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One might also raise the question what the good is that things get
from numbers because their composition is expressible by a number,
either by one which is easily calculable or by an odd number. For in
fact honey-water is no more wholesome if it is mixed in the proportion
of three times three, but it would do more good if it were in no
particular ratio but well diluted than if it were numerically
expressible but strong. Again, the ratios of mixtures are expressed by
the adding of numbers, not by mere numbers; e.g. it is 'three parts to
two', not 'three times two'. For in any multiplication the genus of
the things multiplied must be the same; therefore the product 1X2X3
must be measurable by 1, and 4X5X6 by 4 and therefore all products
into which the same factor enters must be measurable by that factor.
The number of fire, then, cannot be 2X5X3X6 and at the same time
that of water 2X3.
If all things must share in number, it must follow that many
things are the same, and the same number must belong to one thing
and to another. Is number the cause, then, and does the thing exist
because of its number, or is this not certain? E.g. the motions of the
sun have a number, and again those of the moon,-yes, and the life
and prime of each animal. Why, then, should not some of these
numbers be squares, some cubes, and some equal, others double? There
is no reason why they should not, and indeed they must move within
these limits, since all things were assumed to share in number. And it
was assumed that things that differed might fall under the same
number. Therefore if the same number had belonged to certain things,
these would have been the same as one another, since they would have
had the same form of number; e.g. sun and moon would have been the
same. But why need these numbers be causes? There are seven vowels,
the scale consists of seven strings, the Pleiades are seven, at
seven animals lose their teeth (at least some do, though some do not),
and the champions who fought against Thebes were seven. Is it then
because the number is the kind of number it is, that the champions
were seven or the Pleiad consists of seven stars? Surely the champions
were seven because there were seven gates or for some other reason,
and the Pleiad we count as seven, as we count the Bear as twelve,
while other peoples count more stars in both. Nay they even say that
X, Ps and Z are concords and that because there are three concords,
the double consonants also are three. They quite neglect the fact that
there might be a thousand such letters; for one symbol might be
assigned to GP. But if they say that each of these three is equal to
two of the other letters, and no other is so, and if the cause is that
there are three parts of the mouth and one letter is in each applied
to sigma, it is for this reason that there are only three, not because
the concords are three; since as a matter of fact the concords are
more than three, but of double consonants there cannot be more.
These people are like the old-fashioned Homeric scholars, who
see small resemblances but neglect great ones. Some say that there are
many such cases, e.g. that the middle strings are represented by
nine and eight, and that the epic verse has seventeen syllables, which
is equal in number to the two strings, and that the scansion is, in
the right half of the line nine syllables, and in the left eight.
And they say that the distance in the letters from alpha to omega is
equal to that from the lowest note of the flute to the highest, and
that the number of this note is equal to that of the whole choir of
heaven. It may be suspected that no one could find difficulty either
in stating such analogies or in finding them in eternal things,
since they can be found even in perishable things.
But the lauded characteristics of numbers, and the contraries of
these, and generally the mathematical relations, as some describe
them, making them causes of nature, seem, when we inspect them in this
way, to vanish; for none of them is a cause in any of the senses
that have been distinguished in reference to the first principles.

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