## 1. The Philosophy of Pythagoras



See the entry on
 Pythagoras.


## 2. The Most Prominent Pythagoreans of the Fifth and Fourth Century


### 2.1 Philolaus



See the entry on
 Philolaus.


### 2.2 Eurytus



In the ancient sources, Eurytus is most frequently mentioned in the
same breath as Philolaus, and he is probably the student of Philolaus
(Iamblichus, *VP* 148, 139). Aristoxenus (4th c. BCE) presents
Philolaus and Eurytus as the teachers of the last generation of
Pythagoreans (Diogenes Laertius VIII 46) and Diogenes Laertius reports
that Plato came to Italy to meet Philolaus and Eurytus after the death
of Socrates (III 46). In order to be the pupil of Philolaus, who was
born around 470, and teach the last generation of Pythagoreans around
400, Eurytus would need to be born between 450 and 440. The sources
are very confused as to which S. Italian city he was from, Croton
(Iamblichus, *VP* 148), Tarentum (Iamblichus, *VP* 267;
Diogenes Laertius VIII 46) or Metapontum (Iamblichus, *VP* 266
and 267). It may be that the Eurytus from Metapontum is a different
Eurytus. It is possible that Archytas studied with Eurytus, since
Theophrastus (Aristotle's successor in the Lyceum) cites
Archytas as the source for the one testimony we have about the
philosophy of Eurytus (*Metaph*. 6a 19-22). In the
catalogue of Pythagoreans at the end of Iamblichus' *On the
Pythagorean Life* (267), Eurytus appears between Philolaus and
Archytas in the list of Pythagoreans from Tarentum, which may thus
suggest that he was regarded as the pupil of Philolaus and a teacher
of Archytas.



According to Theophrastus (*Metaph*. 6a 19-22), Eurytus
arranged pebbles in a certain way in order to show the number which
defined things in the world, such as a man or a horse. Aristotle
refers to the same practice (*Metaph*. 1092b8 ff.), and
Alexander provides commentary on the Aristotelian passage
(*CAG* I. 827.9). Aristotle introduces Eurytus as someone who
regarded numbers as causes of substances by being the points that
bound spatial magnitudes. He says that Eurytus made likenesses of the
shapes of things in the natural world with pebbles and thus determined
the number which belongs to each thing by the number of pebbles
required. Scholars often treat Eurytus' procedure as puerile and
have sometimes not taken him seriously (Kahn 2001, 33), or suggested
that Theophrastus is ironical in his presentation (e.g., Zhmud 2012,
410-411). There is, however, no obvious irony in
Theophrastus' remarks. He, in fact, presents Eurytus very
positively as someone who showed in detail how specific parts of the
cosmos arose out of basic principles, in contrast to other thinkers,
who posit basic principles but do not go very far in explaining how
the world arises from those principles. This positive presentation may
reflect Theophrastus' source, Archytas, who perhaps saw Eurytus
as attempting to carry out Philolaus' project of determining the
numbers that give us knowledge of things in the world (Huffman 2005,
55; see also Netz 2014, 173-178).



How are we, then, to understand Eurytus' procedure? It does not
seem plausible to suppose that he simply drew a picture or an outline
drawing of a man or a horse and then counted the number of pebbles
required to make the outline (Riedweg 2005, 86) or fill in the
picture, since the number would vary with the size of the drawing and
the size of the pebbles. A large picture of a man would require many
more pebbles than a small one, so that it would seem arbitrary which
number to associate with man. This interpretation treats Eurytus as a
mosaicist and is largely derived from Alexander's testimony.
Aristotle's presentation supports another interpretation. He
draws a parallel with those who arrange numbers of pebbles into
shapes, such as a triangle or a square. This suggests that Eurytus had
observed that, e.g., any three points in a plane determine a triangle
and any four a quadrilateral. He may then have drawn the general
conclusion that any shape or structure was determined by a unique
number of points and tried to represent these by setting out the
necessary number of pebbles. Thus, the complex structure of a
three-dimensional object such as the human body would require a large
number of points, but the number of points required to determine a
human being could be expected to be unique and to differ from the
number that determined any other object in the natural world, such as
a horse (Kirk and Raven 1957, 313 ff.; Guthrie 1962, 273 ff.; Barnes
1982, 390-391; Cambiano 1998). It is important to note that
nothing in these reports suggests that Eurytus thought that things
were composed of numbers or that he regarded the points that defined a
given thing as atoms of which things were made, as has sometimes been
supposed (Cornford 1922-1923, 10-11). Instead, he is best
understood as making a bold attempt to show that the structure of all
things is determined by number and thus to provide specifics for
Philolaus' general thesis that all things are known through
number. Another approach is to argue that no reference is being made
to creating a picture out of pebbles. The pebbles refer instead to
counters on an abacus, which the Greeks used for calculations. In this
case Eurytus can be supposed to have started by identifying certain
basic numerical properties with features of the world and then
deriving the number of man or horse through calculations using the
abacus (Netz 2014, 173-178).


### 2.3 Aristotle's "So-called" Pythagoreans



Aristotle refers to the Pythagoreans frequently in his extant works,
especially in the *Metaphysics*. There are several puzzles
about these references. First, his usual practice is to refer to the
Pythagoreans as a group rather than naming individuals. He mentions
Philolaus and Eurytus by name only once each and Archytas four times.
Yet, the basic Pythagorean system which he describes in most detail in
*Metaphysics* 1.5 shows such strong similarities to the
fragments of Philolaus that Philolaus must be the primary source
(Huffman 1993, 28-94, Schofield 2012, 147), although some
scholars emphasize that Aristotle clearly did use other sources
(Primavesi 2012, 255) and even that Philolaus, while perhaps the acme
of Pythagorean philosophy, might not have represented mainstream
Pythagoreanism thus explaining why Aristotle refers to the
Pythagoreans as a group rather than singling out Philolaus (McKirahan
2013). Second, he frequently refers to the Pythagoreans that he
discusses as the "so-called" Pythagoreans. Why does he add
the qualifying phrase "so-called?" This phrase indicates
not that these are false Pythagoreans in contrast to some other true
Pythagoreans but rather that this is the standard way of referring to
these people, it is what people call them; but the phrase also
indicates that Aristotle has reservations about the name. Aristotle is
expressing his doubts about how or whether these figures are connected
to Pythagoras himself, whom Aristotle regards as a wonder-working
founder of a way of life rather than as participating in the tradition
of Presocratic cosmology (Huffman 1993, 31-34). It could also be
that it is the very variety of sources that Aristotle is using that
leads him to recognize that there are quite different stages in the
develpment of Pythagoreanism and hence to wonder in what sense a
figure like Philolaus who is at the end of that development should
still be called a Pythagorean (Primavesi 2014).



The biggest puzzle, however, concerns the philosophical system that
Aristotle assigns to the Pythagoreans. For the purposes of his
discussion in the *Metaphysics,* he treats most Pythagoreans as
adopting a mainstream system in contrast to another group of
Pythagoreans whose system is based on the table of opposites (see
section 2.4). The central thesis of the mainstream system is stated in
two basic ways: the Pythagoreans say that things are numbers or that
they are made out of numbers. In his most extended account of the
system in *Metphysics* 1.5, Aristotle says that the
Pythagoreans were led to this view by noticing more similarities
between things and numbers than between things and the elements, such
as fire and water, adopted by earlier thinkers. The Pythagoreans thus
concluded that things were or were made of numbers and that the
principles of numbers, the odd and the even, are principles of all
things. The odd is limited and the even unlimited. Aristotle
criticizes the Pythagoreans for being so enamored of numerical order
that they imposed it on the world even where it was not suggested by
the phenomena. Thus appearances suggested that there were nine
heavenly bodies orbiting in the heavens but, since they regarded ten
as the perfect number, they supposed that there must be a tenth
heavenly body, the counter-earth, which we cannot see. Later,
Aristotle is also critical of the Pythagoreans for employing
principles that do not derive from the sensible world, i.e.,
mathematical principles, even though all their efforts were directed
at explaining the physical world (*Metaphysics* 989b29). How
can they explain features of physical bodies such as weight or motion
using principles which have no weight and do not move
(990a8-990a16)? Indeed, it becomes clear that Aristotle
interpreted the Pythagorean cosmogony as starting out by constructing
the number one. The one then draws in the unlimited and produces the
rest of the number series and evidently the cosmos at the same time.
The number one and the other numbers from 1 to 10 are conceived of as
physical entities (*Metaphysics* 1091a13-18). The puzzle
is that Aristotle's description makes clear that he is basically
describing Philolaus' system (e.g., the counter-earth, limit and
unlimited, the generation of a one), yet a number of his central
assertions are flatly contradicted by the surviving fragments of
Philolaus. Most importantly, Philolaus never says that things are
numbers or are made out of numbers. For Philolaus things are composed
of limiters and unlimiteds held together by harmony (Frs. 1, 2 and 6)
and unlimiteds appear to include physical things like fire and breath
(Fr. 7, Aristotle Fr. 201). Numbers and the odd and the even do play a
prominent role in Philolaus (Frs. 4-5), but there is no hint
that they are understood as physical entites. Instead number has an
epistemological role: all things are known through number (Fr. 4). How
are we to explain this tension between what Aristotle reports and the
fragments of Philolaus? One approach is to recognize that Aristotle is
not giving a historical report of what the Pythagoreans said but an
interpretation of what he found in Philolaus and others. He does not
in fact know of any text in which the Pythagoreans said that things
were numbers or were made of numbers. Instead this is a conclusion
drawn by Aristotle; it is his summary statement of what the
Pythagorean system amounts to. That this is what Aristotle is doing is
suggested by another passage in the *Metaphysics* where he
starts out by flatly stating that the Pythagoreans say that all things
are numbers but then goes on to add "at least they apply
mathematical theories to bodies as if they (the bodies) consisted of
those numbers"(*Metaphysics* 1083b16). The "at
least" and "as if" show that Aristotle is drawing an
inference rather than referring to any explicit statement by the
Pythagoreans that things are numbers. Thus for Philolaus there are
analogies between numbers and things and numbers give us knowledge of
things but Aristotle mistakenly takes this to be equivalent to saying
that things are numbers or are made of numbers. Another approach is to
argue that Aristotle was right that Philolaus and other Pythagoreans
thought of the number one and other numbers as physical entities. The
one constructed in Philolaus Fr. 7 is not just the primal physical
unity but also the number one (Schofield 2012). At the opposite
extreme, Zhmud argues that Aristotle has essentially invented this
Pythagorean system with little regard for what any actual Pythagoreans
said in order to serve as background for his account of Plato's
theory of principles (2012a, 438, 394-414). Another approach
tries to mitigate the differences between Philolaus and Aristotle and
suggests that Aristotle's emphasis on number was derived from
Pythagorean numerology that was independent of Philolaus but that was
combined with material from Philolaus as a result of Aristotle's
decision to present one mainstream Pythagorean system (Primavesi
2012). 


### 2.4 The Pythagoreans of the Table of Opposites



At *Metaphysics* 986a22, after presenting his account of the
philosophy of "the so-called" Pythagoreans (985b23), which
has strong connections to the philosophy of Philolaus, Aristotle turns
to "others of this same group" and assigns to them what is
commonly known as the table of opposites (the opposites arranged
according to column [*kata sustoichian*]). These Pythagoreans
presented the principles of reality as consisting of ten pairs of
opposites:




|  |  |
| --- | --- |
| limit | unlimited |
| odd | even |
| unity | plurality |
| right | left |
| male | female |
| rest | motion |
| straight   | crooked |
| light | darkness |
| good | bad |
| square | oblong |



Aristotle then contrasts these Pythagoreans with Alcmaeon of Croton,
who said that the majority of human things come in pairs, and praises
the Pythagoreans for carefully defining the pairs of opposites both in
number and character, whereas Alcmaeon seemed to present a randomly
selected and ill-defined group of opposites. Aristotle suggests that
either Alcmaeon was influenced by these Pythagoreans or they by him.
Aristotle was thus not sure of the date of these Pythagoreans but
seems to entertain the idea that they either lived a little before
Alcmaeon or a little after, which would make them active anywhere from
the late 6th to the mid 5th century. Aristotle's manner of
introducing these Pythagoreans suggests that they are distinct from
Philolaus and his pupil Eurytus and perhaps earlier (Schofield 2012:
156), but it is not possible to be more specific about their identity.
It is possible that Aristotle only knows of the table through oral
transmission and that there were no specific names attached to it.



The table shows a strong normative slant by including good in one
column and bad in the other. In contrast, while Philolaus posits the
first two opposites in the table, limit and unlimited, as first
principles, there is no suggestion in the extant fragments of
Philolaus that limit was good and unlimited bad. Opposites played a
large role in most Presocratic philosophical systems. The Pythagoreans
who posited the table of opposites differed from other early Greek
philosophers not only in the normative view of the opposites but also
by including strikingly abstract pairs such as straight and crooked
and odd and even, in contrast to the more concrete opposites such as
hot and cold, which are typical elsewhere in early Greek philosophy.
Similar tables of opposites appear in the Academy (Aristotle,
*Metaph*. 1093b11; *EN* 1106b29 referring to Speusippus;
Simplicius in *CAG* IX. 247. 30ff.), and Aristotle himself
seems at times to adopt such a table (*Metaph*. 1004b27 ff.;
*Phys*. 201b25). Later Platonists and Neopythagoreans will
continue to develop these tables (see Burkert 1972a, 52, n. 119 for a
list). The table of opposites thus provides one of the clearest cases
of continuity between early Pythagoreanism and Platonism. Zhmud argues
that the table has little to do with early Pythagoreanism and is
largely a product of the Academy (2012: 449-452), but
Aristotle's discussion of it in connection with Alcmaeon clearly
shows that he regarded it as belonging to the fifth-century and it is
implausible to suppose that he confused the work of his contemporaries
in the Academy with Pythagorean ideas that were developed over a
century earlier. It may well be that the similarity between this
Pythagorean table of opposites and later Academic versions led to the
Neopythagorean habit, starting already in the early Academy, of
mistakenly assigning the fundamental pair of opposites in
Plato's late metaphysics, the one and the indefinite dyad, back
to Pythagoras (see on Neopythagoreanism below).


### 2.5 Archytas



See the entry on
 Archytas.


## 3. Other Pythagoreans of the Sixth, Fifth and Fourth Centuries


### 3.1 The Catalogue of Pythagoreans in Iamblichus' *On the Pythagorean Life*: Who Counts as a Pythagorean?



Iamblichus' *On the Pythagorean Life* (4th c. CE) ends
with a catalogue of 218 Pythagorean men organized by city followed by
a list of 17 of the most famous Pythagorean women. Of these 235
Pythagoreans, 145 appear nowhere else in the ancient tradition. This
impressive list of names shows the wide impact of Pythagoreanism in
the fifth and fourth centuries BCE. To what extent is it reliable? A
long line of scholars has argued that the catalogue has close
connections to and is likely to be based on Aristoxenus in the fourth
century BCE and is thus a reasonably accurate reflection of early
Pythagoreanism rather than a creation of the later Neopythagorean
tradition (Rohde 1871-1872, 171; Diels 1965, 23;
Timpanaro-Cardini 1958-1964, III 38 ff.; Burkert 1972a, 105, n. 40;
Zhmud 2012b, 235-244). This is up to a point a reasonable
conclusion, since it is hard to see who would have been better placed
than Aristoxenus to have such detailed information.



The arguments connecting Aristoxenus to the catalogue are not
unassailable, however, and it is likely that the list has been altered
in transmission, so that it cannot simply be accepted as the testimony
of Aristoxenus (Huffman 2008a). No names on the list can be positively
assigned to a date later than Aristoxenus, but this would be likely to
be true, even if the list were compiled at a later date, since
Pythagoreanism appears to have largely died out for the two centuries
immediately following Aristoxenus' death. Thus, Iamblichus does
not mention any Pythagorean who can be positively dated after the time
of Aristoxenus anywhere else in *On the Pythagorean Life*
either. Scholars have also argued that Iamblichus cannot have composed
the catalogue, since he mentions some 18 names that do not appear in
the catalogue. This argument would only work, if Iamblichus were a
careful and systematic author, which the repetitions and
inconsistencies in *On the Pythagorean Life* show that he was
not. While it is unlikely that Iamblichus composed the catalogue from
scratch, it is perfectly possible that he edited it in a number of
ways, while not feeling compelled to make it consistent with
everything he says elsewhere in the text. There are some peculiarities
of the catalogue that suggest a connection to Aristoxenus. Philolaus
and Eurytus are listed not under Croton but under Tarentum, just as
they are in one of the Fragments of Aristoxenus (Fr. 19 Wehrli =
Diogenes Laertius VIII 46). On the other hand, some features of the
catalogue are inconsistent with what we know of Aristoxenus.
Aristoxenus' teacher, Xenophilus, who is identified as from the
Thracian Chalcidice in the Fragments of Aristoxenus (Frs. 18 and 19
Wehrli), is identified as from Cyzicus in the catalogue. Moreover, the
legendary figure, Abaris, is included in the catalogue and even said
to be from the mythical Hyperborea, whereas Aristoxenus is usually
seen as resolutely trying to rationalize the Pythagorean tradition.
Thus, while Aristoxenus is quite plausibly taken to be the author of
the core of the catalogue, it is likely that additions, omissions, and
various changes have been made to the original document and hence it
is impossible to be sure, in most cases, whether a given name has the
authority of Aristoxenus behind it or not.



The catalogue includes several problematic names, such as Alcmaeon,
Empedocles, Parmenides and Melissus. Alcmaeon was active in Croton
when the Pythagoreans flourished there, but Aristotle explicitly
distinguishes Alcmaeon from the Pythagoreans and scholarly consensus
is that he is not a Pythagorean (see the entry on
 Alcmaeon).
 Most scholars would agree that Empedocles was heavily influenced by
Pythagoreanism; in the later tradition fragments of Empedocles are
routinely cited to support the Pythagorean doctrines of metempsychosis
and vegetarianism (e.g., Sextus Empiricus, *Adversus
Mathematicos* IX 126-30). On the other hand, both in the
ancient and in the modern world, Empedocles is not usually labeled a
Pythagorean, because, whatever the initial Pythagorean influences, he
developed a philosophical system that was his own original
contribution. Parmenides is again not usually identified as a
Pythagorean in either the ancient or modern tradition and, although
scholars have speculated on Pythagorean influences on Parmenides,
there is little that can be identified as overtly Pythagorean in his
philosophy. The reason for Parmenides' inclusion in the
catalogue is pretty clearly the tradition that his alleged teacher
Ameinias was a Pythagorean (Diogenes Laertius IX 21). There is no
reason to doubt this story, but it gives us no more reason to call
Parmenides a Pythagorean than to call Plato a Socratic or Aristotle a
Platonist. It would appear that Melissus was included on the list
because he was regarded in turn as the pupil of Parmenides. Inclusion
in the catalogue thus need not indicate that a figure lived a
Pythagorean way of life or that he adopted metaphysical principles
that were distinctively Pythagorean; he need only have had contact
with a Pythagorean teacher. It is possible that Aristoxenus included
Parmenides and Melissus on the list for these reasons or that he had
better reasons for including them (e.g., evidence that they lived a
Pythagorean life), but it is precisely famous names such as these that
would be likely to have been added to the list in later times, and
they may well not have appeared in Aristoxenus' catalogue at
all.



Zhmud (2012a, 109-134) has argued that it begs the question to
use a doctrinal criterion to identify Pythagoreans. We need to first
identify Pythagoreans and then see what their doctrines are.
Aristoxenus' catalogue of Pythagoreans as preserved in
Iamblichus is the crucial source. Zhmud takes the Pythagoreans on this
list whom we can identify (the overwhelming majority are just names
for us) and studies their interests and activities in order to arrive
at a picture of early Pythagoreanism. Of the 235 names Zhmud finds
only 15 about whom we know anything significant. Some of these are
non-controversial (Hippasus, Philolaus, Eurytus and Archytas).
However, Zhmud puts particular emphasis on a series of figures not
typically regarded as Pythagoreans, e.g., Democedes, Alcmaeon, Iccus,
Menestor,and Hippon. The range of interests of these figures leads him
to conclude that there is no one characteristic that is shared by all
Pythagoreans and that Wittgestein's concept of a family
resemblance should be employed to describe Pythagoreanism. Moreover,
his reliance on figures like Alcmaeon and Menestor leads him to the
surprising conclusion that natural science and medicine were more
important than mathematics for the philosophical views of early
Pythagoreans (2012a, 23). The foundation for this view of early
Pythagoreanism is problematic since the scholarly consensus is that
Alcmaeon was not a Pythagorean and it is also far from certain that
Menestor was a Pythagorean (see below). As argued above,
Iamblichus' catalogue cannot be used mechanically as a guarantee
that a given figure was a Pythagorean, because we cannot be sure that
it always reflects Aristoxenus. What criteria should then be used?



First, anyone identified as a Pythagorean by an early source
uncontaminated by the Neopythagorean glorification of Pythagoras (see
below) can be regarded as a Pythagorean. This would include sources
dating before the early Academy (ca. 350 BCE), where Neopythagoreanism
has its origin, and Peripatetic sources contemporary with the early
Academy (ca. 350-300 BCE, e.g., Aristotle, Aristoxenus and
Eudemus), who, under the influence of Aristotle, defined themselves in
opposition to the Academic view of Pythagoras.



Second, a doctrinal criterion is applicable. Anyone who espouses the
philosophy assigned to the Pythagoreans by Aristotle can be regarded
as a Pythagorean, although Aristotle presents that philosophy under an
interpretation that must be taken into account. It is important that
the use of such a doctrinal criterion be limited to quite specific
doctrines such as limiters and unlimiteds as first principles and the
cosmology that includes the counter-earth and central fire.
Particularly to be avoided is the assumption that any early
mathematician or any early figure who assigns mathematical ideas a
role in the cosmos is a Pythagorean. Mathematicians such as Theodorus
of Cyrene (who is included in Iamblichus' catalogue) and
Hippocrates of Chios (who is not) are not treated as Pythagoreans in
the early sources such as Plato, Aristotle and Eudemus, and there is
accordingly no good reason to call them Pythagoreans. Similarly, the
sculptor, Polyclitus of Argos, stated that "the good comes to be
... through many numbers," (Fr. 2 DK), but no ancient
source calls him a Pythagorean (Huffman 2002). As Burkert has
emphasized, mathematics is a Greek and not just a specifically
Pythagorean passion (1972a, 427).



Third, anyone universally (or almost universally) called a Pythagorean
by later sources, and whom early sources do not treat as independent
of Pythagoreanism, explicitly or implicitly, can be regarded as a
Pythagorean. This would include figures embedded in the biographical
tradition about Pythagoras and the early Pythagoreans, such as the
husband and wife, Myllias and Timycha.



This last criterion is more subjective than the first two and
difficult cases arise. The fifth-century botanist Menestor (DK I 375)
is discussed by Theophrastus and called one of "the old natural
philosophers" (*CP* VI 3.5) with no mention of any
Pythagoreanism. In this case, the inclusion of a Menestor in
Iamblichus' catalogue is not enough reason to regard
Theophrastus' Menestor as a Pythagorean. On the other hand,
although Aristotle treats Hippasus separately from the Pythagoreans,
as he does Archytas, the almost universal identification of Hippasus
as a Pythagorean in the later tradition and his deep involvement in
the biography of early Pythagoreanism, show that he should be regarded
as a Pythagorean (on Hippasus, see section 3.4 below). The
fifth-century figure Hippo (DK I 385), who is derided by Aristotle and
paired with Thales as positing water as the first principle
(*Metaph*. 984a3), is a particularly difficult case. An Hippo
is listed in Iamblichus' catalogue under Samos and Censorinus
tells us that Aristoxenus assigned Hippo to Samos rather than
Metapontum (DK I 385.4-5). This makes it look as if Aristoxenus
may be responsible for including Hippo in the catalogue. Burkert has
also tried to demonstrate connections between Hippo's philosophy
and that of the Pythagoreans (1972a, 290, n. 62). On the other hand,
neither Aristotle nor Theophrastus nor any of the Aristotelian
commentators call him a Pythagorean and the doxographers describe this
Hippo as from Rhegium (e.g., Hippolytus in DK I 385.17). It is thus
not clear whether we are dealing with one person or two people named
Hippo and it is doubtful that the Hippo discussed by the Peripatetics
was a Pythagorean (Zhmud regards Hippo as well as Menestor and
Theodorus as Pythagoreans -- 2012a, 126-128). Those figures
of the sixth, fifth and fourth century who have the best claim to be
considered Pythagoreans will be discussed in the following
sections.


### 3.2 The Earliest Pythagoreans: Brontinus, Theano, etc.



In the standard collection of the fragments and testimonia of the
Presocratics, Cercops, Petron, Brontinus, Hippasus, Calliphon,
Democedes, and Parmeniscus are listed as older Pythagoreans (DK I
105-113). Hippasus, who is the most important of these figures,
will be discussed separately below (sect. 3.4). Of the rest only
Brontinus, Calliphon and Parmeniscus appear in Iamblichus'
catalogue.



Brontinus is presented as either the husband or father of Theano (see
section 3.3 below). Brontinus (DK I 106-107) is elsewhere said
to have had a wife Deino and to be either from Metapontum or Croton.
Little is known about him, but his existence appears to be confirmed
by Alcmaeon, writing in the late sixth or early fifth century, who
addresses his book to a Brontinus along with Leon and Bathyllus (Fr. 1
DK). The latter two may also be Pythagoreans, since a Leon is listed
under Metapontum and a Bathylaus (*sic*) under Posidonia, in
Iamblichus' catalogue.



The elusive connection between Orphism and Pythagoreanism rears its
head with Brontinus. In late antiquity there was a consensus that
Pythagoras himself had been initiated into the Orphic mysteries and
derived much of his philosophy from Orphism (Proclus, *Commentary
on Plato's Timaeus*, 3.168.8). Authors of the fifth century
BCE know of no such initiation and often indicate that the influence
went the other way by reporting that Pythagoras was, in fact, the
author of supposed Orphic texts (Ion of Chios as reported in Diog.
Laert. 8.8). Similarly, the fourth-century author, Epigenes, reports
that Brontinus is supposed to be the real author of two works
circulating in the name of Orpheus (West 1983, 9 ff.). In the end it
is impossible to determine the relationship between Pythagoreanism and
Orphism because of the difficulty of defining either movement
precisely (see Betegh 2014).



Cercops (DK I 105-106) is an even more obscure figure who is,
again according to Epigenes, the supposed Pythagorean author of Orphic
texts (West 1983, 9, 248), although Burkert doubts that he was a
Pythagorean (1972a, 130).



To Petron (DK I 106) is ascribed the startling doctrine that there are
183 worlds arranged in a triangle, but he is only known from a passage
in Plutarch, is not called a Pythagorean there and is probably a
literary fiction (Guthrie 1962, 322-323; Burkert 1972a, 114;
Zhmud 2012a, 117).



A Parmeniscus (DK I 112-113) is called a Pythagorean by Diogenes
Laertius (IX 20) and may be the same as the Parmiskos listed under
Metapontum in Iamblichus' catalogue. Athenaeus reports that a
Parmeniscus of Metapontum lost the ability to laugh after descending
into the cave of Trophonius, only to recover it in a temple on Delos,
where the surviving inventory of the temple of Artemis records a
dedication of a cup by a Parmiskos (Burkert 1972a, 154).



There no good reason to think that Democedes (DK I 110-112), the
physician from Croton, was himself a Pythagorean, although he had some
Pythagorean connections. He is famous from Herodotus' account
(III 125 ff.) of his service to the tyrant, Polycrates, and the
Persian king, Darius. One late source names him a Pythagorean (DK I
112.21). Iamblichus mentions a Pythagorean named Democedes, who was
involved in the political turmoil surrounding the conspiracy of Cylon
against the Pythagoreans, but it is far from clear that this was the
physician (*VP* 257-261). Herodotus never calls Democedes
a Pythagorean nor do any other of the later sources (e.g., Aelian,
Athenaeus, the Suda), nor does he appear in Iamblichus'
catalogue. A Calliphon, who could be Democedes' father, is
presented as an associate of Pythagoras by Hermippus (DK I 111.36 ff.)
and appears in Iamblichus' catalogue, so it is reasonable to
regard him as a Pythagorean, although we know nothing more of him. It
is reported (Herodotus III 137) that Democedes married the daughter of
the Olympic victor, Milon, who was the Pythagorean, whose house was
used as a meeting place (Iamblichus, *VP* 249). It was
undoubtedly because Democedes came from Croton at roughly the time
when Pythagoras was prominent there and because of the Pythagorean
connections of his father and father-in-law that late sources came to
label Democedes himself a Pythagorean. For an argument that Democedes
was a Pythagorean see Zhmud 2012a, 120.


### 3.3 Pythagorean Women



Women were probably more active in Pythagoreanism than any other
ancient philosophical movement. The evidence is not extensive but is
sufficient to give us a glimpse of their role. At the end of the
catalogue of Pythagoreans in Iamblichus' *On the Pythagorean
Life*, after the list of 218 male Pythagoreans, the names of 17
Pythagorean women are given (*VP* 267). Since this list is
likely to be based on the work of Aristoxenus, it probably represents
what Aristoxenus learned from fourth-century Pythagoreans, although we
cannot, of course, be certain that some names were not inserted into
the list after the time of Aristoxenus (see section 3.1 above). Eleven
are identified as the wife, daughter or sister of a man but seven are
simply identified by their region or city-state of origin, although
the Echecrateia of Phlius listed seems likely to be connected to the
Echecrates of Phlius who appears in Plato's *Phaedo*. We
know nothing else about most of the names on the list and thus cannot
be sure in individual cases whether they belong to the sixth, fifth or
fourth century. For a speculative reconstruction of the role of women
in the Pythagorean society see Rowett (2014, 122-123), but this
reconstruction partly depends on the speech that Iamblichus reports
Pythagoras gave to the women of Croton upon his arrival (*VP*
54-57); however, while Pythagoras did give speeches to different
groups, including women, the text of the speech in Iamblichus is
probably a later fabrication (Burkert 1972a, 115; Zhmud 2012a, 70).
The Pythagoreans put particular emphasis on marital fidelity on the
part of both men and women (Gemelli Marciano 2014, 145). There is also
no reliable evidence for any writings by these women, although in the
later tradition works were forged in the names of some of them and of
other Pythagorean women not on the list (see section 4.2 below). 



The most famous name on the list is Theano who is here called the wife
of Brontinus but who is elsewhere treated as either the wife, daughter
or pupil of Pythagoras (Diogenes Laertius VIII 42; Burkert 1972a,
114). The role of women in early Pythagoreanism and the centrality of
Theano is further attested by Aristoxenus' contemporary,
Dicaearchus, who reports that Pythagoras had as followers not just men
but also women and that one of these, Theano, became famous (Fr. 40
Mirhday = Porphyry, *VP* 19). It is striking that Dicaearchus
does not identify her as the wife of either Brontius or Pythagoras but
simply as a follower of Pythagoras. In the later tradition a number of
works were forged in her name (see section 4.2 below), but we have
little reliable evidence about her (see Thesleff 1965, 193-201,
for testimonia and texts; Delatte 1922, 246-249; and Montepaone
1993). The second most famous name on the list is Timycha who, when
ten months pregnant, reportedly bit off her own tongue so that she
could not, under torture, reveal Pythagorean secrets to the tyrant
Dionysius (Iamblichus, *VP* 189-194). This story goes
back to Neanthes, writing in the late fourth or early third century
and may rely on local Pythagorean tradition (Schorn 2014, 310).


### 3.4 Hippasus and Other Fifth-century Pythagoreans: *acusmatici* and  *mathematici*



Hippasus is a crucial figure in the history of Pythagoreanism, because
the tradition about him suggests that even in the fifth century there
was debate within the Pythagorean tradition itself as to whether
Pythagoras was largely important as the founder of a set of rules to
follow in living one's life or whether his teaching also had a
mathematical and scientific dimension. Hippasus was probably from
Metapontum (Aristotle, *Metaph*. 984a7; Diogenes Laertius VIII
84), although Iamblichus says there was controversy as to whether he
was from Metapontum or Croton (*VP* 81), and he is listed under
Sybaris in Iamblichus' catalogue (*VP* 267). He is
consistently portrayed as a rebel in the Pythagorean tradition, in one
case a democratic rebel who challenged the aristocratic Pythagorean
leadership in Croton (Iamb. *VP* 257), but more commonly as the
thinker who initiated Pythagorean study of mathematics and the natural
world.



It is in this latter role that he is connected with the split between
two groups in ancient Pythagoreanism, the *acusmatici* (who
emphasized rules for living one's life, including various
taboos) and the *mathematici* (who emphasized study of
mathematics and the natural world). Each group claimed to be the true
Pythagoreans. Our knowledge of this split comes from Iamblichus, who
unfortunately presents two contradictory versions, with the result
that Hippasus is sometimes said to be one of the
*mathematici* and sometimes one of the
*acusmatici*. Burkert has convincingly shown that the correct
version is that reported by Iamblichus at *De Communi Mathematica
Scientia* 76.19 ff. (1972a, 192 ff.). According to this account,
the *acusmatici* denied that the *mathematici*
were Pythagoreans at all, saying that their philosophy derived from
Hippasus instead. The *mathematici* for their part, while
recognizing that the *acusmatici* were Pythagoreans of a sort,
argued that they themselves were Pythagoreans in a truer sense.
Hippasus' supposed innovations, they said, were in fact
plagiarisms from Pythagoras himself. The *mathematici*
explained that, upon Pythagoras' arrival in Italy, the leading
men in the cities did not have time to learn the sciences and the
proofs of what Pythagoras said, so that Pythagoras just gave them
instructions on how to act, without explaining the reasons. The
younger men, who did have the leisure to devote to study, learned the
mathematical sciences and the proofs. The former group were the first
*acusmatici*, who learned the oral instructions of Pythagoras
on how to live (the *acusmata* = "things heard"),
while the latter group were the first *mathematici*.
Hippasus was thus closely connected to the *mathematici*
in this split in Pythagoreanism but ended up being disavowed by both
sides. For an attempt to further characterize the
*mathematici* see Horky 2013.



It is difficult to be sure of Hippasus' dates, but he is
typically regarded as active in the first half of the fifth century
and perhaps early in that period (Burkert 1972a, 206; Zhmud 2012a,
124-125). The split in Pythagoreanism may have occurred after
the main period of his work and was perhaps connected to the attacks
on the Pythagorean societies by outsiders around 450 BCE (Burkert
1972a, 207), but certainty is not possible. Zhmud (2012a,
169-192) has argued that the split is an invention of the later
tradition, appearing first in Clement of Alexandria and disappearing
after Iamblichus. He also notes that the term  *acusmata*
appears first in Iamblichus (*On the Pythagorean Life*
82-86) and suggests that it also is a creation of the later
tradition. He admits that the Pythagorean maxims did exist earlier, as
the testimony of Aristotle shows, but they were known as
*symbola*, were originally very few in number and were mainly a
literary phenomenon rather than being tied to people who actually
practiced them (Zhmud 2012a, 192-195). The consensus view, which
accepts the split, is based on Burkert's argument that
Iamblichus'account of the split between the *acusmatici*
and *mathematici* can be shown to be derived from
Aristotle (1972a, 196). Burkert later reaffirmed this position,
although with a little less confidence, asserting that the
Aristotelian provenance of the text is "as obvious as it is
unprovable" (1998, 315). Indeed the description of the split in
what is likely to be the original version (Iamblichus, *On General
Mathematical Science* 76.16-77.18) uses language in
describing the Pythagoreans that is almost an Aristotelian signature,
"There are two forms of the Italian philosophy which is called
Pythagorean" (76.16). Aristotle famously describes the
Pythagoreans as "those called Pythagoreans" and also as
"the Italians" (e.g., *Mete.* 342b30,
*Cael.* 293a20). Thus, Aristotle remains the most likely
source. Zhmud also argues against the split on the grounds that there
are no individuals in the historical record that can be confidently
identified as *acusmatici*. Since the *acusmatici* were
neither original nor full-time philosophers, however, and simply
preserved the oral taboos handed down by Pythagoras, it is not
surprising that they are not singled out for attention in the sources.
Only a relatively small number of the names in Iamblichus'
catalogue can certainly be identified as *mathematici*
and most of the others, particularly the 145 individuals whose names
are only known from the catalogue, are likely to be
*acusmatici*, who to a greater or lesser degree followed the
Pythagorean *acusmata*, but left no other trace of their
activity. In addition, a number of other Pythagoreans of the fifth and
fourth century, who figure in anecdotes about the Pythagorean life are
likely to be *acusmatici* (see below). 



Hippasus is the first figure in the Pythagorean tradition who can with
some confidence be identified as a natural philosopher, mathematician
and music theorist. His connections are as much with figures outside
the Pythagorean tradition as those within it. This independence may
explain why neither Aristotle nor the doxographical tradition label
him a Pythagorean, but he is too deeply embedded in the traditions
about early Pythagoreanism for there to be any doubt that he was in
some sense a Pythagorean. Aristotle pairs Hippasus with Heraclitus as
positing fire as the primary element (*Metaph*. 984a7) and this
pairing is repeated in the doxography that descends from Theophrastus
(DK I 109. 5-16), according to which Hippasus also said that the
soul was made of fire. Philolaus, who was probably two generations
later than Hippasus, might have been influenced by Hippasus in
starting his cosmology with the central fire (Fr. 7). For Philolaus,
however, the central fire is a compound of limiter and unlimited,
whereas Hippasus is presented as a monist and does not start from
Philolaus' fundamental opposition between limiters and
unlimiteds.



There are only a few other assertions about the cosmology of Hippasus
and most of these seem to be the result of Peripatetic attempts to
classify him, such as the assertions that he makes all things from
fire by condensation and rarefaction and dissolves all things into
fire, which is the one underlying nature and that he and Heraclitus
regarded the universe as one, (always) moving and limited in extent
(DK I 109.8-10). More intriguing is the claim that he thought
there was "a fixed time for the change of the cosmos"
(Diogenes Laertius VIII 84), which might be a reference to a doctrine
of eternal recurrence, according to which events exactly repeat
themselves at fixed periods of time. This doctrine is attested
elsewhere for Pythagoras (Dicaearchus in Porphyry, *VP* 19).
Our information about Hippasus is sketchy, because he evidently did
not write a book. Demetrius of Magnesia (1st century BCE) reports that
Hippasus left nothing behind in writing (Diogenes Laertius VIII 84)
and this is in accord with the tradition that Philolaus was the first
Pythagorean to write a book.



Hippasus originates the early Pythagorean tradition of scientific and
mathematical analysis of music, which reaches its culmination in
Archytas a century later. The correspondence between the central
musical concords of the octave, fifth, and fourth and the whole number
ratios 2 : 1, 3 : 2 and 4 : 3 is reflected in the *acusmata*
(Iamblichus, *VP* 82) and was thus probably already known by
Pythagoras. This correspondence was central to Philolaus'
conception of the cosmos (Fr. 6a). Although the later tradition tried
to assign the discovery to Pythagoras himself (Iamblichus, *VP*
115), the method described in the story would not in fact have worked
(Burkert 1972a, 375-376). Hippasus is the first person to whom
is assigned an experiment demonstrating these correspondences that is
scientifically possible. Aristoxenus (Fr. 90 Wehrli = DK I 109. 31
ff.) reports that Hippasus prepared four bronze disks of equal
diameters, whose thicknesses were in the given ratios, and it is true
that, if free hanging disks of equal diameter are struck, the sound
produced by, e.g., a disk half as thick as another will be an octave
apart from the sound produced by the other disk (Burkert 1972a, 377).
Hippasus, thus, may be the first person to devise an experiment to
show that a physical law can be expressed mathematically (Zhmud 2012a,
310).



Another text associates Hippasus with Lasus of Hermione in an attempt
to demonstrate the correspondence by filling vessels with liquid in
the appropriate ratios. It is less clear whether this experiment would
have worked as described (Barker 1989, 31-32). Lasus was
prominent in Athens in the second half of the sixth century at the
time of the Pisistratid tyranny and was thus probably a generation
older than Hippasus. There is no indication that Lasus was a
Pythagorean and this testimony suggests that the discovery of and
interest in the mathematical basis of the concordant musical intervals
was not limited to the Pythagorean tradition. Lasus and Hippasus are
sometimes said to have been the first to put forth the influential but
mistaken thesis that the pitch of a sound depended on the speed with
which it travels, but it is far more likely that Archytas originated
this view. In the later tradition Hippasus is reported to have ranked
the musical intervals in terms of degrees of concordance, making the
octave the most concordant, followed by the fifth, octave + fifth,
fourth and double octave (Boethius, *Mus*. II 19).



Finally, Iamblichus associates Hippasus with the history of the
development of the mathematics of means (DK I 110. 30-37), which
are important in music theory, but Iamblichus' reports are
confused. It is likely that Hippasus worked only with the three
earliest means (the arithmetic, geometric and subcontrary/harmonic)
and that the changing of the name of the subcontrary mean to the
harmonic mean should be ascribed to Archytas rather than Hippasus
(Huffman 2005, 179-173).



The most romantic aspect of the tradition concerning Hippasus is the
report that he drowned at sea in punishment for the impiety of making
public and giving a diagram of the dodecahedron, a figure with twelve
surfaces each in the shape of a regular pentagon (Iamblichus,
*VP* 88). This is best understood as reflecting some sort of
mathematical analysis of the dodecahedron by Hippasus, but it is
implausible in terms of the history of Greek mathematics to suppose
that he carried out a strict construction of the dodecahedron, which
along with the other four regular solids is most likely to have first
received rigorous treatment by Theaetetus in the fourth century BCE
(Mueller 1997, 277; Waterhouse 1972; Sachs1917, 82). Nor is it clear
why public presentation of technical mathematical analysis should
cause a scandal, since few people would understand it. The most likely
explanation is that the dodecahedron was a cult object for the
Pythagoreans (dodecahedra in stone and bronze have been found dating
back to prehistoric times) and that it was because of these religious
connections that Hippasus' public work on the mathematical
aspects of the solid was seen as impious (Burkert 1972a, 460).



Another late story, which appears first in Plutarch, reports a scandal
which arose when knowledge of irrational magnitudes was revealed,
without specifying any punishment for the one who revealed it
(*Numa* 22). In Pappus' later version of the story, the
person who first spread knowledge of the existence of the irrational
was punished by drowning (Junge and Thomson 1930, 63-64).
Iamblichus knows two different versions of the story, one according to
which the malefactor was banished and a tomb was erected for him,
signifying his expulsion from the community (*VP* 246), but
another according to which he was punished by drowning as was the
person (not specifically said to be Hippasus here) who revealed the
dodecahedron (*VP* 247). Modern scholars have tried to combine
the two stories and suppose that Hippasus discovered the irrational
through his work on the dodecahedron (von Fritz 1945). This is pure
speculation, however, since neither does any ancient source connect
Hippasus to the discovery of the irrational nor does any source relate
the discovery of the irrational to the dodecahedron (Burkert 1972a,
459). Some scholars nonetheless credit Hippasus with the discovery of
irrationality (Zhmud 2012a, 274-278).



Some have argued that Hippasus was an important figure for the early
Academy to whom Academic doctrines were ascribed in order give them
his authority and even that he might be the Prometheus mentioned by
Plato as handing down the method from the gods in the
*Philebus* (Horky 2013). However, there is no explicit mention
of Hippasus by any member of the Academy and he is a minor figure in
fourth-century accounts of early Greek philosophy (e.g., Aristotle) so
it is hard to see what authority he could give to Academic views. 



The other major Pythagoreans of the fifth century were Philolaus and
Eurytus, who are discussed above.



The name, but not too much more, is known of a number of other fifth
century figures, who with varying degrees of probability may be
considered Pythagoreans. To the beginning of the fifth century belongs
Ameinias the teacher of Parmenides (Diogenes Laertius VIII 21). The
athlete and trainer, Iccus of Tarentum, is listed in Iamblichus'
catalogue, but none of the other sources, including Plato, call him a
Pythagorean. In the later tradition, he was famous for the simplicity
of his life and "the dinner of Iccus" was proverbial for
plain fare. Plato praises his self control and reports that he touched
neither women nor boys while training. (*Laws* 839e; see
*Protagoras* 316d and DK I 216. 11 ff.).



Some scholars have treated the Sicilian comic poet Epicharmus as a
Pythagorean and argued that the growing argument which appears in a
fragment of controversial authenticity ascribed to him in Diogenes
Laertius (3.11) is thus Pythagorean in origin (Horky 2013,
131-140). However, no fifth- or fourth-century source identifies
Epicharmus as a Pythagorean and he does not appear in the catalogue of
Iamblichus. The earliest explicit mention of him as a Pythagorean is
in Plutarch (*Numa* 9) in the first century CE. There is no
compelling evidence that the reference to Epicharmus as a Pythagorean
in Iamblichus' *On the Pythagorean Life* 266 derives from
the fourth-century historian Timaeus as Horky proposes (2013, 116).
Burkert suggests that the information on Didorus in 266 might derive
from Timaeus (1972, 203-204) but Iamblichus regularly combines
material from a number of sources so that neither Burkert nor most
scholars regard the passage as a whole as deriving from Timaeus
(Schorn 2014 only mentions VP 254-264 as having material from
Timaeus). Epicharmus has also been thought to be a Pythagorean because
the growing argument which he uses for comic effect uses pebbles to
represent numbers and refers to odd and even numbers. However, neither
of the features is peculiarly Pythagorean; the concept of odd and even
numbers belongs to Greek mathematics in general and not just to the
Pythagoreans and the use of counters (pebbles) on an abacus is the
standard way in which Greeks manipulated numbers (Netz 2014, 178; cf.
Burkert's doubts that there is anything Pythagorean in the
Epicharmus fragment 1972a, 438). Most scholars regard
Epicharmus' Pythagoreanism as a creation of the later tradition
(Zhmud 2012a, 118; Riedweg 2005, 115; Kahn 2001, 87). 



There is no reason to regard the physician Acron of Acragas as a
Pythagorean, as Zhmud does (1997, 73; he appears to have changed his
mind in 2012a, 116). Acron is a contemporary of Empedocles and is
connected to him in the doxographical tradition (DK I 283. 1-9;
Diogenes Laertius VIII 65). No ancient source calls him a Pythagorean.
His name appears in a very lacunose papyrus along with the name of
Aristoxenus (Aristoxenus, Fr. 22 Wehrli), but it is pure speculation
that Aristoxenus labeled him a Pythagorean; Euryphon the Cnidian
doctor of the fifth century, who was not a Pythagorean, also appears
in the papyrus. Acron's father's name was Xenon, and a
Xenon appears in Iamblichus' catalogue, but he is listed as from
Locri and not Acragas, so again this is not good evidence that Acron
was a Pythagorean.



The Pythagorean Paron (DK I 217. 10-15) is probably a fiction
resulting from a misreading of Aristotle (Burkert 1972a, 170).
Aristotle reports the expression of a certain Xuthus, that "the
universe would swell like the ocean," if there were not void
into which parts of the universe could withdraw, when compressed
(*Physics* 216b25). Simplicius says, on unknown grounds, that
this Xuthus was a Pythagorean, and scholars have speculated that he
was responding to Parmenides (DK I. 376. 20-26; Kirk and Raven
1957, 301-302; Barnes 1982, 616).



Aristoxenus reports that two Tarentines, Lysis and Archippus, were the
sole survivors when the house of Milo in Croton was burned, during a
meeting of the Pythagoreans, by their enemies (Iamblichus, *VP*
250). A later romantic version in Plutarch (*On the Sign of
Socrates* 583a) has it that Lysis and Philolaus were the two
survivors, but it appears that the famous name of Philolaus has been
substituted for Archippus, about whom nothing else is known.
Aristoxenus goes on to say that Lysis left southern Italy and went
first to Achaea in the Peloponnese before finally settling in Thebes,
where the famous Theban general, Epaminondas, became his pupil and
called him father. In order to be the teacher of Epaminondas in the
early fourth century, Lysis must have been born no earlier than about
470. Thus the conflagration that he escaped as a young man must have
been part of the attacks on the Pythagoreans around 450, rather than
those that occurred around 500, when Pythagoras himself was still
alive. The later sources often conflate these two attacks on the
Pythagoreans (Minar 1942, 53). Nothing is known of the philosophy of
Lysis, but it seems probable that he should be regarded as one of the
*acusmatici*, since his training of Epaminondas appears to have
emphasized a way of life rather than mathematical or scientific
studies (Diodorus Siculus X 11.2) and Epaminondas' use of the
name father for Lysis suggests a cult association (Burkert 1972a,
179). In the later tradition, Lysis became quite famous as the author
of a spurious letter (Thesleff 1965, 111; cf. Iamblichus, *VP*
75-78) rebuking a certain Hipparchus for revealing Pythagorean
teachings to the uninitiated (see on the Pythagorean pseudepigrapha
below, sect. 4.2).



Zopyrus of Tarentum is mentioned twice, in a treatise on siege-engines
by Biton (3rd or 2nd century BCE), as the inventor of an advanced form
of the type of artillery known as the belly-bow (Marsden 1971,
74-77). Zopyrus' bow used a winch to pull back the string
and hence could shoot a six-foot wooden missile 4.5 inches thick
(Marsden 1969, 14). It is not implausible to suppose that this is the
same Zopyrus as is listed in Iamblichus' catalogue of
Pythagoreans under Tarentum (Diels 1965, 23), although Biton does not
call him a Pythagorean. The traditional dating for Zopyrus puts him in
the first half of the fourth century (Marsden 1971, 98, n. 52), but
Kingsley has convincingly argued that he was in fact active in the
last quarter of the fifth century, when he designed artillery for
Cumae and Miletus (1995, 150 ff.). In a famous passage, Diodorus
reports that in 399 BCE Dionysius I, the tyrant of Syracuse, gathered
together skilled craftsmen from Italy, Greece and Carthage in order to
construct artillery for his war with the Carthaginians (XIV 41.3). It
seems not unlikely that Zopyrus was one of those who came from Italy.
There is no reason to suppose, however, as Kingsley (1995, 146) and
others do, that Zopyrus' interest in mechanics was connected to
his Pythagoreanism or that there was a specifically Pythagorean school
of mechanics in Tarentum (Huffman 2005, 14-17).



It is controversial whether this Zopyrus of Tarentum is the same as
Zopyrus of Heraclea, who is not called a Pythagorean in the sources,
but who is reported in late sources to have written three Orphic
poems, *The Net*, *The Robe* and *The Krater*,
which probably dealt with the structure of human beings and the earth
(West 1983, 10 ff.). This Zopyrus could be from the Heraclea closely
connected to Tarentum, but he might also be from the Heraclea on the
Black Sea. A late source connects Zopyrus of Heraclea with Pisistratus
in the 6th century (West 1983, 249), which would mean that he could
not be the same as Zopyrus of Tarentum in the late 5th century. On the
other hand, Orphic writings are assigned to a number of other
Pythagoreans, and it is not impossible that the same person had
interests both in Orphic mysticism and mechanics. Kingsley supposes
that the myth at the end of Plato's *Phaedo* is based in
minute detail on Zopyrus' *Krater* or an intermediary
reworking of it (1995, 79-171), and tries to connect specific
features of the myth to Zopyrus' interest in mechanics (1995,
147-148), but the parallel which he detects between the
oscillation of the rivers in the mythic account of the underworld and
the balance of opposing forces used in a bow is too general to be
compelling. The connection between Zopyrus and the *Phaedo* is
highly conjectural and must remain so, as long as there are no
fragments of the *Krater*, with which to compare the
*Phaedo*.



A harmonic theorist named Simus is accused of having plagiarized one
of seven pieces of wisdom inscribed on a bronze votive offering, which
was dedicated in the temple of Hera on Pythagoras' native island
of Samos, by Pythagoras' supposed son Arimnestus (Duris of Samos
in Porphyry, *VP* 3). There is a Simus listed under Posidonia
(Paestum in S. Italy) in Iamblichus' catalogue of Pythagoreans,
so that DK treated him as a Pythagorean (I 444-445) who, like
Hippasus, stole some of the master's teaching for his own glory.
There is, however, no obvious connection between the two individuals
named Simus except the name. Most scholars have thus treated Simus as
if he were a harmonic theorist in competition with and independent of
the Pythagorean tradition (Burkert 1972a, 449-450; Zhmud 2012a,
118; West 1992, 79 and 240; Wilamowitz 1962, II 93-94).



What exactly he stole is very unclear. He is said to have removed
seven pieces of wisdom from the monument and put forth one of them as
his own. This is perhaps best understood as meaning that he took an
inscribed piece of metal from the dedicated object, perhaps a cauldron
(see Wilamowitz 1962, II 94). The inscription will have included all
seven pieces of wisdom, but Simus chose to publish only one of them as
his own, the other six being thus lost. The piece of wisdom he put
forth as his own is called a *kanon*
("rule"). West takes this as a reference to the monochord,
which was called the *kanon*, used to determine and
illustrate the numerical ratios, which were related to the concordant
intervals (1992, 240). Since, however, the *kanon* seems
to have been something inscribed on the dedication, along with six
other pieces of wisdom, it is perhaps better to assume that the
*kanon* was a description of a set of ratios determining
a scale (Burkert 1972a, 455; Wilamowitz 1962, 94). There must have
been a scale in circulation associated with the name of Simus. The
story that Duris reports is then an attempt by the Pythagoreans to
claim this scale as, in fact, the work of Pythagoras or his son, which
Simus plagiarized. Duris wrote in the first part of the third century
BCE, so Simus has to be earlier than that. If the son of Pythagoras
really made the dedication in the temple, this would have occurred in
the fifth century, but it is unclear how much later than that
Simus' *kanon* became known. West dates him to the
fifth century, whereas DK places him in the fourth. Zhmud suspects
that he is an invention of the pseudo-Pythagorean tradition
(2012a, 118).



Iamblichus describes an 'arithmetical method' known as the
bloom of Thymaridas (*In Nic.* 62), and elsewhere discusses two
points of terminology in Thymaridas, including his definition of the
monad as "limiting quantity" *(In Nic.* 11 and 27).
Some scholars have dated Thymaridas to the time of Plato or before,
but others argue that the terminology assigned to him cannot be
earlier than Plato and shows connections to Diophantus in the third
century CE (see Burkert 1972a, 442, n. 92 for a summary of the
scholarship). There is also a Thymaridas in the biographical
tradition, who may or may not be the same individual. In a highly
suspect passage in Iamblichus, Thymarides is listed as a pupil of
Pythagoras himself (*VP* 104) and a Thymaridas of Paros appears
in Iamblichus' catalogue and is mentioned in one anecdote
(*VP* 239). There is also a worrisome connection to the
pseudo-Pythagorean literature. A Thymaridas of Tarentum is presented
in an anecdote (Iamblichus, *VP* 145) as arguing that people
should wish for what the gods give them rather than praying that the
gods give them what they want, a sentiment that is also found in a
group of three treatises forged in Pythagoras' name (Diogenes
Laertius VIII 9). The anecdote is drawn from Androcydes' work on
the Pythagorean *symbola* or taboos. If this work could be
dated to the fourth century, it would confirm an early date for
Thymaridas, but all that is certain is that Androcydes' work was
known in the first century BCE and thus that the anecdote originated
before that date (Burkert 1972a, 167). It seems rash, given this
confused evidence, to follow Zhmud and regard Thymaridas as a younger
contemporary or pupil of Archytas (2012a, 131).


### 3.5 The Fourth Century: Aristoxenus, the Last of the Pythagoreans, and the Pythagorists



Aristoxenus (ca. 375- ca. 300 BCE) is most famous as a music theorist
and as a member of the Lyceum, who was disappointed not be to named
Aristotle's successor (Fr. 1 Wehrli). In his early years,
however, he was a Pythagorean, and he is one of the most important
sources for early Pythagoreanism. He wrote five works on
Pythagoreanism, although it is possible that some of these titles are
alternative names for the same work: *The Life of Pythagoras*,
*On Pythagoras and His Associates*, *On the Pythagorean
Life*, *Pythagorean Precepts* and a *Life of
Archytas*. None of these works have survived intact, but portions
of them were preserved by later authors (Wehrli 1945). Aristoxenus is
a valuable source because, as a member of the Lyceum, he is free of
the distorted image of Pythagoras propagated during his lifetime by
Plato's successors in the Academy (see below, sect. 4.1) and
because of his unique connections to Pythagoreanism. 



He was born in Tarentum during the years when the most important
Pythagorean of the fourth century, Archytas, was the leading public
figure and his father, Spintharus, had connections to Archytas (Fr. 30
Wehrli). When Aristoxenus left Tarentum, as a young man, and
eventually came to Athens (ca. 350), his first teacher was the
Pythagorean, Xenophilus, before he went on to become the pupil of
Aristotle (Fr. 1 Wehrli). Some modern scholars are skeptical of
Aristoxenus' testimony, seeing his denial that there was a
prohibition on eating beans and his assertion that Pythagoras was not
a vegetarian and particularly enjoyed eating young pigs and tender
kids (Fr. 25 = Gellius IV 11), as attempts to make Pythagoreanism more
rational than it was (Burkert 1972a, 107, 180). On the other hand, his
*Life of Archytas* is not a simple panegyric; Archytas'
foibles are recognized and his opponents are given a fair hearing. On
Aristoxenus as a source for Pythagoreanism see most recently Zhmud
2012b and Huffman 2014b, 285-295.



Perhaps Aristoxenus' most interesting work on Pythagoreanism is
the *Pythagorean Precepts*, which is known primarily through
substantial excerpts preserved by Stobaeus (Frs. 33-41 Wehrli).
This work does not mention any Pythagoreans by name but presents a set
of ethical precepts that "they" (i.e. the Pythagoreans)
proposed concerning the various stages of human life, education, and
the proper place of sexuality and reproduction in human life. There
are also analyses of concepts important in ethics, such as desire and
luck. Given Aristoxenus' background, the *Precepts* would
appear to be invaluable evidence for Pythagorean ethics in the first
half of the fourth century, when Aristoxenus was studying
Pythagoreanism. They might be expected to partially embody the views
of his teacher Xenophilus. The standard scholarly view of this work,
however, is that Aristoxenus plundered Platonic and Aristotelian ideas
for the glory of the Pythagoreans (Wehrli 1945, 58 ff.; Burkert 1972a,
107-108; Zhmud 2012a, 65). There are serious difficulties with
the standard view, however (Huffman 2019). The analysis of luck that
was supposedly taken from Aristotle is, in fact, in sharp conflict
with Aristotle's view (Mills 1982) and appears to be one of the
views Aristotle was attacking. While the *Precepts* do have
similarities to passages in Plato and Aristotle, they are at a very
high level of generality and are shared with passages in other fifth
and fourth century authors, such as Xenophon and Thucydides; it is the
distinctively Platonic and Aristotelian features that are missing.



The *Precepts* are thus best regarded as what they appear on
the surface to be, an account of Pythagorean ethics of the fourth
century. This ethical system shows a similarity to a conservative
strain of Greek ethics, which is also found in Plato's
*Republic*, but has its own distinctive features (Huffman
2019). The central outlook of the *Precepts* is a distrust of
basic human nature and an emphasis on the necessity for supervision of
all aspects of human life (Fr. 35 Wehrli). The emphasis on order in
life is so marked that the *status quo* is preferred to what is
right (Fr. 34). The Pythagoreans were particularly suspicious of
bodily desire and analyzed the ways in which it could lead people
astray (Fr. 37). There are strict limitations on sexual desire and the
propagation of children (Fr. 39). Despite the best efforts of
humanity, however, many things are outside of human control, so the
Pythagoreans examined the impact of luck on human life (Fr. 41).



Aristoxenus is a source for the famous story of the two Pythagorean
friends Damon and Phintias, which was set during the tyranny of
Dionysius II in Syracuse (367-357). As a test of their
friendship Dionysius falsely accused Phintias of plotting against him
and sentenced him to death. Phintias asked time to set his affairs in
order, and Dionysius was amazed when Damon took his place, while he
did so. Phintias showed his equal devotion to his friend by showing up
on time for his execution. Dionysius cancelled the execution and asked
to become a partner in their friendship but was refused (Iamblichus,
*VP* 234; Porphyry, *VP* 59-60; Diodorus X
4.3).



In Diodorus' version, Phintias is presented as actually engaged
in a plot against Dionysius and some argue that Aristoxenus'
version is an attempt to whitewash the Pythagoreans (Riedweg 2005,
40). On the other hand, Dionysius' eagerness to join in their
friendship, which occurs in both versions, is harder to understand if
there really had been a plot (see Burkert 1972a, 104). There are two
other considerations. First, Aristoxenus cites Dionysius II himself as
his source, whereas it is unclear what source Diodorus used. Second,
it is far from clear that Aristoxenus would object to the Pythagoreans
plotting against a tyrant. Thus, there are good reasons for regarding
Aristoxenus' version as more accurate.



Cleinias and Prorus are another pair of Pythagorean friends, whose
story may have been told by Aristoxenus (Iamblichus, *VP* 127),
although they were not friends in the usual sense. Cleinias, who was
from Tarentum, knew nothing of Prorus of Cyrene other than that he was
a Pythagorean, who had lost his fortune in political turmoil. On these
grounds alone he went to Cyrene, taking the money to restore
Prorus' fortunes (Iamblichus, *VP* 239; Diodorus X 4.1).
Nothing else is known of Prorus, although some pseudepigrapha were
forged in his name (Thesleff 1965, 154.13). It appears that Cleinias
was a contemporary of Plato, since Aristoxenus reports that he and an
otherwise unknown Pythagorean, Amyclas, persuaded Plato not to burn
the books of Democritus, on the grounds that it would do no good,
since they were already widely known (Diogenes Laertius IX 40).
Cleinias was involved in several other anecdotes. Like Archytas he
supposedly refused to punish when angry (*VP* 198) and, when
angered, calmed himself by playing the lyre (Athenaeus XIV 624a).
Asked when one should resort to a woman he said "when one
happens to want especially to be harmed" (Plutarch,
*Moralia* 654b). Several pseudepigrapha appear in
Cleinias' name as well.



Myllias of Croton and his wife Timycha appear in Iamblichus'
catalogue and are known from a famous anecdote of uncertain origin,
which is preserved by Iamblichus (*VP* 189 ff.). They were
persecuted by the tyrant Dionysius II of Syracuse, but Timycha showed
her loyalty and courage by biting off her tongue and spitting it in
the tyrant's face, rather than risk divulging Pythagorean
secrets under torture.



None of the Pythagoreans mentioned in the previous four paragraphs
appear to have to have anything to do with the sciences or natural
philosophy. Since their Pythagoreanism consists exclusively in their
way of life, they are best regarded as examples of the
*acusmatici*. Many scholars have regarded Diodorus of Aspendus
in Pamphylia (southern Asia Minor), as an important example of what
the Pythagorean *acusmatici* were like in the first half of the
fourth century (Burkert 1972a, 202-204). Diodorus is primarily
known through a group of citations preserved by Athenaeus (IV 163c-f),
which describe him as a vegetarian who was outfitted in an outlandish
way, some features of which later became characteristic of the Cynics,
e.g., long hair, long beard, a shabby cloak, a staff and
beggar's rucksack (cf. Diogenes Laertius VI 13). The historian
Timaeus (350-260), however, casts doubt on Diodorus'
credentials as a Pythagorean saying that "he pretended to have
associated with the Pythagoreans" and Sosicrates, another
historian (2nd century BCE; fragments in Jacoby) says that his
outlandish dress was his own innovation, since before this
Pythagoreans had always worn white clothing, bathed and wore their
hair according to fashion (Athenaeus IV 163e ff.). Iamblichus, the
other major source for Diodorus outside Athenaeus, also treats
Diodorus with reserve, saying that he was accepted by the leader of
the Pythagorean school at the time, one Aresas, because there were so
few members of the school. He continues, perhaps again with
disapproval, to report that Diodorus returned to Greece and spread
abroad the Pythagorean oral teachings.



These sources clearly suggest that Diodorus was anything but a typical
Pythagorean, even of the *acusmatic* variety. Burkert has
argued that this reflects a bias of sources such as Aristoxenus, who
wanted to make Pythagoreanism appear reasonable and emphasized the
version of Pythagoreanism practiced by the *mathematici*
rather than the *acusmatici*. In support of this conclusion, he
argues that the two earliest sources present Diodorus as a Pythagorean
without any qualifications (1972a, 204). It is important to look
carefully at those sources, however. First, neither is a philosopher
or a historian, who might be expected to give a careful presentation
of Diodorus. The oldest is a lyre player named Stratonicus (died 350
BCE), who was famous for his witticisms, and the other, Archestratus
(fl. 330 BCE), wrote a book entitled *The Life of Luxury*,
which focused on culinary delights. Such sources might be expected to
accept typical stories that went around about Diodorus without any
close analysis.



In the case of our earliest source, Stratonicus, there is, moreover,
once again evidence suggesting that Diodorus was not regarded as a
typical Pythagorean. In describing Diodorus' relationship to
Pythagoras, Stratonicus does not use a typical word for student or
disciple, but rather the same word (*pelates*) that Plato
used in the *Euthyphro* to describe the day-laborer who died at
the hands of Euthyphro's father. Diodorus is thus being
presented sarcastically as a hired hand in the Pythagorean tradition,
which is very much in accord with the later presentations of him as a
poor man's Pythagoras on the fringes of Pythagoreanism. Thus,
rather than accusing the sources of bias against Diodorus, it seems
better to accept their almost universal testimony that he was not a
typical *acusmatic* but rather a marginal figure, who used
Pythagoreanism in part to try to gain respectability for his own
eccentric lifestyle.



Individuals known as "Pythagorists," i.e. Pythagorizers,
are ridiculed by writers of Greek comedy, such as Alexis, Antiphanes,
Aristophon, and Cratinus the younger, in the middle and second half of
the fourth century (see Burkert 1972a, 198, n. 25 for the evidence and
200, n. 41 for the dating). The most important of the fragments of
these comedies that deal with the Pythagorists are collected by
Athenaeus (IV 160f ff) and Diogenes Laertius (VIII 37-38). The
term "Pythagorist" is usually negative in the comic
writers (Arnott 1996, 581-582) and picks out people who share
some of the same extreme ascetic lifestyle as Diodorus. A fragment of
Antiphanes describes someone as eating "nothing animate, as if
Pythagorizing" (Fr. 133 Kassel and Austin = Athenaeus IV 161a).
In *The Pythagorizing Woman*, Alexis presents the vegetarian
sacrificial feast that is customary for the Pythagoreans as including
dried figs, cheese and olive cakes, and reports that the Pythagorean
life entailed "scanty food, filth, cold, silence, sullenness,
and no baths" as well as drinking water instead of wine (Frs.
201-202 = Athenaeus IV 161c and III 122f).



A number of these characteristics can be connected to the
*acusmata* (Arnott 1996, 583), e.g., the lack of bathing may be
a joke based on the *acusma* that forbids the Pythagoreans from
using the public baths (Iamblichus, *VP* 83), Antiphanes (fr.
158) satirizes the *acusmata's* bizarre list of foods
that can be eaten (D.L. 8.19) by describing his Pythagoreans as
searching for sea orach, and the silence or sullenness ascribed to the
Pythagoreans in comedy accords not just with the *acusmata* but
with early testimony about the Pythagoreans in Isocrates
(*Busiris* 29) and Dicaearchus (Fr. 40 Mirhady). A fragment of
Aristophon's *Pythagorist* suggests that this ascetic
life was based on poverty rather than philosophical scruple and that,
if you put meat and fish in front of these Pythagorists, they would
gobble them down (Fr. 9 = Athenaeus IV 161e). In a fragment of Alexis,
after the speaker reports that the Pythagoreans eat nothing animate,
he is interrupted by someone who objects that "Epicharides eats
dogs, and he is a Pythagorean," to which the response is,
"yes, but he kills them first and so they are not still
animate" (Fr. 223 + Athenaeus 161b). Epicharides and some other
named figures may well be Athenians who are satirized by being
assigned a Pythagorean life (Athenaeus 2006, 272). Another fragment of
Aristophon's *Pythagorist* reports that the Pythagoreans
have a far different existence in the underworld than others, in that
they feast with Hades because of their piety, but this just occasions
the remark that Hades is an unpleasant god to enjoy the company of
such filthy wretches (Fr. 12 = Diogenes Laertius VIII 38).



Both Alexis (Fr. 223 = Athenaeus IV 161b) and Cratinus the younger
(Fr. 7 = Diogenes Laertius VIII 37) wrote plays entitled *The
People of Tarentum*, which, although they may not have been
primarily about Pythagoreans, featured depictions of them (Arnott
1996, 625-626). In this case, the Pythagoreans are again
satirized for their simple diet, bread and water (which is called
"prison fare"), and for drinking no wine. In these plays,
however, the Pythagoreans are also presented as feeding on
"subtle arguments" and "finely honed thoughts"
and as pestering others with them, in a way that is reminiscent of
Aristophanes' treatment of Socrates in the *Clouds*.



Given the fragmentary nature of the evidence, it is unclear whether
these ascetic Pythagoreans who engage in argument are the same as the
Pythagorists in the other comedies, who are characterized by their
filth and eccentric appearance. Certainly the latter are more
reminiscent of Diodorus of Aspendus, while the former might be closer
to what we know of someone like Cleinias. In the first half of the
third century, the poet Theocritus still preserves a memory of these
Pythagorists as "pale and without shoes" (XIV 5). The
scholiast to the passage testifies to the continuing controversy about
the Pythagorists by drawing a distinction between Pythagoreans who
give every attention to their body and Pythagorists who are filthy
(although another scholion reports that others say the opposite, see
Arnott 1996, 581). A passage in Iamblichus (*VP* 80) similarly
argues that the Pythagoreans were the true followers of Pythagoras,
while the Pythagorists just emulated them.



In recent scholarship, the tendency has been to regard Diodorus and
the Pythagorists as legitimate Pythagoreans of the acusmatic stamp,
whose eccentricities are perhaps a little exaggerated in comedy. The
extensive evidence from antiquity which argues that they were not true
Pythagoreans is interpreted as bias on the part of conservative
Pythagoreans of the *hyper-mathematici* sort, such as
Aristoxenus, who wanted to disassociate themselves and Pythagoreanism
in general from such strange people. This is a possible interpretation
of the evidence, but, as the evidence for Diodorus shows, it is also
quite possible that Diodorus and the more extreme Pythagorists
depicted in comedy were in fact people with whom few Pythagoreans
either of the *mathematici* or the *acusmatici*
wanted to associate themselves. Many religious movements have a
radical fringe, and there is little reason to think that
Pythagoreanism should differ in this regard. In connection with his
thesis that the  *acusmata* were a literary phenomenon and that
no one lived a life in accordance with them Zhmud argues that the
Pythagorists of comedy are a creation of the comic stage and do not
provide evidence for Pythagoreans living a life governed by
*acusmata* (Zhmud 2012a, 175-183). It is true that many
of the features of the Pythagorists are shared with Socrates as
presented in the *Clouds* (subtle arguments, plain food, filthy
clothes). Zhmud suggests that vegetarianism was added to this stock
picture of the philosopher to give a Pythagorean color and that this
vegetarianism was derived solely from the eccentric figure of Diodorus
of Aspendus. However, as noted above there are more connections to the
*acusmata* than just vegetarianism and it is hard to believe
that the repeated jokes at the expense of those living a Pythagorean
life had no correlate in reality other than Diodorus. 



Perhaps the best way to evaluate the complicated evidence for
fourth-century Pythagoreanism is to conclude that there were three
main groups, each of which admitted some variation. There were
*mathematici* such as Archytas who did serious research
in the mathematical disciplines and natural philosophy but who also
lived an ascetic life that emphasized self-control and avoidance of
bodily pleasure. Other Pythagoreans such as Cleinias or Xenophilus may
have done no work in the sciences but lived a Pythagorean life, which
was similar to that of Archytas and followed principles similar to
those set out in Aristoxenus' *Pythagorean Precepts*.
They may have observed some mild dietary restrictions and may be
similar to the figures satirized in *The Men of Tarentum* as
eating a simple diet but still engaged in subtle arguments. There was
probably a continuum of people in this category with some following
more or different sets of the *acusmata* than others. Finally
there are the Pythagorean hippies such as Diodorus and the
Pythagorists, who ostentatiously live a life in accord with some of
the *acusmata*, but who take such an extreme interpretation of
them as to be regarded as eccentrics by most Pythagoreans.



Diogenes Laertius reports, evidently on the authority of Aristoxenus,
that the last Pythagoreans were Xenophilus from the Thracian
Chalcidice (Aristoxenus' teacher), and four Pythagoreans from
Phlius: Phanton, Echecrates, Diocles and Polymnastus. These
Pythagoreans are further identified as the pupils of Philolaus and
Eurytus. Little more is known of Xenophilus beyond his living for more
than 105 years (DK I 442-443). The Pythagoreans from Phlius are
just names except Echecrates (DK I 443), to whom Phaedo narrates,
evidently in Phlius, the events of Socrates' last day in
Plato's *Phaedo*. Socrates' interlocutors in the
*Phaedo*, Simmias and Cebes, are often regarded as
Pythagoreans, because they are said to have been pupils of Philolaus
when he was in Thebes. They are also shown to be pupils of Socrates,
however, and it is unclear that their connection to Philolaus was any
closer than their connection to Socrates. They are not listed in
Iamblichus' catalogue as Pythagoreans; Diogenes Laertius
includes them with other followers of Socrates (II 124-125).
Echecrates might have been born around 420 and thus be a young man at
the dramatic date of the *Phaedo*. Aristoxenus' assertion
that these were the last of the Pythagoreans would then suggest that
Pythagoreanism died out around 350, when Echecrates was an old
man.



Riedweg says that this claim is "demonstrably untrue"
pointing to a Pythagorean, Lycon, who criticized Aristotle's
supposed extravagant way of life and to the Pythagorists discussed
above (2005, 106). This seems slender evidence upon which to be so
critical of Aristoxenus. Virtually nothing is known of Lycon, and
Aristocles (1st-2nd c. CE), who recounts the criticism of Aristotle,
says that Lycon "called himself a Pythagorean," thus
expressing some sort of reservation about his credentials (DK I
445-446). Aristoxenus' assertion is probably to be
understood as a general claim that, with the deaths of the
Pythagoreans from Phlius around the middle of the fourth century,
Pythagoreanism as an active movement was dead. This would be
compatible with a few individuals still claiming to be Pythagoreans
after 350.



This is not inconsistent with the existence of a few isolated
individuals, who still claim to be Pythagoreans. Certainly, from the
evidence available to modern scholars, Aristoxenus' claim is
largely true. From about 350 BCE until about 100 BCE, there is a
radical drop in evidence for individuals who call themselves
Pythagoreans. Iamblichus (*In Nic.* 116.1-7) appears to
date the Pythagoreans Myonides and Euphranor, who worked on the
mathematics of means, after the time of Eratosthenes (285-194
BCE) and hence to the second century BCE or later (Burkert 1972a,
442), but Iamblichus' history of the means is very confused and
they might belong to the rise of Neopythagoreanism in the first
centuries BCE and CE. Kahn (2001, 83) sees a hint of Pythagorean cult
activity in the spurious *Pythagorean Memoirs*, which must date
sometime before the first half of the first century BCE, when they are
quoted by Alexander Polyhistor (see section 4.2 below). A few other
Pythagorean pseudepigrapha appear in the period (see further below,
sect. 4.2), although it is unclear what sort of Pythagorean community,
if any, was associated with them. Pythagoreanism is not completely
dead between 350 and 100 (see further below, sect. 3.5), but few
individual Pythagoreans or organized groups of Pythagoreans can be
identified in this period.


### 3.6 Timaeus, Ocellus, Hicetas and Ecphantus



The names Timaeus of Locri and Ocellus of Lucania are famous as the
authors of the two most influential Pythagorean pseudepigrapha (see
below, sect. 4.2). In his catalogue of Pythagoreans, Iamblichus lists
an Ocellus under Lucania and two men named Timaeus, neither under
Locri. The later forgery of works attributed to Timaeus and Ocellus
does not of course mean that Pythagoreans of these names did not
exist, and it is possible that the Timaeus of Locri who is the main
speaker in Plato's *Timaeus* was an historical Timaeus
(some have thought Plato uses him as a mask for Archytas, however). If
they really did exist, however, nothing is known about them, since all
other reports in the ancient tradition are likely to be based on
Plato's *Timaeus* or the spurious works in their
name.



Some scholars have argued that Hicetas and Ecphantus, both of
Syracuse, were not historical figures at all but rather characters in
dialogues written by Heraclides of Pontus, a fourth-century member of
the Academy. By a misunderstanding, they came to be treated as
historical Pythagoreans in the doxographical tradition (see Guthrie
1962, 323 ff. for references). This theory arose because both Hicetas
and Ecphantus are said to have made the earth rotate on its axis,
while the heavens remained fixed, in order to explain astronomical
phenomena, and, in one report, Heraclides is paired with Ecphantus as
having adopted this view (Aetius III 13.3 =DK I 442.23). In addition
Ecphantus is assigned a form of atomism (DK I 442.7 ff.) similar to
that assigned to Heraclides (Fr. 118-121 Wehrli). It is not
uncommon in the doxographical tradition for a report of the form
"x and y believe z" to mean that "y, as reported by
x, believes z," so it is suggested that in this case
"Heraclides and Ecphantus" means "Ecphantus as
presented by Heraclides." There is a serious problem with this
ingenious theory. The doxographical reports about Hicetas and
Ecphantus ultimately rely on Theophrastus (Cicero mentions
Theophrastus by name at DK I 441.27), and it is implausible that
Theophrastus would treat characters invented by his older
contemporary, Heraclides, as historical figures. Theophrastus did
accept the Academic glorification of Pythagoras (see on
Neopythagoreanism below, sect. 4.1), but this provides no grounds for
supposing that he accepted a character in a dialogue as a historical
person (*pace* Burkert 1972a, 341).



The testimonia for Hicetas are meager and contradictory (DK I
441-442). He appears to have argued that the celestial phenomena
are best explained by assuming that all heavenly bodies are stationary
and that the apparent movement of the stars and planets is the result
of the earth's rotation around its own axis. He may also have
followed Philolaus in positing a counter-earth, opposite the earth on
the other side of a central fire, although, if he did, it is unclear
how he would have explained why it and the central fire are not
visible from the rotating earth. In Philolaus' system the
central fire remains invisible because the earth orbits the central
fire as it rotates on its axis, thus keeping one side of the earth
always turned away from the central fire. A little more is known about
Ecphantus (DK I 442). He too is said to have believed that the earth
moved, not by changing its location (as Philolaus proposed, in making
the earth and counter-earth revolve around the central fire: see
Section 4.2 of the entry on
 Philolaus),
 but by rotating on its axis.



Copernicus was inspired by these testimonia about Hicetas and
Ecphantus, as well as those about Philolaus, to consider the motion of
the earth (see below, sect. 5.2). Ecphantus developed his own original
form of atomism. He is best understood as reacting to and developing
the views of Democritus. He agreed with Democritus 1) "that
human beings do not grasp true knowledge of the things that are, but
define them as they believe them to be" (DK I 442.7-8; cf.
Democritus Frs. 6-10) and 2) that all sensible things arise from
indivisible first bodies and void. He differs from Democritus,
however, in supposing that atoms are limited rather than unlimited in
number and that there is just one cosmos rather than many. As in
Democritus, atoms differ in shape and size, but Ecphantus adds power
(*dynamis*) as a third distinguishing factor. He explains
atomic motion not just in terms of weight and external blows, as the
atomists did, but also by a divine power, which he called mind or
soul, so that "the cosmos was composed of atoms but organized by
providence" (DK I 442.21-22). It is because of this divine
power that the cosmos is spherical in shape. This unique spherical
cosmos is reminiscent of Plato's *Timaeus*, but the rest
of Ecphantus' system differs enough from Plato that there is no
question of its being a forgery based on the *Timaeus*. One
testimony says that he was the first to make Pythagorean monads
corporeal, thus differing from the fifth-century Pythagoreans
described by Aristotle, who do not seem to have addressed the question
of whether numbers were physical entities or not.



It is difficult to be sure of the date of either Hicetas or Ecphantus.
Since, however, both seem to be influenced by Philolaus' idea of
a moving earth and since Ecphantus appears to be developing the
atomism of Democritus, it is usually assumed that they belong to the
first half of the fourth century (Guthrie 1962, 325-329; Zhmud
2012a, 130). Hicetas does not appear in Iamblichus' catalogue.
There is an Ecphantus in the catalogue, but he is listed under Croton
rather than Syracuse, so it cannot be certain whether he is the
Ecphantus described in the doxography.


### 3.7 Plato and Pythagoreanism



There is currently a very wide range of opinions about the
relationship of Plato to Pythagoreanism. Many scholars both ancient
and modern have thought that Plato was very closely tied to
Pythagoreanism. In the biography of Pythagoras read by Photius in the
9th century CE (*Bibl.* 249) Plato is presented as a member of
the Pythagorean school. He is the pupil of Archytas and the ninth
successor to Pythagoras himself. If this were true then Plato would
certainly be the most illustrious early Pythagorean after Pythagoras
himself. Some modern scholars, while not going this far, have seen the
connections between Plato and the Pythagoreans to be very close
indeed. Thus, A. E. Taylor in his great commentary on the
*Timaeus* says that his main thesis is that "the teaching
of Timaeus [in Plato's *Timaeus*] can be shown to be in
detail exactly what we should expect from an fifth-century Italian
Pythagorean" (1928, 11). Guthrie in his famous history of
ancient philosophy commented that Pythagorean and Platonic philosophy
were so close that it is difficult to separate them (1975, 35).
Recently it has been argued that Plato was so steeped in
Pythagoreanism that he structured his dialogues by counting numbers of
lines and placing important passages at points in the dialogue that
correspond to important ratios in Pythagorean harmonic theory
(Kennedy, 2010 and 2011). Thus, the vision of the form of beauty
appears 3/4 of the way through the *Symposium* by line count
and the ratio 3 : 4 corresponds to the central musical interval of the
fourth. There are, however, serious questions about the methodology
used (Gregory 2012) and it is a serious problem both that no one in
the ancient world reports that Plato used such a practice and that the
middle of the dialogue, which corresponds to the most concordant
musical interval, the octave (2:1), does not usually contain the most
philosophically important content. Another approach sees Plato as
engaged with and heavily influenced by Pythagorean ideas in passages
where the Pythagoreans are not specifically mentioned in dialogues
such as the *Cratylus* (401b11-d7) and *Phaedo*
(101b10-104c9) (Horky 2013). The problem is that in contrast to
the *Philebus*, where the connection to Philolaus is clear (see
below), the connections to the Pythagoreans in these passages are too
indirect or general (e.g., the concepts odd and even and the number 3
in the *Phaedo* passage are not unique to the Pythagoreans) to
be very convincing and partly depend on the doubtful assumption that
Epicharmus was a Pythagorean (see section 3.4 above). The central text
for many of those who see Plato as closely tied to Pythagoreanism is
Aristotle's comment in *Metaphysics* 1.6 that Plato
"followed these men (i.e. the Pythagoreans according to these
scholars) in most respects" (987a29-31). In contrast to
these attempts to connect Plato closely to Pythagoreanism, most recent
Platonic scholars seem to think Pythagoreanism of little importance
for Plato. Thus two prominent handbooks to Plato's thought
(Kraut 1992; Benson 2006) and another book of essays devoted
specifically to the *Timaeus,* (Mohr and Sattler 2010) hardly
mention the Pythagoreans at all. 



In recent studies of the topic that lie somewhere between these
extremes, one approach is to argue that there is clear Pythagorean
influence on Plato but that its scope is much more limited than often
assumed (Huffman 2013). Plato explicitly mentions Pythagoras and the
Pythagoreans only one time each in the dialogues and this provides
*prima facie* evidence that Pythagorean influence was not
extensive. Moreover, at *Metaphysics* 987a29-31 the
"these men" that Aristole says Plato follows in most
respects may not be the Pythagoreans but the Presocratics in general.
Aristotle's presentation as a whole mainly attests to
Pythagorean influence only on Plato's late theory of principles.
It is often assumed that Plato owes his mathematical conception of the
cosmos and his belief in the immortality and transmigration of the
soul to Pythagoreanism (Kahn 2001, 3-4). However, the role of
Pythagoreanism in Greek mathematics has been overstated and while
Plato had contacts with mathematicians who were Pythagoreans like
Archytas, the most prominent mathematicians in the dialogues,
Theodorus and Theaetetus, are not Pythagoreans. It is thus a serious
mistake to assume that any mention of mathematics in Plato suggests
Pythagorean influence. The same is true of the immortality and
transmigration of the soul in Plato, which are often assumed to be
derived from Pythagoreanism. Some have also thought that Platonic
myths and especially the myth at the end of the *Phaedo* draw
heavily on Pythagoreanism (Kingsley 1995, 79-171). However, most
of the contexts in which Plato mentions the immortality of the soul
including the Platonic myths, suggest that he is thinking of mystery
cults and the Orphics rather than the Pythagoreans (Huffman 2013,
243-254). On the other hand, in the *Philebus* (16c-17a)
Plato gives clear acknowledgement of the debt he owes to men before
his time who posit limit and unlimited as basic principles. The
fragments of Philolaus and Aristotle's reports on Pythagoreanism
make clear that this is a reference to Philolaus and the Pythagoreans.
The principles of limit and unlimited are clearly connected to
Plato's one and indefinite dyad and it is precisely these
principles of Plato that Aristotle connects most closely to
Pythagoreanism (*Metaph.* 987b25-32). Thus Plato's
evidence coheres with Aristotle's to suggest that Pythagoreanism
exerted considerable influence on Plato's late theory of
principles. It is also true that specific aspects of Plato's
mathematical view of the world are owed to the Pythagoreans, e.g., the
world soul in the *Timaeus* is constructed according to the
diatonic scale that is prominent in Philolaus (Fr. 6a). However, most
of the *Timaeus* is not derived from Pythagoreanism and some of
it in fact conflicits with Pythagoreanism (e.g., Archytas famously
argued that the universe was unlimited while Plato's in
limited). The same is true for Plato as a whole. Isolated ideas such
as the one and the dyad and the structure of the world soul show heavy
Pythagorean influence, but there is no evidence that Pythagoreanism
played a central role in the development of the core of Plato's
philosophy (e.g., the theory of forms).



A second approach is to argue that, while it is true that not all
mentions of mathematics or all mentions of the transmigration of the
soul derive from Pythagoreanism, nonetheless a central system of value
that appears early in Plato's work and persists to the end is
derived from Pythagoreanism (Palmer 2014). Already in the
*Gorgias* Plato argues that principles of order and correctness
which are found in the cosmos and explain its goodness also govern
human relations. Socrates here puts forth a much more definite
conception of the good than in earlier dialogues. His complaint that
Callicles pays no attention to the role played by orderliness and
self-control and neglects geometrical equality (507e6-508a8)
mirrors the emphasis on organization and calculation in contemporary
Pythagorean texts such as Archytas Fr. 3 and Aristoxenus'
*Pythagorean Precepts* Fr. 35. It thus appears that
"Socrates'" new insight into the good in
*Gorgias* derives from Plato's contact with the
Pythagoreans after the death of the historical Socrates. Plato never
abandons this Pythagorean conception of value and it can be traced
through the *Phaedo* and *Republic* to late dialogues
such as the *Timaeus*, where the cosmos is embued with
principles of mathematical order, and *Philebus*, where the
highest value is assigned to measure (66a). The question is whether
this emphasis on measure and order is uniquely Pythagorean in
origin.


## 4. Neopythagoreanism



Neopythagoreanism is characterized by the tendency to see Pythagoras
as the central and original figure in the development of Greek
philosophy, to whom, according to some authors (e.g. Iamblichus,
*VP* 1), a divine revelation had been given. This revelation
was often seen as having close affinities to the wisdom of earlier
non-Greeks such as the Hebrews, the Magi and the Egyptians. Because of
the belief in the centrality of the philosophy of Pythagoras, later
philosophy was regarded as simply an elaboration of the revelation
expounded by Pythagoras; it thus became the fashion to father the
views of later philosophers, particularly Plato, back onto Pythagoras.
Neopythagoreans typically emphasize the role of number in the cosmos
and treat the One and Indefinite Dyad as ultimate principles going
back to Pythagoras, although these principles in fact originate with
Plato. The origins of Neopythagoreanism are probably to be found
already in Plato's school, the Academy, in the second half of
the fourth century BCE. There is evidence that Plato's
successors, Speusippus and Xenocrates, both presented Academic
speculations arising in part from Plato's later metaphysics as
the work of Pythagoras, who lived some 150 years earlier. After a
decline in interest in Pythagoreanism for a couple of centuries,
Neopythagoreanism emerged again and developed further starting in the
first century BCE and extending throughout the rest of antiquity and
into the middle ages and Renaissance. During this entire period, it is
the Neopythagorean construct of Pythagoras that dominates, a construct
that has only limited contact with early Pythagoreanism; there is
little interest in an historically accurate presentation of Pythagoras
and his philosophy. In reading the following account of
Neopythagoreanism, it may be helpful to refer to the
 Chronological Chart of Sources for Pythagoras,
 in the entry on Pythagoras.


### 4.1 Origins in the Early Academy: Speusippus, Xenocrates and Heraclides in Contrast to Aristotle and the Peripatetics



The evidence for Speusippus, Plato's successor as head of the
Academy, is fragmentary and second hand, so that certainty in
interpretation is hardly possible. In one passage, however, he assigns
not just Plato's principles, the one and the dyad, to "the
ancients," who in context seem likely to be the Pythagoreans,
but also a development of the Platonic system according to which the
one was regarded as beyond being (Fr. 48 Taran; see Burkert 1972a,
63-64; Dillon 2003, 56-57). Some scholars reject this
widely held view on the grounds that this fragment of Speusippus is
spurious (Zhmud 2012a, 424--425, who cites other scholars; Taran
1981, 350ff.; for a response see Dillon 2014, 251) and if this were
true it would seriously weaken the case for supposing that
Neopythagoreanism began already in the Academy. Speusippus also wrote
a book *On Pythagorean Numbers* (Fr. 28 Taran), which builds on
ideas attested for the early Pythagoreans (e.g., ten as the perfect
number, although Zhmud regards the perfection of ten as a Platonic
rather than a Pythagorean doctrine 2012a, 404-09). We cannot be
sure, however, either that the title goes back to Speusippus or that
he assigned all ideas in it to the Pythagoreans. Aristotle twice cites
agreement between Speusippus and the Pythagoreans (*Metaph*.
1072b30 ff.; *EN* 1096b5-8), which might suggest that
Speusippus himself had identified the Pythagoreans as his predecessors
in these areas. Speusippus and Xenocrates denied that the creation of
the universe in Plato's *Timaeus* should be understood
literally; when the view that the cosmos was only created in thought
and not in time is assigned to Pythagoras in the later doxography
(Aetius II 4.1 -- Diels 1958, 330), it certainly looks as if
an idea which had its origin in the interpretation of Plato's
*Timaeus* in the Academy is being assigned back to Pythagoras
(Burkert 1972a, 71). The evidence is not sufficient to conclude that
Speusippus routinely assigned Platonic and Academic ideas to the
Pythagoreans (Taran 1981, 109), but there is enough evidence to
suggest that he did so in some cases.



Speusippus' successor as head of the Academy, Xenocrates, may
actually have followed some version of the Pythagorean way of life,
e.g., he was apparently a vegetarian, refused to give oaths, was
protective of animals and followed a highly structured daily regimen,
setting aside time for silence (Dillon 2003, 94-95 and 2014,
254-257; Burkert, however, argues that he rejected
metempsychosis [1972a, 124]). He wrote a book entitled *Things
Pythagorean*, the contents of which are unfortunately unknown
(Diogenes Laertius IV 13). In the extant fragments of his writings, he
refers to Pythagoras by name once, reporting that "he discovered
that the musical intervals too did not arise apart from number"
(Fr. 9 Heinze). Several doctrines of Xenocrates are also assigned to
Pythagoras in the doxographical tradition, e.g., the definition of the
soul as "a number moving itself," which Xenocrates clearly
developed on the basis of Plato's *Timaeus* (Plutarch,
*On the Generation of the Soul* 1012d; Aetius IV
2.3-4). This suggests that Xenocrates, like Speusippus, may have
assigned his own teachings back to Pythagoras or at least treated
Pythagoras as his precursor in such as way that it was easy for others
to do so (Burkert 1972a, 64-65; Dillon 2003, 153-154;
Zhmud [2012a, 55 and 426-427] disputes this interpretation).



Yet another member of the early Academy, Heraclides of Pontus
(Gottschalk 1980), in a series of influential dialogues, further
developed the presentation of Pythagoras as the founder of philosophy.
In the dialogue, *On the Woman Who Stopped Breathing*,
Pythagoras is presented as the inventor of the word
"philosophy" (Frs. 87-88 Wehrli = Diogenes Laertius
Proem 12 and Cicero, *Tusc*. V 3.8). Although some scholars
have tried to find a kernel of truth in the story (e.g., Riedweg 2005,
90 ff., for a response see Huffman 2008b), its definition of the
philosopher as one who seeks wisdom rather than possessing it is
regarded by many scholars as a Socratic/Platonic formulation, which
Heraclides, in his dialogue, is assigning to Pythagoras as part of a
literary fiction (Burkert 1960 and 1972a, 65). Heraclides also assigns
to Pythagoras a definition of happiness as "the knowledge of the
perfection of the numbers of the soul" (Fr. 44 Wehrli), in which
again the Platonic account of the numerical structure of the soul in
the *Timaeus* appears to be fathered on Pythagoras. Other
fragments show Heraclides' further fascination with the
Pythagoreans. He developed what would become one of the canonical
accounts of Pythagoras' previous incarnations (Fr. 89 Wehrli).
Perhaps on the basis of the Pythagorean Philolaus' astronomical
system, he developed the astronomical theory, later to be championed
by Copernicus, according to which the apparent daily motion of the sun
and stars was to be explained by the rotation of the earth (Frs.
104-108; see on Hicetas and Ecphantus above, sect. 3.6). For a
different view of Heraclides' relation to the Pythagoreans see
Zhmud 2012a, 427-432.



In contrast to the fascination with and glorification of Pythagoras in
the Academy after Plato's death, Aristotle did not treat
Pythagoras as part of the philosophical tradition at all. In the
surveys of his predecessors in his extant works, Aristotle does not
include Pythagoras himself and he evidently presented him in his lost
special treatises on the Pythagoreans only as a wonder-worker and
founder of a way of life. While Aristotle did acknowledge close
connections between Plato's late theory of principles (One and
Indefinite Dyad) and fifth-century Pythagoreans, he also sharply
distinguished Plato from the Pythagoreans on a series of important
points (*Metaph*. 987b23 ff.), perhaps in response to the
Academy's tendency to assign Platonic doctrines to Pythagoras.
Aristotle's students Eudemus, in his histories of arithmetic,
geometry and astronomy and Meno, in his history of medicine, follow
Aristotle's practice of not mentioning Pythagoras himself,
referring to individual Pythagoreans such as Philolaus or to the
Pythagoreans as a group. Eudemus assigns the Pythagoreans a number of
important contributions to the sciences but does not give them the
decisive or foundational role found in the Neopythagorean tradition.
Aristotle's pupils Dicaearchus (Porphyry, *VP* 19) and
Aristoxenus do mention Pythagoras but this is because they are
focusing on the the Pythagorean way of life and the history of the
Pythagorean communities. Neither assign to Pythagoras or the
Pythagoreans the characteristics of Neopythagoreanism. Aristoxenus is
one of the most important and extensive sources for Pythagoreanism
(see 3.5 above). He presents Pythagoras and the Pythagoreans in a
positive manner but avoids the hagiography and extravagant claims of
the later Neopythagorean tradition. The standard view is that he tries
to emphasize the rational as opposed to the religious side of
Pythagoras (e.g. Burkert 1972a, 200-205), but several fragments
do highlight the religious aspect of Pythagoras' work, assigning
him the doctrine of metempsychosis (fr. 12) and associating him with
the Chaldaean Zaratas (Fr. 13) and the Delphic oracle (Fr. 15). It is
only by rejecting the authenticity of such fragments (as does Zhmud
2012a, 88-91) that Aristoxenus' account is purged of
religious elements. Dicaearchus' account of Pythagoreas is also
usually viewed as positive. He is supposed to have presented
Pythagoras as the model of the practical life as opposed to the
contemplative life (Jaeger 1948, 456; Kahn 2001, 68). However,
Dicaearchus presents a very sarcastic account of Pythagoras'
rebirths according to which he was reborn as the beautiful prostitute
Alco (Fr. 42) and careful reading of his other accounts of Pythagoras
suggests that he may have presented him as a charismatic charlatan who
bewitched his hearers (Fr. 42) and was seen as a threat to the
established laws of the state and hence was refused entrance by such
city-states as Locri (Fr. 41a). Thus, Aristoxenus and Dicaearchus were
as divided in their interpretation of Pythagoras as were Heraclitus
and Empedocles in earlier centuries. The Peripatetic tradition as a
whole is in strong contrast, then, with the Academy insofar as it
emphasizes Pythagoreans rather than Pythagoras himself. When
Pythagoras is mentioned, it is mostly in connection with the way of
life, and interpretations range from positive to strongly satirical
but in either case avoid the hagiography of the Neopythagorean
tradition. 



It is then one of the great paradoxes of the ancient Pythagorean
tradition that Aristotle's successor, Theophrastus, evidently
accepted the Academic lionization of Pythagoras, and identifies
Plato's one and the indefinite dyad as belonging to the
Pythagoreans (*Metaph*. 11a27 ff.), although Aristotle is
emphatic that this pair of principles in fact belong to Plato
(*Metaph*. 987b25-27). Since Theophrastus' work,
*Tenets in Natural Philosophy*, was the basis of the later
doxographical tradition, it may be that Theophrastus is responsible
for the Neopythagorean Pythagoras of the Academy dominating the later
doxography, the Pythagoras who originated the one and the indefinite
dyad (Aetius I 3. 8), but it may also be that the Pythagorean
sections of the doxography were rewritten in the first century BCE,
under the influence of the Neopythagoreanism of that period (Burkert
1972a, 62; Zhmud 2012a, 455).



The standard view has thus been that the Academy was the origin of
Neopythagoreanism with its glorification of Pythagoras and its
tendency to assign mature Platonic views back to Pythagoras and the
Pythagoreans. Aristotle and the Peripatetics on the other hand
diminish the role of Pythagoras himself and, while noting connections
between Plato and the Pythagoreans, carefully distinguish Pythagorean
philosphy from Platonism. Zhmud has recently put forth a challenge to
this view arguing the situation is almost the reverse: the Academy in
general regards Pythagoras and Pythagoreans favorably but does not
assign mature Platonic views to them, it is rather Aristotle who ties
Plato closely to the Pythagoreans (2012a, 415-456).


### 4.2 The Pythagorean Pseudepigrapha



Although the origins of Neopythagoreanism are thus found in the fourth
century BCE, the figures more typically labeled Neopythagoreans belong
to the upsurge in interest in Pythagoreanism that begins in the first
century BCE and continues through the rest of antiquity. Before
turning to these Neopythagoreans, it is important to discuss another
aspect of the later Pythagorean tradition, the Pythagorean
pseudepigrapha. Many more writings forged in the name of Pythagoras
and other Pythagoreans have survived than genuine writings. Most of
the pseudepigrapha themselves only survive in excerpts quoted by
anthologists such as John of Stobi, who created a collection of Greek
texts for the edification of his son in early fifth century CE. The
modern edition of these Pythagorean pseudepigrapha by Thesleff (1965)
runs to some 245 pages.



There is much uncertainly as to when, where, why and by whom these
works were created. No one answer to these questions will fit all of
the treatises. Most scholars (e.g., Burkert 1972b, 40-44;
Centrone 1990, 30-34, 41-44 and 1994) have chosen Rome or
Alexandria between 150 BCE and 100 CE as the most likely time and
place for these compositions, since there was a strong resurgence of
interest in Pythagoreanism in these places at these times (see below).
Thesleff's view that the majority were composed in the third
century BCE in southern Italy (1961 and 1972, 59) has found less
favor. Centrone argues convincingly that a central core of the
pseudepigrapha were forged in the first centuries BCE and CE in
Alexandria, because of their close connection to Eudorus and Philo,
who worked in Alexandria in that period (Centrone 2014a). For an
overview of the Pythagorean pseudepigrapha see Centrone 2014a and
Moraux 1984, 605-683.



A number of motives probably led to the forgeries. The existence of
avid collectors of Pythagorean books such as Juba, King of Mauretania
(see below), and the scarcity of authentic Pythagorean texts will have
led to forgeries to sell for profit to the collectors. Other short
letters or treatises may have originated as exercises for students in
the rhetorical schools (e.g., the assignment might have been to write
the letter that Archytas wrote to Dionysius II of Syracuse asking that
Plato be freed; see Diogenes Laertius III 21-22). The contents
of the treatises suggest, however, that the primary motivation was to
provide the Pythagorean texts to support the Neopythagorean position,
first adumbrated in the early Academy, that Pythagoras was the source
of all that is true in the Greek philosophical tradition. The
pseudepigrapha show the Pythagoreans anticipating the most
characteristic ideas of Plato and Aristotle. Most of the treatises are
composed in the Doric dialect (spoken in Greek S. Italy) but, apart
from that concession to verisimilitude, there is little other attempt
to make them appear to be archaic documents that anticipated Plato and
Aristotle. Instead, Plato's and Aristotle's philosophical
positions are stated in a bald fashion using the exact Platonic and
Aristotelian terminology. In many cases, however, this glorification
of Pythagoras may not have been the final goal. The ancient authority
of Pythagoras was sometimes used to argue for a specific
interpretation of Plato, often an interpretation that showed Plato as
having anticipated and having responded to criticisms of Aristotle.
For example, in defense of the interpretation of Plato's
*Timaeus*, which defends Plato against Aristotle's
criticisms by claiming that the creation of the world in the
*Timaeus* is metaphorical, a Platonist could point to the
forged treatise of Timaeus of Locri which does present the generation
as metaphorical but which can also be regarded as Plato's
source. These pseudo-Pythagorean treatises are adopting the same
strategy as Eudorus of Alexandria and thus may be more important for
debates within later Platonism than for Pythagoreanism *per se*
(Bonazzi 2013). 



One plausible explanation of the sudden proliferation of Pythagorean
pseudepigrapha in the first century BCE and first century CE is the
reappearance of Aristotle's esoteric writings in the middle of
the first century BCE (Kalligas 2004, 39-42). In those treatises
Plato is presented as adopting a pair of principles, the one and the
indefinite dyad, which are not obvious in the dialogues, but which
Aristotle compares to the Pythagorean principles limit and unlimited
(e.g., *Metaph.* 987b19-988a1). Aristotle can be read,
although probably incorrectly, as virtually identifying Platonism and
Pythagoreanism in these passages. Thus, Pythagorean enthusiasts may
have felt emboldened by this reading of Aristotle to create the
supposed original texts upon which Plato drew. They may also have
found support for this in Plato's making the south-Italian
Timaeus his spokesman in the dialogue of the same name. It is thus not
surprising that the most famous of the pseudepigrapha is the treatise
supposedly written by this Timaeus of Locri (Marg 1972), which has
survived complete and which is clearly intended to represent the
original document on which Plato drew, although it, in fact, also
responds to criticisms made of Plato's dialogue in the first
couple of centuries after it was written (Ryle 1965, 176-178).
The treatise of Timaeus of Locri is first mentioned by Nicomachus in
the second century CE (*Handbook* 11) and is thus commonly
dated to the first century CE. Another complete short treatise (13
pages in Thesleff) is *On the Nature of the Universe*
supposedly by the Pythagorean Ocellus (Harder 1966), which has
passages that are almost identical to passages in Aristotle's
*On Generation and Corruption*. Since Ocellus' work is
first mentioned by the Roman polymath, Varro, scholars have dated it
to the first half of the first century BCE. Although Plato was in
general more closely associated with the Pythagorean tradition than
Aristotle, a significant number of Pythagorean pseudepigrapha follow
'Ocellus' in drawing on Aristotle (see Karamanolis 2006,
133-135).



It is likely that in some cases letters were forged in order to
authenticate these forged treatises. Thus a correspondence between
Plato and Archytas dealing with the acquisition of the writings of
Ocellus (Diogenes Laertius VIII 80-81) may be intended to
validate the forgery in Ocellus' name (Harder 1966, 39ff). A
letter from Lysis to Hipparchus (Thesleff 1965, 111-114), which
enjoyed considerable fame in the later tradition and is quoted by
Copernicus, urges that the master's doctrines not be presented
in public to the uninitiated and recounts Pythagoras'
daughter's preservation of his "notebooks"
(*hypomnemata*) in secrecy, although she could have sold
them for much money (see Riedweg 2005, 120-121). Burkert (1961,
17-28) has argued that this letter was forged to authenticate
the "Pythagorean Notes" from which Alexander Polyhistor
(1st century BCE) derived his influential account of Pythagoreanism
(Diogenes Laertius VIII 24-36 -- see the end of this
section and for Alexander see section 4.5 below). While some of
Pythagoras' teachings were undoubtedly secret, many were not,
and the claim of secrecy in the letter of Lysis is used to explain
both the previous lack of early Pythagorean documents and the recent
"discovery" of what are in reality forged documents, such
as the notebooks.



There are fewer forged treatises in Pythagoras' name than in the
name of other Pythagoreans and they are a very varied group suggesting
different origins. Callimachus, in the third century BCE, knew of a
spurious astronomical work circulating in Pythagoras' name
(Diogenes Laertius IX 23) and there may have been a similar work
forged in the second century (Burkert 1961, 28-42). A group of
three books, *On Education*, *On Statesmanship* and
*On Nature*, were forged in Pythagoras' name sometime
before the second century BCE (Diogenes Laertius VIII 6 and 9; Burkert
1972a, 225). Heraclides Lembus, in the second century BCE, knew of at
least six other works in Pythagoras' name, all of which must
have been spurious, including a *Sacred Discourse* (Diogenes
Laertius VIII 7). The thesis that the historical Pythagoras wrote a
*Sacred Discourse* should be rejected (Burkert 1972a, 219).
There was also a spurious treatise on the magical properties of plants
and the *Golden Verses*, which are discussed further below
(sect. 4.5). On the spurious treatises assigned to Pythagoras see
Centrone 2014a, 316-318.



Archytas
 appears to have been the most popular name in which to forge
treatises. Some 45 pages are devoted to pseudo-Archytan treatises in
Thesleff's collection as compared to 30 pages for Pythagoras.
The most famous of the pseudo-Archytan texts is *The Whole System
of Categories*, which, along with *On Opposites*,
represents the attempt to claim Aristotle's system of categories
for the Pythagoreans. The pseudo-Archytan works on categories are very
frequently cited by the commentators on Aristotle's
*Categories* (e.g., Simplicius and Syrianus) and were regarded
as authentic by them, but in fact include modifications made to
Aristotle's theory in the first century BCE and probably were
composed in that century (Szlezak 1972). Another treatise, *On
Principles*, is full of Aristotelian terminology such as
"form," "substance," and "what
underlies"; *On Intelligence and Perception* contains a
paraphrase of the divided line passage in Plato's
*Republic*. There are also a series of pseudepigrapha on ethics
by Archytas and other authors (Centrone 1990). Philolaus, the third
most famous Pythagorean after Pythagoras and Archytas, also turns up
as the author of several spurious treatises, but a number of the
forgeries were in the names of obscure or otherwise unknown
Pythagoreans. Thus, Callikratidas and Metopos are presented as
anticipating Plato's doctrine of the tripartite soul and as
using Plato's exact language to articulate it (Thesleff 1965,
103.5 and 118.1-4). Although there are indications that some
ancient scholars had doubts about the authenticity of the
pseudo-Pythagorean texts, for the most part they succeeded in their
purpose all too well and were accepted as genuine texts on which Plato
and Aristotle drew.



Although the pseudepigrapha are too varied to admit of one origin,
Centrone has recently argued that a core group of pseudepigrapha do
appear to be part of a single project (2014a). They are written in
Doric Greek (the dialect used in southern Italy where the Pythagoreans
flourished) in order to give them the appearance of authenticity and
share a common style. There are some twenty-five treatises belonging
to this group and they include some of the most famous pseudepigrapha,
including the work by ps.-Timaeus that was supposed to be
Plato's model, ps.-Archytas' works on categories and
ps.-Ocellus *On the Universe*. These treatises espouse the same
basic system and seem designed to cover all the basic fields of
knowledge. The system is based on theory of principles in which God is
the supreme entity above a pair of principles, one of which is limited
and the other unlimited, and which are identified with Aristotelian
form and matter. This system is very similar to what is found in
Eudorus, a Platonist working in Alexandria in the fist cenutury BCE.
Starting from these principles a common system is then developed which
applies to theology, cosmology, ethics, and politics. The connections
to Eudorus and to Philo who also worked in Alexandria, very much
suggest that this group of treatises was developed as a coherent
project in Alexandria sometime in the first century BCE or the first
century CE.



One important group of Pythagorean pseudepigrapha are those forged in
the names of Pythagorean women. Although some work has been done on
them there is still a pressing need for a comprehensive collection of
these texts and a study of them in light of the most recent
scholarship on Pythagoreanism. Pomeroy 2013 provides some useful
commentary but has serious drawbacks (see Centrone 2014b and Brodersen
2014). Many of the texts are collected in Thesleff 1965 under the
names Theano, Periktione, Melissa, Myia and Phintys and taken together
occupy about 15 pages of text. To Periktione are assigned two
fragments from a treatise *On the Harmony of a Woman*.
Periktione is the name of Plato's mother and it is probable that
hers is the famous name in which these works were forged. Two further
fragments from *On Wisdom* are also assigned to her. These
fragments show a strong similarity to fragments from a treatise with
identical title by Archytas and are likely to have been assigned to
Periktione by mistake. Two fragments from a work *On the Temperance
of a Woman* are assigned to Phintys. For Theano, the most famous
Pythagorean woman (see 3.3 above), one fragment of a work *On
Piety* is preserved as well as the titles of several other works,
numerous apophthegms and a number of letters. On Theano in the
pseudepigraphal tradition see Huizenga 2013, 96-117. Melissa and
Myia are represented by one letter each. With few exceptions the works
focus on female virtue, proper marital conduct, and practical issues
such as how to choose a wet nurse and how to deal with slaves. The
advice is quite conservative, stressing obedience to one's
husband, chastity and temperance. There is little that is specifically
Pythagorean. Since the authors are pseudonymous it is impossible to be
sure whether they were in fact written by women using female
pseudonyms or men using female pseudonyms (Huizenga 2013, 116). In the
case of the letters Stadele's edition (1980) is to be
preferred to Thesleff (1965). The letters of Melissa and Myia along
with three letters of Theano are often found together in the
manuscript tradition and may have come to be seen as offering a
curriculum for the moral training of women (Huizenga 2013). Due to the
dearth of preserved writings by women from the ancient world some have
been tempted to suppose that the writings are genuine works by the
named authors. However, as demonstrated above, Pythagorean
pseudepigrapha were very widespread and more common than genuine
Pythagorean works. In such a context the onus of proof is on someone
who wants to show that a work is genuine. The content of the writings
by Pythagorean women is simply too general to make a convincing case
that a specific writing could only have been written by the supposed
author rather than by a later forger. In fact, the writings by women
fit the pattern of the rest of the pseudepigrapha very well. They are
generally forged in the name of famous Pythagorean women, whose names
give authority to the advice imparted (Huizenga 2013, 117). How better
could one impart force to advice to women than to assign that advice
to women who belonged to the philosophical school that gave most
prominence to women? The pseudepigrapha written in the names of
Pythagorean women probably mostly date to the first centuries BCE and
CE like the other Pythagorean pseudepigrapha, but certainty is not
possible. 



One of the most discussed treatises among the pseudepigrapha are the
*Pythagorean Notes*, which were excerpted by Alexander
Polyhistor in the first century BCE, who was in turn quoted by
Diogenes Laertius in his *Life of Pythagoras* (VIII
24-33). Thus the *Notes* date before the middle of the
first century BCE (probably towards the end of the third century BCE
[Burkert 1972a, 53]) and are earlier than most pseudepigrapha. In
Diogenes' life the *Pythagorean Notes* serve as the main
statement of Pythagoras' philosophical views. The treatise is
wildly eclectic, drawing from Plato's *Timaeus*, the
early Academy and Stocisim and the scholarly consensus is that the
treatise is a forgery (Burkert 1961, 26ff., Long 2013, Laks 2014). It
is tempting to suppose that some early material may be preserved
amidst later material, but the text is such an amalgam that it is in
practice impossible to identify securely any early material (Burkert
1961, 26; Laks 2014, 375). The *Notes* are well organized and
present a complete if compressed philosophy organized around the
concept of purity (Laks 2014). Starting from basic principles (the
Platonic monad and dyad) they give an account of the world, living
beings, and the soul ending with moral precepts (some of the
Pythagorean *acusmata*). Kahn thought that the treatise
reflected a Pythagorean community that was active in the Hellenistic
period (2001, 83) but Long is more likely to be right that its learned
eclecticism suggests that it is a scholarly creation (Long 2013,
158-159).


### 4.3 Neopythagorean Metaphysics: Eudorus, Moderatus, Numenius and Hippolytus



"Neopythagorean" is a modern label, which overlaps with
two other modern labels, "Middle Platonist" and
"Neoplatonist," so that a given figure will be called a
Neoplatonist or Middle Platonist by some scholars and a Neopythagorean
by others. There are several different strands in Neopythagoreanism.
One strand focuses on Pythagoras as a master metaphysician. In this
guise he is presented as the author of a theory of principles, which
went even beyond the principles of Plato's later metaphysics,
the one and the indefinite dyad, and which shows similarities to the
Neoplatonic system of Plotinus. The first Neopythagorean in this sense
is Eudorus of Alexandria, who was active in the middle and later part
of the first century BCE. He evidently presented his own innovations
as the work of the Pythagoreans (Dillon 1977, 119). According to
Eudorus, the Pythagoreans posited a single supreme principle, known as
the one and the supreme god, which is the cause of all things. Below
this first principle are a second one, which is also called the monad,
and the indefinite dyad. These latter two are Plato's principles
in the unwritten doctrines, but Eudorus says they are properly
speaking elements rather than principles (Simplicius, *in
Phys*., *CAG* IX 181. 10-30). The system of
principles described by Eudorus also appears in the pseudo-Pythagorean
writings (e.g., pseudo-Archytas, *On Principles*; Thesleff
1965, 19) and it is hard to be certain in which direction the
influence went (Dillon 1977, 120-121). On Eudorus'
connection to the pseudo-Pythagorean writings see also Bonazzi 2013
and Centrone 2014. A generation after Eudorus, another Alexandrian,
the Jewish thinker Philo, used a Pythagorean theory of principles,
which is similar to that found in Eudorus, and Pythagorean number
symbolism in order to give a philosophical interpretation of the
*Old Testament* (Kahn 2001, 99-104; Dillon 1977,
139-183). Philo's goal was to show that Moses was the
first philosopher. For Philo Pythagoras and his travels to the east
evidently played a crucial role in the transmission of philosophy to
the Greeks (Dillon 2014). Philo like Eudorus has close connections to
the Pythagorean pseudepigrapha (Centrone 2014). 



Moderatus of Gades (modern Cadiz in Spain), who was active in the
first century CE, shows similarities to Eudorus in his treatment of
Pythagorean principles. Plutarch explicitly labels him a Pythagorean
and presents his follower, Lucius, as living a life in accord with the
Pythagorean taboos, known as *symbola* or *acusmata*
(*Table Talk* 727b). It is thus tempting to assume that
Moderatus too lived a Pythagorean life (Dillon 1977, 345). His
philosophy is only preserved in reports of other thinkers, and it is
often difficult to distinguish what belongs to Moderatus from what
belongs to the source.



He wrote a comprehensive eleven volume work entitled *Lectures on
Pythagoreanism* from which Porphyry quotes in sections 48-53
of his *Life of Pythagoras*. In this passage, Moderatus argues
that the Pythagoreans used numbers as a way to provide clear teaching
about bodiless forms and first principles, which cannot be expressed
in words. In another excerpt, he describes a Pythagorean system of
principles, which appears to be developed from the first two
deductions of the second half of Plato's *Parmenides*. In
this system there are three ones: the first one which is above being,
a second one which is identified with the forms and which is
accompanied by intelligible matter (i.e. the indefinite dyad) and a
third one which is identified with soul. The first two ones show
connections to Eudorus' account of Pythagorean first principles;
the whole system anticipates central ideas of the most important
Neoplatonist, Plotinus (Dillon 1977, 346-351; Kahn 2001,
105-110).



Moderatus was a militant Neopythagorean, who explicitly charges that
Plato, Aristotle and members of the early academy claimed as their own
the most fruitful aspects of Pythagorean philosophy with only small
changes, leaving for the Pythagoreans only those doctrines that were
superficial, trivial and such as to bring discredit on the school
(Porphyry, *VP* 53). These trivial doctrines have been thought
to be the various taboos preserved in the *symbola*, but, since
his follower Lucius is explicitly said to follow the *symbola*,
it seems unlikely that Moderatus was critical of them. The charge of
plagiarism might suggest that Moderatus was familiar with the
pseudo-Pythagorean treatises, which appear to have been forged in part
to show that Pythagoras had anticipated the main ideas of Plato and
Aristotle (see Kahn 2001, 105).



It is with Numenius (see Dillon 1977, 361-379 and Kahn 2001,
118-133, and the entry on
 Numenius,
 especially section 2), who flourished ca. 150 CE in Apamea in
northern Syria (although he may have taught at Rome), that
Neopythagoreanism has the clearest direct contact with the great
Neoplatonist, Plotinus. Porphyry reports that Plotinus was, in fact,
accused of having plagiarized from Numenius and that, in response,
Amelius, a devotee of Numenius' writings and follower of
Plotinus, wrote a treatise entitled *Concerning the Difference
Between the Doctrines of Plotinus and Numenius* (*Life of
Plotinus* 3 and 17). The third century Platonist, Longinus, to a
degree describes Plotinus himself as a Neopythagorean, saying that
Plotinus developed the exegesis of Pythagorean and Platonic first
principles more clearly than his predecessors, who are identified as
Numenius, his follower Cronius, Moderatus and Thrasyllus, all
Neopythagoreans (Porphyry, *Life of Plotinus* 20). Numenius
also had considerable influence on Porphyry (Macris 2014, 396),
Iamblichus (O'Meara 2014, 404-405) and Calcidius (Hicks
2014, 429).



Numenius is regularly described as a Pythagorean by the sources that
cite his fragments such as Eusebius (e.g. Fr. 1, 4b, 5 etc. Des
Places). He presents himself as returning to the teaching of Plato and
the early Academy. That teaching is in turn presented as deriving from
Pythagoras. Plato is described as "not better than the great
Pythagoras but perhaps not inferior to him either" (Fr. 24 Des
Places). Strikingly, Numenius presents Socrates too as a Pythagorean,
who worshipped the three Pythagorean gods recognized by Numenius (see
below). Thus Plato derived his Pythagoreanism both from direct contact
with Pythagoreans and also from Socrates (Karamanolis 2006,
129-132). For Numenius a true philosopher adheres to the
teaching of his master, and he wrote a polemical treatise, directed
particularly at the skeptical New Academy, with the title *On the
Revolution of the Academics against Plato* (Fr. 24 Des Places).
Numenius presents the Pythagorean philosophy to which Plato adhered as
ultimately based on a still earlier philosophy, which can be found in
Eastern thinkers such as the Magi, Brahmans, Egyptian priests and the
Hebrews (Fr. 1 Des Places). Thus, Numenius was reported to have asked
"What else is Plato than Moses speaking Greek?" (Fr. 8 Des
Places).



Numenius presents his own doctrine of matter, which is clearly
developed out of Plato's *Timaeus*, as the work of
Pythagoras (Fr. 52 Des Places). Matter in its disorganized state is
identified with the indefinite dyad. Numenius argues that for
Pythagoras the dyad was a principle independent of the monad; later
thinkers, who tried to derive the dyad from the monad (he does not
name names but Eudorus, Moderatus and the Pythagorean system described
by Alexander Polyhistor fit the description), were thus departing from
the original teaching. In emphasizing that the monad and dyad are
independent principles, Numenius is indeed closer to the Pythagorean
table of opposites described by Aristotle and to Plato's
unwritten doctrines. Since it is in motion, disorganized matter must
have a soul, so that the world and the things in it have two souls,
one evil derived from matter and one good derived from reason.
Numenius avoids complete dualism in that reason does have ultimate
dominion over matter, thus making the world as good as possible, given
the existence of the recalcitrant matter.



The monad, which is opposed to the indefinite dyad, is just one of
three gods for Numenius (Fr. 11 Des Places), who here follows
Moderatus to a degree. The first god is equated with the good, is
simple, at rest and associates only with itself. The second god is the
demiurge, who by organizing matter divides himself so that a third god
arises, who is either identified with the organized cosmos or its
animating principle, the world soul (Dillon 1977, 366-372).
Numenius is famous for the striking images by means of which he
elucidated his philosophy, such as the comparison of the helmsman, who
steers his ship by looking at the heavens, to the demiurge, who steers
matter by looking to the first god (Fr. 18 Des Places).
Numenius' argument that there is a first god above the demiurge
is paralleled by a passage in another treatise, which shows
connections to Neopythagorean metaphysics, *The Chaldaean
Oracles* (Majercik 1989), which were published by Julian the
Theurgist, during the reign of Marcus Aurelius (161-180 CE) and
thus at about the same time as Numenius was active. It is hard to know
which way the influence went (Dillon 1977, 363).



In *The Refutation of all Heresies*, the Christian bishop
Hippolytus (died ca. 235 CE) adopts the strategy of showing that
Christian heresies are in fact based on the mistaken views of pagan
philosophers. Hippolytus spends considerable time describing
Pythagoreanism, since he regards it as the primary source for gnostic
heresy (see Mansfeld 1992 for this and what follows).
Hippolytus' presentation of Pythagoreanism, which groups
together Pythagoras, Plato, Empedocles and Heraclitus into a
Pythagorean succession, belongs to a family of Neopythagorean
interpretations of Pythagoreanism developed in the first century BCE
and the first two centuries CE and which also appear in later
commentators such as Syrianus and Philoponus. Hippolytus'
interpretation shows similarities to material in Eudorus, Philo
Judaeus, Plutarch and Numenius among others, although he adapts the
material to fit his own purposes. He regards Platonism and
Pythagoreanism as the same philosophy, which ultimately derives from
Egypt. Empedocles is regarded as a Pythagorean and is quoted,
sometimes without attribution, as evidence for Pythagorean views.
According to Hippolytus the Monad and the Dyad are the two Pythagorean
principles, although the Dyad is derived from the Monad. The
Pythagoreans recognize two worlds, the intelligible, which has the
Monad as its principle, and the sensible, whose principle is the
*tetraktys*, the first four numbers, which correspond to the
point, line, surface and solid. The *tetraktys* contains the
decad, since the sum of 1, 2, 3 and 4 is 10, and this is embodied in
the ten Aristotelian categories, which describe the sensible world.
The pseudo-Archytan treatise,  *The Whole System of
Categories*, had already claimed this Aristotelian doctrine for
the Pythagoreans (see 4.2 above). Finally, the intelligible world is
equated with Empedocles' sphere controlled by the uniting power
of Love in contrast to the world of sense perception in which the
dividing power of Strife plays the role of the demiurge
(*Refutation of all Heresies* 6, 23-25).


### 4.4 Neopythagorean Mathematical Sciences: Nicomachus, Porphyry and Iamblichus



A second strand of Neopythagoreanism, while maintaining connection to
these metaphysical speculations, emphasizes Pythagoras' role in
the mathematical sciences. Nicomachus of Gerasa (modern Jerash in
Jordan) was probably active a little before Numenius, in the first
half of the second century CE. Unlike Neopythagoreans such as Eudorus,
Moderatus and Numenius, whose works only survive in fragments, two
complete works of Nicomachus survive, *Introduction to
Arithmetic* and *Handbook of Music*. More than anyone else
in antiquity he was responsible for popularizing supposed Pythagorean
achievements in mathematics and the sciences. The *Handbook of
Music* gives the canonical but scientifically impossible story of
Pythagoras' discovery of the whole number ratios, which
correspond to the basic concordant intervals in music: the octave
(2:1), fifth (3:2), and fourth (4:3); he supposedly heard the concords
in the sounds produced by hammers of varying weights in a
blacksmith's shop, which he happened to be passing (Chapter 6
-- translation in Barker 1989, 256 ff.). In the next century,
Iamblichus took this chapter over virtually verbatim and without
acknowledgement in his *On the Pythagorean Life* (Chapter 26)
and it was repeated in many later authors. The harmonic theory
presented by Nicomachus in the *Handbook* is not original and
is, in fact, somewhat retrograde. It is tied to the diatonic scale
used by Plato in the *Timaeus* (35b-36b), which was previously
used by the Pythagorean Philolaus in the fifth-century (Fr. 6a) and
shows no awareness of or interest in the more sophisticated analysis
of Archytas in the fourth century BCE. Nicomachus is not concerned
with musical practice but with "what pure reasoning can reveal
about the properties of a rationally impeccable and unalterable system
of quantitative relations" (Barker 2007, 447). Nicomachus also
relies heavily and without acknowledgement on a non-Pythagorean
treatment of music, Aristoxenus' *Elementa Harmonica*,
many of the ideas of which he assigns to the Pythagoreans (e.g., in
Chapter 2; see Barker 1989, 245 ff.).



The *Handbook* was influential because it put forth an
accessible version of Pythagorean harmonics (Barker 2014,
200-202). Nicomachus provided a more detailed treatment of
Pythagorean harmonics in his lost *Introduction to Music*. Most
scholars agree that Books I-III and perhaps Book IV of Boethius'
*De Institutione Musica* are a close paraphrase, which is often
essentially a translation, of Nicomachus' lost work (see Bower
in Boethius 1989, xxviii and Barker 2007, 445). Even more influential
than his work on harmonics was Nicomachus' *Introduction to
Arithmetic*. Again Nicomachus was not an original or particularly
talented mathematician, but this popularizing textbook was widely
influential. There were a series of commentaries on it by Iamblichus
(3rd CE), Asclepius of Tralles (6th CE), and Philoponus (6th CE) and
it was translated into Latin already in the second half of the second
century by Apuleius. Most importantly, Boethius (5th-6th CE) provides
what is virtually a translation of it in his *De Institutione
Arithmetica*, which became the standard work on arithmetic in the
middle ages. On Boethius' use of Nicomachus see Hicks 2014,
422-424.



In the *Introduction to Arithmetic*, Nicomachus assigns to
Pythagoras the Platonic division between the intelligible and sensible
world, quoting the *Timaeus* as if it were a Pythagorean text
(I 2). He also assigns Aristotelian ideas to Pythagoras, in particular
a doctrine of immaterial attributes with similarities to the
Aristotelian categories (I 1). Nicomachus divides reality into two
forms, magnitude and multitude. Wisdom is then knowledge of these two
forms, which are studied by the four sciences, which will later be
known as the *quadrivium*: arithmetic, music, geometry and
astronomy. He quotes a genuine fragment of Archytas (Fr. 1) in support
of the special position of these four sciences. Nicomachus presents
arithmetic as the most important of the four, because it existed in
the mind of the creating god (the demiurge) as the plan which he
followed in ordering the cosmos (I 4), so that numbers thus appear to
have replaced the Platonic forms as the model of creation (on forms
and numbers in Nicomachus see Helmig 2007). It is striking that, along
with this Platonization of Pythagoreanism, Nicomachus does give an
accurate presentation of Philolaus' basic metaphysical
principles, limiters and unlimiteds, before attempting to equate them
with the Platonic monad and dyad (II 18).



Another work by Nicomachus, *The Theology of Arithmetic*, which
can be reconstructed from a summary by Photius and an anonymous work
sometimes ascribed to Iamblichus and known as the *Theologoumena
Arithmeticae* (Dillon 1977, 352-353), suggests that he
largely returned to the system of principles found in Plato's
unwritten doctrines and did not follow Eudorus and Moderatus in
attempts to place a supreme god above the demiurge. Nicomachus
apparently presents the monad as the first principle and demiurge,
which then generates the dyad, but much is unclear (Dillon 1977,
353-358). The *Theology of Arithmetic* may have been most
influential in its attempt to set up an equivalence between the pagan
gods and the numbers in the decad, which was picked up later by
Iamblichus and Proclus (Kahn 2001, 116). Nicomachus also wrote a
*Life of Pythagoras*, which has not survived but which Porphyry
(e.g., *VP* 59) and Iamblichus used (Rohde 1871-1872;
O'Meara 2014, 412-413).



After Plotinus (205-270 CE), Neopythagoreanism becomes absorbed
into Neoplatonism. Although Plotinus was clearly influenced by
Neopythagorean speculation on first principles (see above), he was not
a Neopythagorean himself, in that he did not assign Pythagoras a
privileged place in the history of Greek philosophy. Plotinus treats
Pythagoras as just one among many predecessors, complains of the
obscurities of his thought and labels Plato and not Pythagoras as
divine (*Enneads* IV 8.11 ff.).



The earliest extant *Life of Pythagoras* is that of Diogenes
Laertius, who was active ca. 200 CE. The most recent treatment of
Diogenes' life is Laks 2014, on which much of what follows
depends. Unlike his successors Porphyry and Iamblichus (see below)
Diogenes had no philosophical affiliation and hence no philosophical
axe to grind in presenting the life of Pythagoras. Indeed, it is
striking that his life shows little influence from the Neopythagorean
authors discussed above. Diogenes draws on a wide variety of important
sources, some going back to the fourth century and others deriving
from the Hellenistic period. This material is put together in a very
loose, sometimes undetectable, organizational structure. There is a
notable section on Pythagoras' supposed writings (VIII,
6-7). He shows particular interest in the Pythagorean way of
life and quotes a large number of Pythagorean *symbola* for
some of which his source was Aristotle (VIII 34-35). The main
section on Pythagoras' philosophical doctrines is a long
quotation from the first-century polymath Alexander Polyhistor who
claims to be in turn drawing on a treatise called *Pythagorean
Notes* (VIII 24-33). For more on this treatise see the
section on Pythagorean pseudepigrapha above (4.2). Diogenes quotes a
number of passages satirizing Pythagoras, including Xenophanes'
famous puppy fragment, and presents some of his own epigrams making
fun of the Pythagorean way of life (VIII, 36). However, other parts of
his life present Pythagoras in a quite postive light so that it is
hard to determine precisely what attitude Diogenes took towards
Pythagoras (Laks 2014, 377-380).



The *Life of Pythagoras* by Plotinus' pupil and editor,
Porphyry (234-ca. 305) is one of our most important sources for
Pythagoreanism (For what follows see Macris 2014). It was originally
part of his now lost *Philosophical History*. Continuing
interest in Pythagoras in later centuries led the *Life of
Pythagoras* to be preserved separately and it is the only large
section of the *Philosophical History* to survive. The
*Philosophical History*  ended with Plato and clearly regarded
Platonic philosophy as the true philosophy so that Pythagoras seems to
have been highlighted as a key figure in the development of
Plato's philosophy. Porphyry's *Life of Pythagoras*
is particularly valuable, because he often clearly identifies his
sources. This same penchant for identifying and seeking out important
Pythagorean sources can be seen in his commentary on Ptolemy's
*Harmonics* (2nd CE), in which he preserves several genuine
fragments of the early Pythagorean Archytas, along with some
pseudo-Pythagorean material. In the *Life of Pythagoras*
Porphyry does not structure his information according to any
overarching theme but instead sets out the information derived from
other sources in a simple and orderly way with the minimum of
editorial intervention. Although he cites some fifteen sources, some
going back to the fourth century BCE, it is likely that he did not use
most of these sources but rather found them quoted in the four main
sources, which he used directly: 1) Nicomachus' *Life of
Pythagoras*, 2) Moderatus' *Lectures on
Pythagoreanism*, 3) Antonius Diogenes' novel
*Unbelievable Things Beyond Thule*, and 4) a handbook of some
sort. Since these sources come from the first and second centuries CE,
Porphyry basically provides us with the picture of Pythagoras common
in Middle Platonism. This Pythagoras is the prototype of the sage of
old who was active as a teacher and tied to religious mystery.
However, he is not yet Iamblichus' priviliged soul sent to save
humanity (Macris, 2014, 390). Porphyry provides little criticism of
his sources and, although his life has a neutral factual tone, in
contrast to Diogenes Laertius in his *Life of Pythagoras*, he
includes no negative reports about Pythagoras.



It would appear, however, that Pythagoras was not made the source of
all Greek philosophy, but was rather presented as one of a number of
sages both Greek and non-Greek (e.g., Indians, Egyptians and Hebrews),
who promulgated a divinely revealed philosophy. This philosophy is, in
fact, Platonic in origin as it relies on the Platonic distinction
between the intelligible and sensible realms; Porphyry unhistorically
assigns it back to these earlier thinkers, including Pythagoras.
Pythagoras' philosophy is thus said to aim at freeing the mind
from the fetters of the body so that it can attain a vision of the
intelligible and eternal beings (*Life of Pythagoras*
46-47). O'Meara thus seems correct to conclude that
Porphyry was "...not a Pythagoreanizing Platonist ...
but rather a universalizing Platonist: he finds his Platonism both in
Pythagoras and in very many other quarters" (1989, 25-29).
Porphyry himself lived an ascetic life that was probably largely
inspired by Pythagoreanism (Macris 2014, 393-394).



Porphyry's pupil, Iamblichus (ca. 245- ca. 325 CE), from Chalcis
in Syria, opposed his teacher on many issues in Neoplatonic philosophy
and was responsible for a systematic Pythagoreanization of
Neoplatonism (see O' Meara 1989 and 2014), particularly under
the influence of Nicomachus' earlier treatment of Pythagorean
work in the *quadrivium*. Iamblichus wrote a work in ten books
entitled *On Pythagoreanism*. The first four books have
survived intact and excerpts of Books V-VII are preserved by the
Byzantine scholar Michael Psellus. Book One, *On the Pythagorean
Life*, has biographical aspects but is primarily a detailed
description of and a protreptic for the Pythagorean way of life. It
might be that Iamblichus' Pythagoras is intended in part as a
pagan rival to Christ and to Christianity, which was gaining strength
at this time. Porphyry, indeed, had written a treatise *Against the
Christians*, now lost. In Iamblichus, Pythagoras' miraculous
deeds include a meeting at the beginning of his career with fishermen
hauling in a catch (*VP* 36; cf. Matthew 1. 16-20; see
Iamblichus, *On the Pythagorean Life*, Dillon and Hershbell
(eds.) 1991, 25-26). O'Meara, on the other hand, doubts
this connection to Christ (2014, 405 n. 21) and suggests that
Iamblichus may have constructed Pythagoras as a rival to
Porphyry's presentation of Plotinus as the model philosopher
(1989, 214-215). In the end we cannot be certain whether
Iamblichus is responding to Porphyry or Porphyry to Iamblichus, but
they can be seen as battling over Plato's legacy (O'Meara
2014, 403). Porphyry in his *Life of Plotinus* and edition of
his works is promoting Plotinus' interpretation of Plato.
Iamblichus, on the other hand, advocates a return to the philosophy
that inspired Plato, Pythagoreanism. Pythagorean philosophy is
portrayed by Iamblichus as a gift of the gods, which cannot be
comprehended without their aid; Pythagoras himself was sent down to
men to provide that aid (*VP* 1).



Iamblichus' *On the Pythagorean Life* is largely a
compilation of earlier sources but, unlike Porphyry, he does not
usually identify them. Rohde (1871-1872) argued influentially
that  *On the Pythagorean Life* was largely a compilation from
two sources: Nicomachus' *Life of Pythagoras* and a life
of Pythagoras by Apollonius of Tyana. O'Meara argues that this
underestimates both the extent to which Iamblichus reworked his
sources for his own philosophical purposes and the variety of sources
that he used (O'Meara 2014, 412-415). A particularly clear
example of Iamblichus' distintive development of ideas found in
earlier sources can be seen in his treatment of the doctrine of the
harmony of the spheres (O'Meara 2007). It is also true that the
remaining books of *On Pythgoreanism* use a variety of sources.
Book Two, *Protreptic to Philosophy*, is an exhortation to
philosophy in general and to Pythagorean philosophy in particular and
relies heavily on Aristotle's lost *Protrepticus*. Book
Three, *On General Mathematical Science*, deals with the
general value of mathematics in aiding our comprehension of the
intelligible realm and is followed by a series of books on the
specific sciences. The treatment of arithmetic in Book IV takes the
form of a commentary on Nicomachus' *Introduction to
Arithmetic*. Books V-VII then dealt with arithmetic in physics,
ethics and theology respectively and were followed by treatments of
the other three sciences in the quadrivium: *On Pythagorean
Geometry*, *On Pythagorean Music* and *On Pythagorean
Astronomy*. Iamblichus was particularly interested in Pythagorean
numerology and his section on arithmetic in theology is probably
reflected in the anonymous treatise which has survived under the title
*Theologoumena Arithmeticae* and which has sometimes been
ascribed to Iamblichus himself. It appears that here again Iamblichus
relied heavily on Nicomachus, this time on his *Theology of
Arithmetic*.



It is possible that Iamblichus used the ten Books *of On
Pythagoreanism* as the basic text in his school, but we know that
he went beyond these books to the study of Aristotelian logic and the
Platonic dialogues, particularly the *Timaeus* and
*Parmenides* (Kahn 2001, 136-137). Nonetheless, it was
because of Iamblichus that Pythagoreanism in the form of numerology
and mathematics in general was emphasized by later Neoplatonists such
as Syrianus (fl. 430 CE) and Proclus (410/412-485 CE). Proclus
is reported to have dreamed that he was the reincarnation of
Nicomachus (Marinus, *Life of Proclus* 28). Proclus did treat
Plato's writings as clearer than the somewhat obscure writings
of the Pythagoreans but his Platonism is still heavily Pythagorean
(O' Meara 2014, 415). The successors of Proclus appear to follow
his and Iamblichus' interpretation of Pythagoras (O'Meara
2013).


### 4.5 Pythagoreans as Relgious Experts, Magicians and Moral Exemplars: Pythagoreanism in Rome, *The Golden Verses* and Apollonius of Tyana



A third strand in Neopythagoreanism emphasizes Pythagoras'
practices rather than his supposed metaphysical system. This
Pythagoras is an expert in religious and magical practices and/or a
sage who lived the ideal moral life, upon whom we should model our
lives. This strand is closely connected to the striking interest in
and prominence of Pythagoreanism in Roman literature during the first
century BCE and first century CE. Cicero (106-43 BCE) in
particular refers to Pythagoras and other Pythagoreans with some
frequency. In *De Finibus* (V 2), he presents himself as the
excited tourist, who, upon his arrival in Metapontum in S. Italy and
even before going to his lodgings, sought out the site where
Pythagoras was supposed to have died. At the beginning of Book IV
(1-2) of the *Tusculan Disputations*, Cicero notes that
Pythagoras gained his fame in southern Italy at just the same time
that L. Brutus freed Rome from the tyranny of the kings and founded
the Republic; there is a clear implication that Pythagorean ideas,
which reached Rome from southern Italy, had an influence on the early
Roman Republic. Cicero goes on to assert explicitly that many Roman
usages were derived from the Pythagoreans, although he does not give
specifics. According to Cicero, it was admiration for Pythagoras that
led Romans to suppose, without noticing the chronological
impossibility, that the wisest of the early Roman kings, Numa, who was
supposed to have ruled from 715-673 BCE, had been a pupil of
Pythagoras.



In addition to references to Pythagoras himself, Cicero refers to the
Pythagorean Archytas some eleven times, in particular emphasizing his
high moral character, as revealed in his refusal to punish in anger
and his suspicion of bodily pleasure (*Rep*. I 38. 59;
*Sen*. XII 39-41). Cicero's own philosophy is not
much influenced by the Pythagoreans except in *The Dream of
Scipio* (*Rep*. VI 9), which owes even more to Plato.



The interest in Pythagoras and Pythagoreans in the first century BCE
is not limited to Cicero, however. Both a famous ode of Horace (I 28)
and a brief reference in Propertius (IV 1) present Archytas as a
master astronomer. Most striking of all is the speech assigned to
Pythagoras that constitutes half of Book XV of Ovid's
*Metamorphoses* (early years of the first century CE) and that
calls for strict vegetarianism in the context of the doctrine of
transmigration of souls. These latter themes are true to the earliest
evidence for Pythagoras, but the rest of Ovid's presentation
assigns to Pythagoras a doctrine that is derived from a number of
early Greek philosophers and in particular the doctrine of flux
associated with Heraclitus (Kahn 2001, 146-149).



This flourishing of Pythagoreanism in Roman literature of the golden
age has its roots in one of the earliest Roman literary figures,
Ennius (239-169 BCE), who, in his poem *Annales*, adopts
the Pythagorean doctrine of metempsychosis, in presenting himself as
the reincarnation of Homer, although he does not mention Pythagoras by
name in the surviving fragments. Roman nationalism also played a role
in the emphasis on Pythagoreanism at Rome. Since Pythagoras did his
work in Italy and Aristotle even referred to Pythagoreanism in some
places as the philosophy of the Italians (e.g., *Metaph*.
987a10), it is not surprising that the Romans wanted to emphasize
their connections to Pythagoras. This is particularly clear in
Cicero's references to Pythagoreanism but once again finds its
roots even earlier. In 343 BCE during the war with the Samnites,
Apollo ordered the Romans to erect one statue of the wisest and
another of the bravest of the Greeks; their choice for the former was
Pythagoras and for the latter Alcibiades. Pliny, who reports the story
(*Nat*. XXXIV 26), expresses surprise that Socrates was not
chosen for the former, given that, according to Plato's
*Apology*, Apollo himself had labeled Socrates the wisest; it
is surely the Italian connection that explains the Romans'
choice of Pythagoras. Cicero (not Aristoxenus as suggested by Horky
2011) connects the great wisdom assigned to the Samnite Herrenius
Pontius to his contact with the Pythagorean Archytas (*On Old
Age* 41). This Roman attempt to forge a connection with Pythagoras
can also be seen in the report of Plutarch (*Aem. Paul.* 1)
that some writers traced the descent of the Aemelii, one of
Rome's leading families, to Pythagoras, by claiming
Pythagoras' son Mamercus as the founder of the house.



Although Rome's special connection to Pythagoras thus had
earlier roots, those roots alone do not explain the efflorescence of
Pythagoreanism in golden age Latin literature; some stimulus probably
came from the rebirth of what were seen as Pythagorean practices in
the way certain people lived. The two most learned figures in Rome of
the first century BCE, Nigidius Figulus and Varro, both have
connections to Pythagorean ritual practices. Thus we are told that
Varro (116-27 BCE) was buried according to the Pythagorean
fashion in myrtle, olive and black poplar leaves (Pliny, *Nat*.
XXXV 160). Amongst Varro's voluminous works was the
*Hebdomades* ("*Sevens*"), a
collection of 700 portraits of famous men, in the introduction to
which Varro engaged in praise for the number 7, which is similar to
the numerology of later Neopythagorean works such as Nicomachus'
*Theology of Arithmetic*; in another work Varro presents a
theory of gestation, which has Pythagorean connections, in that it is
based on the whole number ratios that correspond to the concordant
intervals in music (Rawson 1985, 161).



It is Nigidius Figulus, praetor in 58, who died in exile in 45,
however, who is usually identified as the figure who was responsible
for reviving Pythagorean practices. In the preface to his translation
of Plato's *Timaeus*, which is often treated as virtually
a Pythagorean treatise by the Neopythagoreans, Cicero asserts of
Nigidius that "following on those noble Pythagoreans, whose
school of philosophy had to a certain degree died out, ... this
man arose to revive it." Some scholars are dubious about this
claim of Cicero. They point to the evidence cited above for the
importance of Pythagoreanism in Rome in the two centuries before
Nigidius and suggest that Cicero may be illegitimately following
Aristoxenus' claim that Pythagoreanism died out in the first
half of the fourth century (Riedweg 2005, 123-124). While there
may be some evidence that there were practicing Pythagoreans in the
second half of the fourth century (see above section 3.5), it is hard
to find anyone to whom to apply that label in the third and second
centuries, so that, from the perspective of the evidence available to
us at present, Cicero may well be right that Nigidius was the first
person in several centuries to claim to follow Pythagorean practices.
However, the sources for Nigidius are meager and there is no evidence
that he was the leader of a large and powerful group. If there was an
organized group at all, it is more likely to have been a smaller
circle (Flinterman 2014, 344).



It is difficult to be sure in what Nigidius' Pythagoreanism
consisted. There is no mention of Pythagoras or Pythagoreans in the
surviving fragments of his work nor do they show him engaging in
Pythagorean style numerology as Varro did (Rawson 1985, 291 ff.). In
Jerome's chronicle, Nigidius is labeled as Pythagorean and
*magus*; the most likely suggestion, thus, is that his
Pythagoreanism consisted in occult and magical practices. Pliny treats
Nigidius alongside the *Magi* and also presents Pythagoras and
Democritus as having learned magical practices from the *Magi*.
Cicero describes Nigidius as investgating matters that nature had
hidden and this may be a reference to such magical lore (Flinterman
2014, 345). Nigidius' expertise as an astrologer (he is reported
to have used astrology to predict Augustus' future greatness on
the day of his birth [Suetonius, *Aug*. 94.5]) may be another
Pythagorean connection; Propertius' reference (IV 1) to Archytas
shows that Pythagorean work in astronomy was typically connected to
astrology in first century Rome.



What led Nigidius and Varro to resurrect purported Pythagorean cult
practices? One important influence may have been the Greek scholar
Alexander Polyhistor, who was born in Miletus but was captured by the
Romans during the Mithridatic wars and brought to Rome as a slave and
freed by Sulla in 80 BCE. He taught in Rome in the 70s. It is an
intriguing suggestion that Nigidius learned his Pythagoreanism from
Alexander (Dillon 1977, 117; For critiques of this suggestion see
Flinterman 2014, 349-350 and Long 2013, 145). There is no
evidence that Alexander himself followed Pythagorean practices, but he
wrote a book *On Pythagorean Symbols*, which was presumably an
account of the Pythagorean *acusmata* (or *symbola*),
which set out the taboos that governed many aspects of the Pythagorean
way of life. In addition, in his *Successions of the
Philosophers*, he gave a summary of Pythagorean philosophy, which
he supposedly found in the *Pythagorean Notes* (See section 4.2
above) and which has been preserved by Diogenes Laertius (VIII
25-35). The basic principles assigned to Pythagoras are those of
the Neopythagorean tradition that begins in the early Academy, i.e.,
the monad and the indefinite dyad. Since Alexander also assigns to the
Pythagoreans the doctrine that the elements change into one another,
we might suppose that Ovid also used Alexander directly or indirectly,
since he assigns a similar doctrine to Pythagoras in the
*Metamorphoses* (XV 75 ff., Rawson 1985, 294).



It is necessary to look in a slightly different direction, in order to
see how magical practices came to be particularly associated with
Pythagoras and thus why Nigidius was called Pythagorean and
*magus*. In the first century, it was widely believed that
Pythagoras had studied with the Magi (Cicero, *Fin*. V 87),
i.e. Persian priests/wise men. What Pythagoras was thought to have
learned from the Magi most of all were the magical properties of
plants. Pliny the elder (23-79 CE) identifies Pythagoras and
Democritus as the experts on such magic and the Magi as their teachers
(*Nat*. XXIV 156-160). Pliny goes on to give a number of
specific examples from a book on plants ascribed to Pythagoras. This
book is universally regarded as spurious by modern scholars, and even
Pliny, who accepts its authenticity, reports that some people ascribe
it to Cleemporus. We can date this treatise on plants to the first
half of the second century or earlier, since Cato the elder
(234-149 BCE) appears to make use of it in his *On
Agriculture* (157), when he discusses the medicinal virtues of a
kind of cabbage, which was named after Pythagoras (*brassica
Pythagorea*).



A clearer understanding of this pseudo-Pythagorean treatise on plants
and a further indication of its date can be obtained by looking at the
work of Bolus of Mendes, an Egyptian educated in Greek (see Dickie
2001, 117-122, to whom the following treatment of Bolus is
indebted). Bolus composed a work entitled *Cheiromecta*, which
means "things worked by hand" and may thus refer to
potions made by grinding plants and other substances (Dickie 2001,
119). Bolus discussed not just the magical properties of plants but
also those of stones and animals. Pliny regarded the
*Cheiromecta* as composed by Democritus on the basis of his
studies with the Magi (*Nat.* 24. 160) and normally cites its
contents as what Democritus or the Magi said. Columella, however,
tells us what was really going on (*On Agriculture* VII 5.17).
The work was in fact composed by Bolus, who published it under the
name of Democritus. Bolus thus appears to have made a collection of
magical recipes, some of which do seem to have connections to the
Magi, since they are similar to recipes found in 8th century cuneiform
texts (Dickie 2001, 121). In order to gain authority for this
collection, he assigned it to the famous Democritus.



Since Democritus was sometimes regarded as the pupil of Pythagoreans
(Diogenes Laertius IX 38), Bolus' choice of Democritus to give
authority to his work may suggest that someone else (the Cleemporus
mentioned by Pliny?) had already used Pythagoras for this purpose and
that the pseudo-Pythagorean treatise on the magical properties of
plants was thus already in existence when Bolus wrote, in the first
half of the second century BCE. An example of the type of recipe
involved is Pliny's ascription to Democritus of the idea that
the tongue of a frog, cut out while the frog was still alive, if
placed above the heart of a sleeping woman, will cause her to give
true answers (*Nat*. XXXII 49). Thus, the picture of Pythagoras
the magician, which may lie behind a number of the supposed
Pythagorean practices of Nigidius Figulus, is based on little more
than the tradition that Pythagoras had traveled to Egypt and the east,
so that he became the authority figure, to whom the real collectors of
magical recipes in the third and second century BCE ascribed their
collections.



Nigidius' revival of supposed Pythagorean practices spread to
other figures in first century Rome. Cicero attacked Vatinius, consul
in 48 and a supporter of Caesar, for calling himself a Pythagorean and
trying to shield his scandalous practices under the name of Pythagoras
(*Vat*. 6). The scandalous practices involved necromancy,
invoking the dead, by murdering young boys. Presumably this method of
necromancy would not be ascribed to Pythagoras, but the suggestion is
that some methods of consulting the dead were regarded as Pythagorean.
Cicero later ended up defending this same Vatinius in a speech which
has not survived but some of the contents of which we know from the
ancient scholia on the speech against Vatinius. In this speech Cicero
defended Vatinius' habit of wearing a black toga, which he
attacked in the earlier speech (*Vat*. 12), as a harmless
affectation of Pythagoreanism (Dickie 2001, 170). Thus, the title of
Pythagorean in first century Rome carried with it associations with
magical practices, not all of which would have been widely approved.




Another example of the connection between Pythagoreanism and magic and
its possible negative connotations is Anaxilaus of Larissa (Rawson
1985, 293; Dickie 2001, 172-173). In his chronicle, Jerome
describes him with the same words as he used for Nigidius, Pythagorean
and *magus*, and reports that he was exiled from Rome in 28
BCE. We know that Anaxilaus wrote a work entitled *Paignia*
("tricks"), which seems to have consisted of some rather
bizarre conjuring tricks for parties. Pliny reports one of
Anaxilaus' tricks as calling for burning the discharge from a
mare in heat in a flame, in order to cause the guests to see images of
horses' heads (*Nat*. XXVIII 181). The passion for things
Pythagorean can also be seen in the figure of king Juba of Mauretania
(ca. 46 BCE - 23 CE), a learned and cultured man, educated at
Rome and author of many books. Olympiodorus describes him as "a
lover of Pythagorean compositions" and suggests that Pythagorean
books were forged to satisfy the passion of collectors such as Juba
(*Commentaria in Aristotelem Graeca* 12.1, p. 13).



The connection between Pythagoreanism and astrology visible in
Nigidius can perhaps also be seen in Thrasyllus of Alexandria (d. 36
CE), the court astrologer and philosopher, whom the Roman emperor
Tiberius met in Rhodes and brought to Rome. Thrasyllus is famous for
his edition of Plato's dialogues arranged into tetralogies, but
he was a Platonist with strong Pythagorean leanings. Porphyry in his
*Life of Plotinus* (20) quotes Longinus as saying that
Thrasyllus wrote on Platonic and Pythagorean first principles (Dillon
1977, 184-185). Most suggestive of all is the quotation from
Thrasyllus preserved by Diogenes Laertius (Diogenes Laertius IX 38),
in which Thrasyllus calls Democritus a zealous follower of the
Pythagoreans and asserts that Democritus drew all his philosophy from
Pythagoras and would have been thought to have been his pupil, if
chronology did not prevent it. It is impossible to be sure what
Thrasyllus had in mind here, but one very plausible suggestion is that
he is thinking of Democritus as a sage, who practiced magic, the
Democritus created by Bolus, who was the successor to the arch mage
Pythagoras, the supposed author of the treatise on the magical uses of
plants (Dickie 2001, 195). Some have argued that the subterranean
basilica discovered near the Porta Maggiore and dating to the first
century CE was the meeting place of a Pythagorean community but the
evidence for this suggestion is very weak (Flinterman 2014).



We cannot be sure whether the Pythagoreanism of Nigidius, Varro and
their successors was limited to such things as burial ritual, magical
practices and black togas or whether it extended to less spectacular
features of a "Pythagorean" life. Q. Sextius, however,
founded a philosophical movement in the time of Augustus, which
prescribed a vegetarian diet and taught the doctrine of transmigration
of souls, although Sextius presented himself as using different
arguments than Pythagoras for vegetarianism (Seneca, *Ep*. 108.
17 ff.). One of these Sextians, as they were known, was Sotion, the
teacher of Seneca, and it is Seneca who gives us most of the
information we have about them. It is also noteworthy that Sextius is
also reported to have asked himself at the end of each day "What
bad habit have you cured today? What vice have you resisted? In what
way are you better" (Seneca, *De Ira* III 36). Cicero
tells us that it was "the Pythagorean custom" to call to
mind in the evening everything said, heard or done during the day
(*Sen*. 38, cf. Iamblichus, *VP* 164). The practice
described by Cicero is directed at training the memory in contrast to
Sextius' questions, which call for moral self-examination. On
Pythagoreanism in Rome see further Flinterman 2014.



Something similar to the Sextian version of the practice is found in
lines 40-44 of the *Golden Verses*, a pseudepigraphical
treatise consisting of 71 Greek hexameter verses, which were ascribed
to Pythagoras or the Pythagoreans. The poem is a combination of
materials from different dates, and it is uncertain when it took the
form preserved in manuscripts and called the *Golden Verses*;
dates ranging from 350 BCE to 400 CE have been suggested (see Thom
1995). It is not referred to by name until 200 CE. The *Golden
Verses* are frequently quoted in the first centuries CE and thus
constitute one model of the Pythagorean life in Neopythagoreanism, one
that is free from magical practices. Much of the advice is common to
all of Greek ethical thought (e.g., honoring the gods and parents;
mastering lust and anger; deliberating before acting, following
measure in all things), but there are also mentions of dietary
restrictions typical of early Pythagoreanism and the promise of
leaving the body behind to join the aither as an immortal.



Our most detailed account of a Neopythagorean living a life inspired
by Pythagoras is Philostratus' *Life of Apollonius of
Tyana*. Apollonius was active in the second half of the first
century CE and died in 97; Philostratus' life, which was written
over a century later at the request of the empress Julia Domna and
completed after her death in 217 CE, is more novel than sober
biography. According to Philostratus, Apollonius identified his wisdom
as that of Pythagoras, who taught him the proper way to worship the
gods, to wear linen rather than wool, to wear his hair long, and to
eat no animal food (I 32). Some have wondered if Apollonius'
Pythagoreanism is largely the creation of Philostratus, but the
standard view has been that Apollonius wrote a life of Pythagoras used
by Iamblichus (*VP* 254) and Porphyry (Burkert 1972, 100), and
the fragment of his treatise *On Sacrifices* has clear
connections to Neopythagorean philosophy (Kahn 2001, 143-145).
Rohde thought that large parts of Apollonius's *Life of
Pythagoras* could be found in Iamblichus' *On the
Pythagorean Life*, but recently more and more doubt has arisen as
to whether the Apollonius who wrote the *Life of Pythagoras*
used by Iamblichus is really Apollonius of Tyana (Flinterman 2014,
357). 



Like Pythagoras, Apollonius journeys to consult the wise men of the
east and learns from the Brahmins in India that the doctrine of
transmigration, which Apollonius inherited from Pythagoras, originated
in India and was handed on to the Egyptians from whom Pythagoras
derived it (III 19). Philostratus (I 2) emphasizes that Apollonius was
not a magician, thus trying to free him from the more disreputable
connotations of Pythagorean practices associated with figures such as
Anaxilaus and Vatinius (see above). Nonetheless, Philostratus'
life does recount a number of Apollonius' miracles, such as the
raising of a girl from the dead (IV 45). On Apollonius as a
Pythagorean see further Flinterman 2014.



These miracles made Apollonius into a pagan counterpart to Christ. The
emperor Alexander Severus (222-235 CE) worshipped Apollonius
alongside Christ, Abraham and Orpheus (*Hist. Aug., Vita Alex.
Sev.* 29.2). Hierocles, the Roman governor of Bithynia, who was
rigorous in his persecution of Christians, championed Apollonius at
the expense of Christ, in *The Lover of Truth,* and drew as a
response Eusebius' *Against Hierocles*. As mentioned
above, there is some probability that Iamblichus intends to elevate
Pythagoras himself as a pagan counterpart to Christ in his *On the
Pythagorean Life* (Dillon and Hershbell 1991, 25-26).



The satirist Lucian (2nd CE) provides us with a hostile portrayal of
another holy man with Pythagorean connections, Alexander of
Abnoteichus in Paphlagonia, who was active in the middle of the second
century CE. In *Alexander the False Prophet*, Lucian reports
that Alexander compared himself to Pythagoras (4), could remember his
previous incarnations (34) and had a golden thigh like Pythagoras
(40). Lucian shows the not often seen negative side to both
Pythagoras' and Alexander's reputations when he reports
that, if one took even the worst things said about Pythagoras,
Alexander would far outdo him in wickedness (4). Some have seen
Alexander as largely a literary construction by Lucian with little
historical basis but other evidence confirms that there were traveling
Pythagorean wonder-workers in the early imperial period (Flinterman
2014, 359). 



Despite these attacks on figures such as Apollonius and Alexander who
modeled themselves on Pythagoras, the Pythagorean way of life was in
general praised; the Neopythagorean tradition which portrays
Pythagoras as living the ideal life on which we should model our own
reaches its culmination in Iamblichus' *On the Pythagorean
Life* and Porphyry's *Life of Pythagoras*


## 5. Pythagoreanism in the Middle Ages and Renaissance



The influence of Pythagoreanism in the Middle Ages and Renaissance was
extensive and was found in most disciplines, in literature and art as
well as in philosophy and science. Here only the highlights of that
influence can be given (see further Heninger 1974, Celenza 1999,
Celenza 2001, Kahn 2001, Riedweg 2005, Hicks 2014 and Allen 2014, to
all of whom the following account is indebted). It is crucial to
recognize from the beginning that the Pythagoras of the Middle Ages
and Renaissance is the Pythagoras of the Neopythagorean tradition, in
which he is regarded as either the most important or one of the most
important philosophers in the Greek philosophical tradition. Thus,
Ralph Cudworth, in *The True Intellectual System of the
Universe* asserted that "Pythagoras was the most eminent of
all the ancient Philosophers" (1845, II 4). This is a far cry
from the Pythagoras that can be reconstructed by responsible
scholarship. Riedweg has put it well: "Had Pythagoras and his
teachings not been since the early Academy overwritten with
Plato's philosophy, and had this 'palimpsest' not in
the course of the Roman empire achieved unchallenged authority among
Platonists, it would be scarcely conceivable that scholars from the
Middle Ages and modernity down to the present would have found the
pre-Socratic charismatic from Samos so fascinating" (2005,
128).


### 5.1 Boethius/Nicomachus, Calcidius, Macrobius and the Middle Ages



In the Middle Ages Pythagoras and Pythagorean philosophy were regarded
as the height of Greek philosophical achievement, although, somewhat
paradoxically Pythagoreanism was not still an active philosophy as
were Platonism and Aristotelianism but instead belonged to an
"imagined history" of philosophy (Hicks 2014, 420). The
view of Pythagoreanism in the Middle Ages was heavily determined by
three late ancient Latin writers: Calcidius, Macrobius and Boethius.
It was in particular the mathematical Pythagoreanism of Nicomachus as
transmitted by Boethius that determined the medieval picture of
Pythagoras. In ethics, Christians were able to embrace some
Pythagorean maxims such as the principle labeled Pythagorean by
Boethius: "Follow God" (*Consolation of Philosophy*
1.4). Some attention was also paid to other Pythagorean
*symbola* (see section 5.2 below). On the other hand the
doctrine of metempsychosis with its idea that human beings were born
again as animals was repugnant to Christian doctrine (John of
Salisbury, *Policraticus* 7.10). When it comes to
Pythagoras' life it is crucial to recognize that
Iamblichus' and Porphyry's lives of Pythagoras were not
known in the Middle Ages so that Pythagoras' activities were
mostly known through passages from classical authors and church
fathers (Hicks 2014, 421). Pythagoras was included in medieval
encyclopedic works and was given particularly thorough treatment by
Vincent of Beauvais (before 1200-1264) in his *Speculum
historiale* (3.24-26), by John of Wales (fl.
1260-1283) in *Compendiloquium* (3.6.2) and in *The
Lives and Habits of the Philosophers* ascribed to, but probably
not actually composed by, Walter Burley (1275-1344; see Riedweg
2005, 129; Heninger 1974, 47; Hicks 2014, 421). 



The most influential texts for the conception of Pythagoras in the
Latin Middle Ages and early Renaissance were Boethius'
(480-524 CE) *De Institutione Arithmetica* and *De
Institutione Musica*, which are virtually translations of the
Neopythagorean Nicomachus' (second century CE) *Introduction
to Arithmetic* and *Introduction to Music* (this larger
work is now lost, but a smaller *Handbook of Harmonics*
survives). Boethius followed Nicomachus' classification of four
mathematical sciences depending on the nature of their objects
(arithmetic deals with multitude in itself, music with relative
multitude, geometry with unmoving magnitudes and astronomy with
magnitude in motion). Boethius introduced the term
*quadrivium*, "fourfold road" to understanding, to
refer to these four sciences. In music theory, Boethius presents the
Pythagoreans as taking a middle position, which gives a role in
harmonics to both reason and perception. His presentation of the
Pythagorean position was central to music theory for over a thousand
years (Hicks 2014, 424). Boethius recounts the apocryphal story of
Pythagoras' discovery in a blacksmith's shop of the ratios
that govern the concordant intervals (*Mus*. I 10). 



The medieval picture of Pythagoras as a natural philosopher and the
medieval understanding of his theory of the nature of the soul were
heavily influenced by the Latin commentary on Plato's
*Timaeus* by Calcidius (4th century CE) and the *Commentary
on the Dream of Scipio* by Macrobius (5th century CE). Calcidius
regarded Plato's *Timaeus* as a heavily Pythagorean
document. Under the influence of the Neopythagorean Numenius,
Calcidius assigned to Pythagoras the view that god was unity and
matter duality (Hicks 2014, 429). Calcidius describes Plato's
World-Soul in a way that highlights its harmonic structure and
Macrobius explicitly ascribes to Pythagoras the view that the soul is
a harmony (*Commentary on the Dream of Scipio* 1.14.19). The
doctrine of the harmony of the spheres, which portrays the cosmos as a
harmony that is expressed in the music made by the revolutions of the
planets, follows from the numerical structure of the World-Soul and
was also assigned to Pythagoras by Calcidius. Most medieval
Neoplatonic cosmoligies adopted the doctrine, but the reintroduction
of Aristotle's criticism of it in the thirteenth century caused
many to abandon the theory until it was revived in the Renaissance by
Ficino (Hicks 2014, 434). Later, Shakespeare refers to the doctrine
memorably in *The Merchant of Venice* (V i. 54-65).
Cicero's presentation of it in the *Dream of Scipio* was
also influential in the Renaissance (Heninger 1974, 3).



Pythagorean influence also appeared at less elevated levels of
medieval culture. A fourteenth-century manual for preachers, which
contained lore about the natural world and is known as *The Light
of the Soul*, ascribes a series of odd observations about nature
to Archita Tharentinus, who is presumably intended to be the fourth
century BCE Pythagorean, Archytas of Tarentum. These are mostly cited
from a book, which was evidently forged in Archytas' name and
known as *On Events in Nature*. Some of the observations are
plausible enough, e.g., that a person at the bottom of a well sees
stars in the middle of the day, others more puzzling, e.g., that a
dying man emits fiery rays from his eyes at death, while still others
may have connections to magic, e.g., "if someone looks at a
mirror, before which a white flower has been placed, he cries."
Some magical lore ascribed to an Architas is also found in the
thirteenth-century *Marvels of the World* (ps.-Albertus
Magnus), e.g., "if the wax of the left ear of a dog be taken and
hung on people with periodic fever, it is beneficial..."
These texts seem to continue the connection between Pythagoreanism and
magic, which developed in the third and second centuries BCE, and is
prominent in Rome during the first-century BCE (see above section
4.5).


### 5.2 The Renaissance: Ficino, Pico, Reuchlin, Copernicus and Kepler



In the Renaissance, Pythagoreanism played an important role in the
thought of fifteenth- and sixteenth century Italian and German
humanists. The Florentine Marsilio Ficino (1433-1499) is most
properly described as a Neoplatonist. He made the philosophy of Plato
available to the Latin-speaking west through his translation of all of
Plato into Latin. In addition he translated important works of writers
in the Neoplatonic and Neopythagorean tradition, such as Plotinus,
Porphyry, Iamblichus and Proclus. From that tradition he accepted and
developed the view that Plato was heir to an ancient
theology/philosophy (*prisca theologia*) that was derived from
earlier sages including Pythagoras, who immediately preceded Plato in
the succession (Allen 2014, 435-436). Ficino like the
Neopythagoreans had no conception of an early and a late
Pythagoreanism, for him Pythagoreanism was a unity as indeed was the
entire tradition of ancient theology (Celenza, 1999, 675-681).
Ficino regarded works ascribed to the Chaldaean Zoroaster, the
Egyptian Hermes Trismegistus, Orpheus and Pythagoras, which modern
scholarship has shown to be forgeries of late antiquity, as genuine
works on which Plato drew (Kristeller 1979, 131). Ficino provided a
complete translation of the writings ascribed to Hermes Trismegistus
into Latin as well as translations of 39 of the short Pythagorean
sayings known as *symbola*, many of which are ancient, and
Hierocles' commentary on the pseudo-Pythagorean *Golden
Verses* (Heninger 1974, 63 and 66). The *Golden Verses*
(see Thom 1995) were, in fact, one of the most popular Greek texts in
the Renaissance and were commonly used in textbooks for learning
Greek; other pseudo-Pythagorean texts, such as the treatises ascribed
to Timaeus of Locri and Ocellus, were translated early and regarded as
genuine texts on which Plato drew (Heninger 1974, 49, 55-56).
Ficino thought, moreover, that this whole pagan tradition could be
reconciled with Christian and Jewish religion and accepted the view
that Pythagoras was born of a Jewish father (Heninger 1974, 201). For
Ficino and the Renaissance as a whole, Pythagoras was the most
important of the Presocratic philosophers but he never overshadowed
Plato, who was the highest authority, in part because there was no
extensive body of texts by Pythagoras himself to compete with the
Platonic dialogues (Allen 2014, 453).



Ficino translated Iamblichus' four works on Pythagoreanism for
his own use and Iamblichus' *On the Pythagorean Life* had
particular influence on him. Ficino felt that in his time there was a
need for a divinely inspired guide on earth and fashioned himself as
such a prophet under the influence of Iamblichus' presentation
of Pythagoras as a divine guide sent by the gods to save mankind
(Celenza 1999, 667-674). The Pythagorean musical practice that
he found in Iamblichus' *On the Pythagorean Life* , with
its emphasis on the impact of music on the soul, shaped his own music
making and his presentation of himself as a Pythagorean and Orphic
holy man (Allen 2014, 436-440). Ficino and other Renaissance
thinkers grappled with the challenge that the Pythagorean notion of
*metempsychosis* presented to Christiantiy and how it might be
reconciled with Christian views (Allen 2014, 440-446). Ficino
was eager to absolve Plato from such a heresy. He does this in part by
treating metempsychosis metaphorically as referring to the
soul's ability to remake itself, but he also emphasized that
metempsychosis was not present in Plato's latest work,
*Laws*, and made the Pythagoreans scapegoats by suggesting that
other passages in Plato refer not to Plato's own doctrines but
the Pythagoreans (Celenza 1999, 681-691). Ficino saw his own
arithmology as Pythgorean and study of Neopythagorean mathematical
treatises by Nicomachus and Theon led Ficino to conclude that
Plato's nuptial number in Book 8 of the *Republic* was 12
(Allen 2014, 446-450). He also mistakenly and paradoxically
followed the Neopythagoreans in thinking that the Pythagoreans
occupied the crucial position in the history of philosophy of the
first philosophers to distinguish between the corporeal and
incorporeal and to assert the superiority of the latter, an
achievement that is more reasonably assigned to Ficino's hero
Plato (Celenza 1999, 699-706). 



The Pythagorean *symbola* were important to Ficino and the
Renaissance. They had already been interpreted as moral maxims by the
early church fathers (e.g., Clement, Origen and Ambrose). Ambrose, for
example, interpreted the Pythagorean "do not take the public
path" to mean that priests should live lives of exceptional
purity (*Ep. 81*). Jerome discussed 13 symbola in his
*Epistle Against Rufinus* and this list became the basis for
medieval discussions of the *symbola* in texts such as the
*Speculum historiale* of Vincent of Beauvais and the *Lives
and Habits of the Philosophers*  of Walter Burley (Celenza 2001,
11-12). Ficino particularly encountered them in
Iamblichus' *On the Pythagorean Life* and
*Protrepticus*. For Ficino, their brevity was appropriate to
revealing the supreme reality, since he argued that the closer the
mind approaches to the One the fewer words it needs (Allen 2014,
450-451). In addition, he found them relevant to the preparation
and purification of the soul (Celenza, 1999, 693). They were widely
discussed by Ficino's contemporaries and successors (Celenza
2001, 52-81). Some figures wrote treatises devoted to their
interpretation (Ficino's mentor Antonio degli Agli, his follower
Giovanni Nesi [for an edition of Nesi's work see Celenza 2001],
Filippo Beroaldo the Elder and Lilio Gregorio Giraldi), while others
discussed them as part of larger works (Erasmus and Reuchlin). Not
everyone took the *symbola* seriously; Angelo Poliziano, the
great Florentine philologist and professor, presents a satire on them
in the fashion of Lucian, joking about Pythagoras' ability to
talk to animals and ridiculing the prohibition on beans (Celenza 2001,
33).



Ficino's friend and younger contemporary, Giovanni Pico della
Mirandola (1463-1494), advanced an even more radical doctrine of
universal truth, according to which all philosophies had a share of
truth and could be reconciled in a comprehensive philosophy
(Kristeller 1979, 205). His *Oration on the Dignity of Man*
shows the variety of ways in which he was influenced by the
Pythagorean tradition. He equates the friendship that the Pythagoreans
saw as the goal of philosophy (see, e.g., Iamblichus, *VP* 229)
with the peace that the angels announced to men of good will (1965,
11-12); the Pythagorean *symbola* forbidding urinating
towards the sun or cutting the nails during sacrifice are interpreted
allegorically as calling on us to relieve ourselves of excessive
appetite for sensual pleasures and to trim the pricks of anger (1965,
15); the practice of philosophizing through numbers is assigned to
Pythagoras along with Philolaus, Plato and the early Platonists (1965,
25-26); Pythagoras is said to have modeled his philosophy on the
Orphic theology (1965, 33). Finally, on the basis of the
pseudo-Pythagorean letter of Lysis to Hipparchus, Pythagoras is said
to have kept silent about his doctrine and left just a few things in
writing to his daughter at his death. In observing such silence,
Pythagoras is portrayed as following an earlier practice symbolized by
the sphinx in Egypt and most of all by Moses, who indeed published the
law to men but supposedly kept the interpretation of that law a
secret. Pico equates this secret interpretation of the law with the
Cabala, an esoteric doctrine in which the words and numbers of Hebrew
scripture are interpreted according to a mystical system (1965, 30;
see also *Heptaplus* 1965, 68).



Pico's interest in reconciling the Cabala with Christianity and
the pagan philosophical tradition, including Pythagoreanism, was
further developed by the German humanist, Johannes Reuchlin
(1445-1522). In the dedicatory letter for his *Three
Books* *On the Art of the Cabala* (1517), which was
addressed to Pope Leo X, Reuchlin says that as Ficino has restored
Plato for Italy so he will "offer to the Germans Pythagoras
reborn," although he cannot "do this without the cabala of
the Hebrews, because the philosophy of Pythagoras took its beginning
from the precepts of the cabalists" (tr. Heninger 1974, 245).
Thus, in an earlier work (*De verbo mirifico*) he had equated
the four consonants in the Hebrew name for God, JHVH, with the
Pythagorean *tetraktys*, and gave to each of the letters, which
are equated with numbers as in Greek practice, a mystical meaning. The
first H, which also stands for the number five that the Pythagoreans
equated with marriage, is thus taken to symbolize the marriage of the
trinity with material nature, which was equated with the dyad by the
Neopythagoreans (Riedweg 2005, 130).



At the level of popular culture, several fortune-telling devices were
tied to Pythagoras, the most famous of which went under the name of
the Wheel of Pythagoras (Heninger 1974, 237). Pythagoras was probably
most widely known, however, through Ovid's presentation of him
at the beginning of Book XV of the *Metamorphoses*, which was
immensely popular in the Renaissance (Heninger 1974, 50). Ovid
recounts the story, which had already been recognized as apocryphal by
Cicero (*Tusc*. IV 1), that the second Roman king, Numa,
studied with Pythagoras. Pythagoras is presented inaccurately by Ovid
as a great natural philosopher, who discovered the secrets of the
universe and who believed in a doctrine of the flux of four elements.
On the other hand, Ovid's emphasis on the prohibition on eating
animal flesh and on the immortality of the soul have some connection
to the historical Pythagoras. In the Renaissance, Pythagoras was not
primarily known for the "Pythagorean Theorem," as he is
today. Better known was the doubtful anecdote (Burkert 1960, Riedweg
2005, 90-97), going back ultimately to Heraclides of Pontus but
known to the Renaissance mainly through Cicero (*Tusc*. V
3-4), that he was the first to coin the word
"philosopher" (Heninger 1974, 29).



In the sixteenth century, Pythagorean influence was particularly
important in the development of astronomy. The Polish astronomer
Copernicus (1473-1543), in the *Preface and Dedication to
Pope Paul III* attached to his epoch making work*, On the
Revolution of the Heavenly Spheres*, reports that, in his
dissatisfaction with the commonly accepted geocentric astronomical
system of Ptolemy (2nd century CE), he laboriously reread the works of
all the philosophers to see if any had ever proposed a different
system. This labor led him to find inspiration not from Pythagoras
himself but rather from later Pythagoreans and in particular from
Philolaus. Copernicus found in Cicero (*Ac*. II 39. 123) that
the Pythagorean Hicetas (4th century BCE -- Copernicus mistakenly
calls him Nicetas) had proposed that the earth revolved around its
axis at the center of the universe and in pseudo-Plutarch (Diels 1958,
378) that another Pythagorean, Ecphantus, and Heraclides of Pontus
(both 4th century BCE), whom Copernicus regarded as a Pythagorean, had
proposed a similar view. More importantly, he also found in
pseudo-Plutarch that the Pythagorean, Philolaus of Croton (5th century
BCE), "held that the earth moved in a circle ... and was
one of the planets" (*On the Revolutions of the Heavenly
Spheres* 1. 5, tr. Wallis).



Copernicus reports to the Pope that he was led by these earlier
thinkers "to meditate on the mobility of the earth."
Pythagorean influence on Copernicus was not limited to the notion of a
moving earth. In the same preface he explains his hesitation to
publish his book in light of the pseudo-Pythagorean letter of Lysis to
Hipparchus, which recounts the supposed reluctance of the Pythagoreans
to divulge their views to the common run of people, who had not
devoted themselves to study (for further Pythagorean influences on
Copernicus see Kahn 2001, 159-161). A number of the followers of
Copernicus saw him as primarily reviving the ancient Pythagorean
system rather than presenting anything new (Heninger 1974, 130 and
144, n. 131); Edward Sherburne reflects the common view of the late
17th century in referring to the heliocentric system as "the
system of Philolaus and Copernicus" (Heninger 1974,
129-130), although in the Philolaic system it is, in fact, a
central fire and not the sun that is at the center of the
universe.



The last great Pythagorean was Johannes Kepler (1571-1630
-- see Kahn 2001, 161-172 for a good brief account of
Kepler's Pythagoreanism). Kepler began by developing the
Copernican system in light of the five regular solids (tetrahedron,
cube, octahedron, dodecahedron and icosahedron), to which Plato
appealed in his construction of matter in the *Timaeus* (see
especially 53B-55C). He followed the Renaissance practice illustrated
above of regarding Greek philosophy as closely connected to the wisdom
of the Near East, when he asserted that the *Timaeus* was a
commentary on the first chapter of *Genesis* (Kahn 2001, 162).
In the preface to his early work, *Mysterium Cosmographicum*
(1596), Kepler says that his purpose is to show that God used the five
regular bodies, "which have been most celebrated from the time
of Pythagoras and Plato," as his model in constructing the
universe and that "he accommodated the number of heavenly
spheres, their proportions, and the system of their motions" to
these five regular solids (tr. Heninger 1974, 110-111).



In ascribing geometrical knowledge of the five regular solids to
Pythagoras, Kepler is following an erroneous Neopythagorean tradition,
although the dodecahedron may have served as an early Pythagorean
symbol (see on Hippasus in section 3.4 above and Burkert 1972,
70-71, 404, 460). Thus, this aspect of Kepler's work is
more Platonic than Pythagorean. The five solids were conceived of as
circumscribing and inscribed in the spheres of the orbits of the
planets, so that the five solids corresponded to the six planets known
to Kepler (Saturn, Jupiter, Mars, Earth, Venus, Mercury). There were
six planets, because there were precisely five regular bodies to be
used in constructing the universe, corresponding to the five intervals
between the planets. This view was overthrown by the later discovery
of Uranus as a seventh planet. Kepler's cosmology was, however,
far from a purely *a priori* exercise. Whereas his
contemporary, Robert Fludd, developed a cosmology structured by
musical numbers, which could in no way be confirmed by observation,
Kepler strove to make his system consistent with precise observations.
Kahn suggests that we here see again the split "between a
rational and an obscurantist version of Pythagorean thought,"
which is similar to the ancient split in the school between
*mathematici* and *acusmatici* (2001, 163).



Close work with observational data collected by Tycho Brahe led Kepler
to abandon the universal ancient view that the orbits of the planets
were circular and to recognize their elliptical nature. More clearly
Pythagorean is Kepler's consistent belief that the data show
that the motions of the planets correspond in various ways to the
ratios governing the musical concords (see Dreyer 1953,
405-410), so that there is a heavenly music, a doctrine attested
for Philolaus and Archytas, which probably goes back to Pythagoras as
well. For Kepler, however, the music produced by the heavenly motions
was "perceived by reason, and not expressed in sound"
(*Harmonice Mundi* V 7). In his attempt to make the numbers of
the heavenly music work, he joked that he would appeal to the shade of
Pythagoras for aid, "unless the soul of Pythagoras has migrated
into mine" (Koestler 1959, 277).



Kepler has been described "as the last exponent of a form of
mathematical cosmology that can be traced back to the shadowy figure
of Pythagoras" (Field 1988, 170). It is true that Kepler's
work led the way to Newton's mechanics, which cannot be
described in terms of ancient geometry and number theory but relies on
the calculus and which relies on a theory of physical forces that is
alien to ancient thought. On the other hand, many modern scientists
accept the basic tenet that knowledge of the natural world is to be
expressed in mathematical formulae, which is rightly regarded as a
central Pythagorean thesis, since it was first rigorously formulated
by the Pythagoreans Philolaus ( Fr. 4) and Archytas and may, in a
rudimentary form, go back to Pythagoras himself.