but this entanglement with politics led in the end to the dismemberment
and suppression of the society. The authorities
differ hopelessly in chronology, but according to the balance of
evidence the first reaction against the Pythagoreans took place
in the lifetime of Pythagoras after the victory gained by Crotona
over Sybaris in 510. Dissensions seem to have arisen about the
allotment of the conquered territory, and an adverse party was
formed in Crotona under the leadership of Cylon. This was
probably the cause of Pythagoras’s withdrawal to Metapontum,
which an almost unanimous tradition assigns as the place of
his death in the end of the 6th or the beginning of the 5th
century. The order appears to have continued powerful in
Magna Graecia till the middle of the 5th century, when it was
violently trampled out. The meeting-houses of the Pythagoreans
were everywhere sacked and burned; mention is made
in particular of “the house of Milo” in Crotona, where fifty or
sixty leading Pythagoreans were surprised and slain.
The persecution to which the brotherhood was subjected throughout Magna Graecia was the immediate cause of the spread of the Pythagorean philosophy in Greece proper. Philolaus, who resided at Thebes in the end of the 5th century (cf. Plato, Phaedo, 61 D), was the author of the first written exposition of the system. Lysis, the instructor of Epaminondas, was another of these refugees. This Theban Pythagoreanism had an important influence upon Plato's thought, and Philolaus had also disciples, in the stricter sense. But as a philosophic school Pythagoreanism became extinct in Greece about the middle of the 4th century. In Italy—where, after a temporary suppression, it attained a new importance in the person of Archytas of Tarentum-the school finally disappeared about the same time.
Aristotle in his accounts of Pythagorean doctrines never refers to Pythagoras but always with a studied vagueness to “the Pythagoreans” (οἱ καλούμονοι Πυθαγορειοι). Nevertheless, certain doctrines may be traced to the founder's teaching. Foremost among these is the 'theory of the immortality and transmigration of the soul (see Metempsychosis). Pythagoras’s teaching on this point is connected by one of the most trustworthy authorities with the doctrine of the kinship of all living beings; and in the light of anthropological research it is easy to recognize the close relationship pf the two beliefs. The Pythagorean rule of abstinence from fies is thus, in its origin, a taboo resting upon the blood-brotherhood of men and beasts; and the same line of thought shows a number of the Pythagorean rules of life which we find embedded in the different traditions to be genuine taboos belonging to a similar level of primitive thought. The moral and religious application which Pythagoras gave to the; doctrine of transmigration continued to be the teaching of the school. The view of the body ({{Greek|σῶμα) as the tomb (σῆμα) of the soul, and the account of philosophy in the Phaedo as a meditation of death, are expressly connected by Plato with the teaching of Philolaus; and the strain of asceticism and other worldliness which meets us here and elsewhere in Plato is usually traced to Pythagorean influence. Plato’s mythical descriptions of a future life of retribution and purificatory wandering can also be shown to reproduce Pythagorean teaching, though the substance of them may have been drawn from a common source in the Mysteries.
The scientific doctrines of the Pythagorean school have no
apparent connexion with the religious mysticism of the society pr
their rules of living. They have their origin in the same disinterested
desire of knowledge which gave rise to the other philosophical
schools of Greece, and the idea of “philosophy” or the
“theoretic life” as a method of emancipation from the evils of
man's present state of existence, though a genuine Pythagorean
conception, is clearly an afterthought. The discourses and speculations
of the Pythagoreans all connect themselves with the idea
of number, and the school holds an important place in the history
of mathematical and astronomical science. An unimpeached
tradition carries back the Pythagorean theory of numbers to the
teaching of the founder himself. Working on hints contained in
the oldest traditions, recent investigators have shown that the discoveries
attributed to Pythagoras connect themselves with a pflmltive
numerical symbolism, according to which numbers were represented
by dots arranged in symmetrical patterns, such as are still
to be seen in the marking of dice or dominoes. Each pattern of
units becomes on this plan a fresh unit. The “holy tetractys,”
by which the later Pythagoreans used to swear, was a figure of
this kind 
representing the number 10 as the triangle of 4.
and showing at a glance that 1 + 2 + 3 + 4 = 10. The sums
of the series of any successive numbers may be graphically
represented in a similar way, and are hence spoken of as “triangular
numbers,” while the sums of the series of successive odd numbers are
called “square numbers,” and those of successive even numbers
oblong numbers; thus 3 and 5 added to the unit give a figure
of this description 
while 4 and 6, added to 2, are thus
represented 
. Such a method of representing number
in areas leads naturally to problems of a geometrical nature, and as
the practical use of the right-angled triangle was already familiar in
the arts and crafts, there is no reason to dispute the well-established
tradition which assigns to Pythagoras the discovery of the proposition
that in such a triangle the square on the hypotenuse is
equal to the sum of the squares on the other two sides. And it
is probably also correct to attribute to him the discovery of the
harmonic intervals which underlie the production of musical
sounds. Impressed by this reduction of musical sounds to numbers
and by the presence of numerical relations in every department
of phenomena, Pythagoras and his early followers enunciated the
doctrine that “all things are numbers.” Numbers seemed to them,
as Aristotle put it, to be the first things in the whole of nature,
and they supposed the elements of numbers to be the elements of
all things, and the whole heaven to be a musical scale and a number
(Meta. A. 986a). Numbers, in other words, were conceived at that
early stage of thought not as relations or qualities predicable of
things, but as themselves constituting the substance or essence
of the phenomena-the rational reality to which the appearances of
sense are reducible.
But the development of these ideas into a comprehensive metaphysical system was no doubt the work of Philolaus in the latter part of the 5th century. His formulation of the theory implies a knowledge of the teaching of Parmenides and Empedocles, and had itself in turn a great influence upon Plato. The “elements of numbers,” of which Aristotle speaks in the passage quoted above, were, according to the Pythagoreans, the Odd and the Even, which they identified with the Limit and the Unlimited; and Aristotle distinctly says that they did not treat these as “ priorities of certain other substances” such as fire, water or anything else of that sort, but that the unlimited itself and the one were the reality of the things of which they were predicated, and that is why they said that number was “the reality of everything” (Meta. A. 587). Numbers, therefore, are spatially conceived, “one” being identified with the point in the sense of a unit having position and magnitude. From combinations of such units the higher numbers and geometrical figures arise—“two” being identified with the line, “ three ” with the surface, and “ four ” with the solid-and the Pythagoreans proceeded to explain the elements of Empedocles as built up out of geometrical figures in the manner followed by Plato in the Timaeus. The identification of the numerical opposites, the Odd and the Even, with the Limit and the Unlimited-otherwise difficult to explain-may perhaps be understood, as Burnet suggests, by reference to the arrangement of the units or “terms” (ὅροι) in patterns. “When the odd is divided into two equal parts,” he quotes from Stobaeus, “a unit is left over in the middle; but when the even is so divided, an empty field is left over, without a master and without a number, showing that it is defective and incomplete.” The idea of opposites, derived, perhaps, originally from Heracleitus, was developed by the Pythagoreans in a list of ten fundamental oppositions, bearing a certain resemblance to the tables of categories framed by later philosophers, but in its arbitrary mingling of mathematical, physical and ethical contrasts characteristic of the uncritical beginnings of speculative thought: (1) limited and unlimited, (2) odd and even, (3) one and many, (4) right and left, (5) male and female, (6) rest and motion, (7) straight and curved, (8) light and darkness, (9) good and evil, (10) square and oblong. To the Pythagoreans, as to Heracleitus, the universe was in a sense the realized union of these opposites, but interpretations of Pythagoreanism which represent the whole system as founded on the opposition of unity and duality, and proceed to identify this with the opposition of form and matter, of divine activity and passive material, betray on the surface their post-Platonic origin. Still more is this the case when in Neoplatonic fashion they go on to derive this original opposition from the supreme unity or God. The further speculations of the Pythagoreans on the subject of number rest mainly on analogies, which often become capricious and tend to lose themselves at last in a barren symbolism. “Seven” is called παρθένος and Ἀθἠνη, because within the decade it has neither factors nor product. “Five,” on the other hand, signifies marriage, because it is the union of the first masculine with the first feminine number (3+2, unity being considered as a number apart). The thought already becomes more fanciful when “one” is identified with reason, because it is unchangeable; “two” with opinion, because it is unlimited and indeterminate; “four” with justice, because it is the first square number, the product of equals.
The astronomy of the Pythagoreans was their most notable contribution to scientific thought, and its importance lies in the fact that they were the first to conceive the earth as a globe,
