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[1082a] [1] For example, in the Ideal number 10 there are ten units, and 10 is composed both of these and of two 5's. Now since the Ideal 10 is not a chance number,1 and is not composed of chance 5's, any more than of chance units, the units in this number 10 must be different;for if they are not different, the 5's of which the 10 is composed will not be different; but since these are different, the units must be different too. Now if the units are different, will there or will there not be other 5's in this 10, and not only the two? If there are not, the thing is absurd2; whereas if there are, what sort of 10 will be composed of them? for there is no other 10 in 10 besides the 10 itself:

Again, it must also be true that 4 is not composed of chance 2's. For according to them the indeterminate dyad, receiving the determinate dyad, made two dyads; for it was capable of duplicating that which it received.3

Again, how is it possible that 2 can be a definite entity existing besides the two units, and 3 besides the three units? Either by participation of the one in the other, as "white man" exists besides "white" and "man," because it partakes of these concepts; or when the one is a differentia of the other, as "man" exists besides "animal" and "two-footed."

[20] Again, some things are one by contact, others by mixture, and others by position; but none of these alternatives can possibly apply to the units of which 2 and 3 consist. Just as two men do not constitute any one thing distinct from both of them, so it must be with the units.The fact that the units are indivisible will make no difference; because points are indivisible also, but nevertheless a pair of points is not anything distinct from the two single points.

Moreover we must not fail to realize this: that on this theory it follows that 2's are prior and posterior, and the other numbers similarly.Let it be granted that the 2's in 4 are contemporaneous; yet they are prior to those in 8, and just as the <determinate> 2 produced the 2's in 4, so4 they produced the 4's in 8. Hence if the original 2 is an Idea, these 2's will also be Ideas of a sort.And the same argument applies to the units, because the units in the original 2 produce the four units in 4; and so all the units become Ideas, and an Idea will be composed of Ideas. Hence clearly those things also of which these things are Ideas will be composite;

1 I think Ross's interpretation of this passage must be right. The Ideal 10 is a unique number, and the numbers contained in it must be ideal and unique; therefore the two 5's must be specifically different, and so must their units—which contradicts the view under discussion.

2 i.e., it is only reasonable to suppose that other 5's might be made up out of different combinations of the units.

3 Cf. Introduction.

4 In each case the other factor is the indeterminate dyad (cf. sect. 18).

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