^{1}And they point out that the interval from α to ω in the alphabet is equal to that from the lowest note of a flute to the highest, whose number is equal to that of the whole system of the universe.

^{2}We must realize that no one would find any difficulty either in discovering or in stating such correspondences as these in the realm of eternal things, since they occur even among perishable things.

As for the celebrated characteristics of
number, and their contraries, and in general the mathematical
properties, in the sense that some describe them and make them out to
be causes of the natural world, it would seem that if we examine them
along these lines, they disappear; for not one of them is a cause in
any of the senses which we distinguished with until respect to the
first Principles.^{3}
There is a sense,
however, in which these thinkers make it clear that goodness is
predicable of numbers, and that the odd, the straight, the
equal-by-equal,^{4} and the powers^{5} of
certain numbers, belong to the series of the Beautiful.^{6} For the seasons are connected with a
certain kind of number^{7}; and the other examples which they adduce from mathematical
theorems all have the same force.Hence they would seem to be mere coincidences,
for they are accidental; but all the examples are appropriate to each
other, and they are one by analogy. For there is analogy between all
the categories of Being—as "straight" is in
length,
[20]
so is "level"
in breadth, perhaps "odd" in number, and "white" in color.

Again, it is not the Ideal numbers that are the causes of harmonic
relations, etc. (for Ideal numbers, even when they are equal, differ
in kind, since their units also differ in kind)^{8}; so on this ground at least we need not posit
Forms.

Such, then, are the consequences of the theory, and even more might be adduced. But the mere fact that the Platonists find so much trouble with regard to the generation of Ideal numbers, and can in no way build up a system, would seem to be a proof that the objects of mathematics are not separable from sensible things, as some maintain, and that the first principles are not those which these thinkers assume.