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[1093b] [1] and the line scans in the first half with nine syllables, and in the second with eight.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 powers5 of certain numbers, belong to the series of the Beautiful.6 For the seasons are connected with a certain kind of number7; 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.

1 i.e., a dactylic hexameter whose sixth foot is always a spondee or trochee has nine syllables in the first three feet and eight in the last three. For τὸ δεξιόν meaning "the first part" of a metrical system see Bassett,Journal of Classical Philology 11.458-460.

2 Alexander suggests that the number 24 may have been made up of the 12 signs of the zodiac, the 8 spheres (fixed stars, five planets, sun and moon) and 4 elements.

3 Cf. Aristot. Met. 1.3.1, Aristot. Met. 5.1, 2.

4 i.e., square.

5 Probably their "power" of being represented as regular figures; e.g. the triangularity of 3 or 6.

6 Cf. Aristot. Met. 1.5.6.

7 i.e., 4.

8 Aristotle has argued (Aristot. Met. 13.6-8.) that if the Ideal numbers differ in kind, their units must differ in kind. Hence even equal numbers, being composed of different units, must be different in kind. In point of fact, since each ideal number is unique, no two of them could be equal.

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