Early Greek Philosophy by John Burnet, with Burnet's notes
159. What Is the Unit? 161. The Unit

From Chapter VIII., The Younger Eleatics

160. The Fragments
The fragments of Zeno himself also show that this was his line of argument. I give them according to the arrangement of Diels.

(1)

If what is had no magnitude, it would not even be . . . . But, if it is, each one must have a certain magnitude and a certain thickness, and must be at a certain distance from another, and the same may be said of what is in front of it; for it, too, will have magnitude, and something will be in front of it.28 It is all the same to say this once and to say it always; for no such part of it will be the last, nor will one thing not be as compared with another.29

So, if things are a many, they must be both small and great, so small as not to have any magnitude at all, and so great as to be infinite. R. P. 134.

(2)

For if it were added to any other thing it would not make it any larger; for nothing can gain in magnitude by the addition of what has no magnitude, and thus it follows at once that what was added was nothing.30 But if, when this is taken away from another thing, that thing is no less; and again, if, when it is added to another thing, that does not increase, it is plain that, what was added was nothing, and what was taken away was nothing. R. P. 132.

(3)

If things are a many, they must be just as many as they are, and neither more nor less. Now, if they are as many as they are, they will be finite in number.

If things are a many, they will be infinite in number; for there will always be other things between them, and others again between these. And so things are infinite in number. R. P. 13331

Burnet's Notes

.

28. I formerly rendered "the same may be said of what surpasses it in smallness; for it too will have magnitude, and something will surpass it in smallness." This is Tannery's rendering, but I now agree with Diels in thinking that ἀπέχειν refers to μέγεθος and προέχειν to πάχος. Zeno is showing that the Pythagorean point must have three dimensions.

29. Reading, with Diels and the MSS., οὔτε ἕτερον πρὸς ἕτερον οὐκ ἔσται.. Gomperz's conjecture (adopted in R. P.) seems to me arbitrary.

30. Zeller marks a lacuna here. Zeno must certainly have shown that the subtraction of a point does not make a thing less; but he may have done so before the beginning of our present fragment.

31. This is what Aristotle calls "the argument from dichotomy" (Phys. A, 3. 187 a 2 ; R. P. 134 b). If a line is made up of points, we ought to be able to answer the question, "How many points are there in a given line?" On the other hand you can always divide a line or any part of it into two halves; so that, if a line is made up of points, there will always be more of them than any number you assign.






















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