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.
In a sense, however, they make it plain that goodness belongs to
numbers, and that the odd, the straight, the square, the potencies
of certain numbers, are in the column of the beautiful. For the
seasons and a particular kind of number go together; and the other
agreements that they collect from the theorems of mathematics all have
this meaning. Hence they are like coincidences. For they are
accidents, but the things that agree are all appropriate to one
another, and one by analogy. For in each category of being an
analogous term is found-as the straight is in length, so is the
level in surface, perhaps the odd in number, and the white in colour.

    Again, it is not the ideal numbers that are the causes of
musical phenomena and the like (for equal ideal numbers differ from
one another in form; for even the units do); so that we need not
assume Ideas for this reason at least.

    These, then, are the results of the theory, and yet more might
be brought together. The fact that our opponnts have much trouble with
the generation of numbers and can in no way make a system of them,
seems to indicate that the objects of mathematics are not separable
from sensible things, as some say, and that they are not the first
principles.

                                   -THE END-
.