbers including
*arithmetike*.



Why does Archytas think that logistic is superior to the other
sciences? In B4, he particularly compares it to geometry, arguing that
logistic (1) "deals with what it wishes more vividly than
geometry" and (2) "completes demonstrations" where
geometry cannot, even "if there is any investigation concerning
shapes." This last remark is surprising, since the study of
shapes would appear to be the proper domain of geometry. The most
common way of explaining Archytas' remark is to suppose 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 sees this as a reason for doubting the authenticity of the
fragment, since the exact opposite seems to be true. Archytas could
determine the cube root of two geometrically, through his solution to
the duplication of the cube, but could not do 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 in geometry 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 diagonal with the side of the square, since incommensurability
arises when two magnitudes "have not to one another the ratio
which number has to number" (Euclid X 7). These
suggestions show that logistic can be superior to geometry in certain
cases, but they do not explain Archytas' more general assertion
that logistic 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 sciences as 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 most
commonly used of ethical treatises. Moreover, it is specifically with
regard to wisdom (*sophia*) that logistic is said to be
superior, and, while *sophia* can refer to technical expertise,
it more commonly refers to the highest sort of intellectual
excellence, 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 makes us wiser than the other sciences? Since
Archytas evidently agreed with Philolaus that we only understand
individual things in the world insofar as we grasp the numbers that
govern them, it seems quite plausible that Archytas would regard
logistic 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 the Argive sculptor Polyclitus,
which he also referred to as the Canon (i.e., the standard). Although
Polyclitus undoubtedly made use of geometry in constructing this
magnificent shape, in a famous sentence from his book, also entitled
*Canon*, he asserts that his statue came to be not through many
shapes but "through many numbers" (DK40 B2, see
Huffman 2002a). Geometrical relations alone will not determine the
form of a given object, we have to assign specific proportions,
specific numbers. Archytas also thought that numbers and logistic were
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 state depends. Justice is a relation that needs
to be stated numerically and it is through such a statement that rich
and poor can live together, each seeing that he has what is fair.
Logistic will always be superior to the other sciences, because those
sciences will in the end rely on numbers to give us knowledge of the
sounds we hear, the shapes we see and the movements of the heavenly
bodies which we observe.


### 3.3 Optics and Mechanics



Aristotle is the first Greek author to mention the sciences of optics
and mechanics, describing optics as a subordinate science to geometry
and mechanics as a subordinate science to solid geometry
(*APo.* 78b34). Archytas does not mention either of these
sciences in B1, when describing the work of his predecessors in the
sciences, nor does Plato mention them. This silence suggests that the
two disciplines may have first developed in the first half of the
fourth century, when Archytas was most active, and it is possible that
he played an important role in the development of both of them. In a
recently 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 in
Pythagoreanism. Just as the Pythagoreans were impressed with the fact
that musical intervals were based on whole number ratios, so they were
impressed that the phenomena of optics could be explained in terms of
geometrical 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 visual ray, which proceeded from the eye,
requires the support of and coalesces with external light, Archytas
explained vision in terms of the visual ray alone (A25). It is
tempting, then, to suppose that Archytas played a major role in the
development of the mathematically based Pythagorean optics, to which
Aristotle refers. On the other hand, when Aristotle refers to
Pythagoreans, he generally means Pythagoreans of the fifth century.
Elsewhere he treats Archytas independently of the Pythagorean
tradition, writing works on Archytas which were distinct from his work
on the Pythagoreans. It would thus be more natural to read
Aristotle's reference to Pythagorean optics as alluding to
fifth-century Pythagoreans such as Philolaus. Archytas will then have
been responsible for developing an already existing Pythagorean
optical tradition into a science, rather than 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 science of mechanics. There is a
puzzle, however, since, no ancient Greek author in the later
mechanical tradition (e.g., Heron, Pappus, Archimedes, Philon) ever
ascribes any work in the field to Archytas. What did the ancients mean
by mechanics? A rough definition would be "the description and
explanation of the operation of machines" (Knorr 1996). The
earliest treatise in mechanics, the *Mechanical Problems*
ascribed to Aristotle, begins with problems having to do with a simple
machine, the lever. Pappus (AD 320) refers to machines used to lift
great weights, machines of war such as the catapult, water lifting
machines, amazing devices (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, there is a theoretical part that is heavily
mathematical (1022. 13-15). Given his interest in describing
physical phenomena in mathematical terms, it might seem logical that
Archytas would make important contributions to mechanics. 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 to
Plutarch's story about the quarrel between Plato and Archytas
over Archytas' supposed mechanical solution to the problem of
doubling the cube. This story is likely to be false (see 2.1 above).
Some scholars have argued that Archytas devised machines of war (Diels
1965; Cambiano 1998), as Archimedes did later, but this conclusion is
based on questionable inferences and no ancient source ascribes such
machines to Archytas. The only mechanical device that can with some
probability be assigned to Archytas, apart from the children's
toy known as a "clapper" (A10), is an automaton in the
form of a wooden dove, which was connected 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). Kingsley is deeply
critical of scholars who see these inventions as toys and suggests
connections to earlier Chinese inventors who produced a wooden bird
that could fly for military purposes (2014: 155-9). Others have
suggested that, since ancient siege devices were called by the names
of animals (e.g., "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 misunderstood to be a mechanical dove (Berryman 2003: 355; 2009:
78). However, no ancient source explains the dove in this way. A
complicating factor here is that Diogenes Laertius reports (A1) that
there was a book on mechanics in circulation, which some thought to be
by a different Archytas, so that it is possible that the flying dove
is, in fact, the work of a separate Archytas. Archytas' solution
to the duplication of the cube, although it was not mechanical itself,
was of enormous importance for mechanics, since the solution to the
problem allows one not just to double a cube but also to construct
bodies that are larger or smaller than a given body in any given
ratio. Thus, the solution permits the construction of a full-scale
machine on the basis of a working model. Pappus cites the solution to
the duplication of the cube as one of the three most crucial
geometrical theorems for practical mechanics (*Math. Coll.*
1028. 18-21). It may then be that Archytas' primary
contribution to mechanics was precisely his solution to the
duplication of the cube and that it is this solution which constituted
the mathematical first principles which Archytas provided for
mechanics. For a recent discussion of these issues seen Berryman 2009:
87-97. It is more doubtful that Archytas wrote a treatise on
mechanics. Schofield has recently argued that the evidence for
Archytas' work both in optics and in mechanics is so meagre that
we should "be skeptical about Archytas' alleged
role" (2014: 86).


## 4. Definitions



In the *Metaphysics*, Aristotle praises Archytas for having
offered definitions which took account of both form and matter
(1043a14-26 = A22). The examples given are
"windlessness" (*nenemia*), which is defined
as "stillness [the form] in a quantity of air [the
matter]," and "calm-on-the-ocean"
(*galene*), which is defined as "levelness
[the form] of sea [the matter]." The terms form and matter are
Aristotle's, and we cannot be sure how Archytas conceptualized
the two parts of his definitions. A plausible suggestion is that he
followed his predecessor Philolaus in adopting limiters and unlimiteds
as his basic metaphysical principles and that he saw his definitions
as combinations of limiters, such as levelness and stillness, 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 of some other
sort of investigation, e.g. cosmology, but were chosen precisely to
illustrate principles of definition. Archytas may thus have devoted a
treatise to the topic. Aristotle elsewhere comments on the use of
proportion in developing definitions and uses these same examples
(*Top.* 108a7). The ability to recognize likeness 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 expressed in the following proportion: as
*nenemia* is to the air so *galene* is
to the sea. It is tempting to suppose that Archytas, who saw the world
as explicable in terms of number and proportion, also saw proportion
as the key in developing definitions. This would explain another
reference to Archytas in Aristotle. At *Rhetoric*
1412a9-17 (= A12) Aristotle praises Archytas precisely for his
ability to see similarity, even in things which differ greatly, and
gives as an example Archytas' assertion that an arbitrator and
an altar are the same. DK oddly include this 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 to their common functions as a refuge, while recognizing
the different context and way in which this function is carried out.
For doubts about this reconstruction of Archytas' theory of
definition, see Barker 2006, 314-318 and Schofield 2014, 80.


## 5. Cosmology and Physics



We have very little evidence for Archytas' cosmology, yet he was
responsible for one of the most famous cosmological arguments in
antiquity, an argument which has 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 to Archytas 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 to engage in a
thought experiment (this is one of the first recorded thought
experiments in antiquity): "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 be able 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 that there will
always be something, into which his staff can be extended, beyond the
supposed limit, and hence that something is clearly unlimited. Neither
Plato nor Aristotle accepted this argument, and both believed that the
universe was limited. Nonetheless, Archytas' argument had great
influence and was taken over and adapted by the Stoics, Epicureans
(Lucretius I 968-983), Locke and Newton, among others, while
eliciting responses from Alexander and Simplicius (Sorabji 1988,
125-141). For a discussion of the relation between
Archytas' thought experiment and other thought experiments in
antiquity see Ierodiakonou 2011. Not all scholars have been impressed
by the argument (see Barnes 1982, 362), and modern notions of space
allow for it to be finite without having an edge, and without an edge
Archytas' argument cannot get started (but see Sorabji 1988,
160-163). Beyond this argument, there is only exiguous evidence
for Archytas' system of the physical world. Eudemus praises
Archytas for recognizing that the unequal 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 all things are moved in accordance with
proportion (Arist., *Prob*. 915a25-32 = A23a). The same
testimonium indicates that different sorts of proportion defined
different sorts of motion. Archytas asserted that "the
proportion of equality" (arithmetic proportion?) defined natural
motion, which he regarded as curved motion (for a different account of
the proportion of equality see De Groot 2014: 195-207). This
explanation of natural motion is supposed to explain why certain parts
of plants and animals (e.g. the stem, thighs, arms and trunk) are
rounded rather than triangular or polygonal. Some scholars argue that
it was the influence of Archytas that led Plato and Eudemus to
emphasize uniform circular movement in explaining 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 remain
obscure.


## 6. Ethics and Political Philosophy



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 in terms of number and proportion. In
Fr. 3, rational calculation is identified as the basis of the stable
state:



Once calculation (*logismos*) was discovered, it stopped
discord and increased concord. For people do not want more than their
share, and equality exists, once this has come into being. For by
means of calculation we will seek reconciliation in 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 will
have what is fair on account of this.



This glorification of calculation is reminiscent of Plato's
praise for "geometrical equality" in the *Gorgias*
(507e6-508a8) and Plato may be thinking of Archytas Fr. 3
(Palmer 2014: 205-6; Huffman 2013: 259-61). The emphasis
on equality (*isotas*) and fairness (*to ison*) 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 to understand numerical
calculations, including basic proportions, an ability that is shared
by most human beings. It is the clarity of calculation and proportion
that does away with the constant striving for more
(*pleonexia*), which produces discord in the state. Since the
state is based on a widely shared human ability to calculate, an
ability that the rich and poor share, Archytas was led to support a
more democratic constitution (see 1.3 above) than Plato, who
emphasizes the expert mathematical knowledge of a few (*R*.
546a ff.). Zhmud (2006: 60-76) points out connections between B3
and Isocrates and argues that Isocrates is referring to Archytas, when
he says that some praise the sciences for their utility and others try
to demonstrate that they contribute greatly to virtue
(*Busiris* 23). However, Archytas seems to accept both of these
views about the sciences, while Isocrates refers to two different
groups of people. Isocrates' reference is also very general and
makes no allusion to the central terms of B3 so that it is doubtful
that he has Archytas in mind. For further discussion of the argument
of B3 see Huffman 2005: 182-224 and Schofield 2008. 



Most of our evidence for Archytas' ethical views is,
unfortunately, not based on fragments of his writings but rather on
anecdotes, which probably ultimately derive from Aristoxenus'
*Life of Archytas*. The good life of the individual, no less
than the stability of the state, appears to have been founded on
rational calculation. Aristoxenus presented a confrontation between
the Syracusan hedonist, Polyarchus, and Archytas. Polyarchus'
long speech is preserved by Athenaeus and Archytas' response by
Cicero (A9 = *Deip.* 545a and *Sen.* XII 39-41
respectively). Schofield (2014: 70, n.2) raises doubts about whether
Cicero really preserves Archytas' response (See Huffman 2005:
323-37). Horky (2011: 120) assumes without argument that
Aristoxenus presented Archytas as delivering his speech in the
presence of Plato and the Samnite C. Pontius, but this is part of
Cicero's frame story and there is no evidence that it derives
from Aristoxenus. Polyarchus' defense of always striving for
more (*pleonexia*) and of the pursuit of pleasure is
reminiscent of Plato's presentations of Callicles and
Thrasymachus, but is not derived from those presentations and is
better seen as an important parallel development (Huffman 2002b).
Archytas bases his response on 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, since his 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 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 in obtaining it, the less we are
able to reason. Aristotle appears to refer to this argument in the
*Nicomachean Ethics* (1152b16-18). Archytas'
argument is specifically directed against bodily pleasure and he did
not think that all pleasure was disruptive; he enjoyed playing with
children (A8) and recognized that the pleasures of friendship were
part of a good life (Cicero, *Amic.* XXIII 88). Other anecdotes
emphasize that our actions must be governed by reason rather than the
emotions: Archytas refused to punish the serious misdeeds of his
slaves, because he had become angry and did not want to act out of
anger (A7); he restrained himself from swearing aloud by writing his
curses on a wall instead (A11). Palmer argues that Archytas was
working with a Pythagorean conception of the soul as divided into two
parts: the intellect and a part responsible for the affective states
such as emotions and appetites (2014: 209). 


## 7. Importance and Influence



Archytas fits the common stereotype of a Pythagorean better than
anyone else does. He is by far the most accomplished Pythagorean
mathematician, making important contributions to geometry,
logistic/arithmetic and harmonics. He was more successful as a
political leader than any other ancient philosopher, and there is a
rich anecdotal tradition about his personal self-control. It is
striking, however, that there are essentially no testimonia connecting
Archytas to metempsychosis or the religious aspect of Pythagoreanism.
Archytas is a prominent figure in the rebirth of interest in
Pythagoreanism in first century BC Rome: Horace, Propertius and Cicero
all highlight him. As the last prominent member of the early
Pythagorean tradition, more pseudo-Pythagorean works came to be forged
in his name than any other Pythagorean, including Pythagoras himself.
His name, with the spelling Architas, continued to exert power in
Medieval and Renaissance texts, although the accomplishments assigned
to him in 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 in serious disagreement on a number of issues.
Plato's only certain reference to Archytas is part of a
criticism of his approach to harmonics in Book VII of the
*Republic*, where there is probably also a criticism of his
work in solid geometry. 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 protreptic directed at Archytas, who
refused to separate numbers from things. It is sometimes thought that
the eponymous primary speaker in Plato's *Timaeus*, who
is 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 (Huffman
2013: 263-8). It is based on the split between the sensible and
intelligible world, 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. Plato constructs the world soul according
to ratios that are important in harmonic theory, but he uses
Philolaus' ratios rather than Archytas'. Plato does adopt
Archytas' theory of pitch with some modification, but Archytas
and Plato disagree on the explanation of sight. For a different view
of the relation between Plato and Archytas see Kingsley 2014:
156-7. Archytas' refusal to split the intelligible from
the sensible may have made him a more attractive figure to Aristotle,
who devoted four books to him (Huffman 2005: 583-594) and
praised his definitions for 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 is
closer to Aristotle's mathematical account of distributive and
redistributive justice (*EN* 1130b30 ff.) than to Plato's
emphasis on the expert mathematical knowledge of the guardians.
Clearly Archytas was an important influence on both Plato and
Aristotle, but the exact nature of those philosophical relationships
is complex.