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the whole of nature, 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. And all the properties of numbers and scales which they
could show to agree with the attributes and parts and the whole
arrangement of the heavens, they collected and fitted into their
scheme; and if there was a gap anywhere, they readily made additions
so as to make their whole theory coherent. E.g. as the number 10 is
thought to be perfect and to comprise the whole nature of numbers,
they say that the bodies which move through the heavens are ten, but
as the visible bodies are only nine, to meet this they invent a
tenth--the 'counter-earth'. We have discussed these matters more
exactly elsewhere.
But the object of our review is that we may learn from these
philosophers also what they suppose to be the principles and how these
fall under the causes we have named. Evidently, then, these thinkers
also consider that number is the principle both as matter for things
and as forming both their modifications and their permanent states,
and hold that the elements of number are the even and the odd, and
that of these the latter is limited, and the former unlimited; and
that the One proceeds from both of these (for it is both even and
odd), and number from the One; and that the whole heaven, as has
been said, is numbers.
Other members of this same school say there are ten principles,
which they arrange in two columns of cognates-limit and unlimited, odd
and even, one and plurality, right and left, male and female,
resting and moving, straight and curved, light and darkness, good
and bad, square and oblong. In this way Alcmaeon of Croton seems
also to have conceived the matter, and either he got this view from
them or they got it from him; for he expressed himself similarly to
them. For he says most human affairs go in pairs, meaning not definite
contrarieties such as the Pythagoreans speak of, but any chance
contrarieties, e.g. white and black, sweet and bitter, good and bad,
great and small. He threw out indefinite suggestions about the other
contrarieties, but the Pythagoreans declared both how many and which
their contraricties are.
From both these schools, then, we can learn this much, that the
contraries are the principles of things; and how many these principles
are and which they are, we can learn from one of the two schools.
But how these principles can be brought together under the causes we
have named has not been clearly and articulately stated by them;
they seem, however, to range the elements under the head of matter;
for out of these as immanent parts they say substance is composed
and moulded.
From these facts we may sufficiently perceive the meaning of the
ancients who said the elements of nature were more than one; but there
are some who spoke of the universe as if it were one entity, though
they were not all alike either in the excellence of their statement or
in its conformity to the facts of nature. The discussion of them is in
no way appropriate to our present investigation of causes, for. they
do not, like some of the natural philosophers, assume being to be
one and yet generate it out of the one as out of matter, but they
speak in another way; those others add change, since they generate the
universe, but these thinkers say the universe is unchangeable. Yet
this much is germane to the present inquiry: Parmenides seems to
fasten on that which is one in definition, Melissus on that which is
one in matter, for which reason the former says that it is limited,
the latter that it is unlimited; while Xenophanes, the first of
these partisans of the One (for Parmenides is said to have been his
pupil), gave no clear statement, nor does he seem to have grasped
the nature of either of these causes, but with reference to the
whole material universe he says the One is God. Now these thinkers, as
we said, must be neglected for the purposes of the present inquiry-two
of them entirely, as being a little too naive, viz. Xenophanes and
Melissus; but Parmenides seems in places to speak with more insight.

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