will be the last, nor will one thing not be as compared with another.[1] 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.[2] 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. 133.[3]
161.The unit. If we hold that the unit has no magnitude—and this is required by what Aristotle calls the argument from dichotomy,[4]—then everything must be infinitely small. Nothing made up of units without magnitude can itself have any magnitude. On the other hand, if we insist that the units of which things are built up are something and not nothing, we must hold that everything is infinitely great.
- ↑ Reading, with Diels and the MSS., οὔτε ἕτερον πρὸς ἕτερον οὐκ ἔσται. Gomperz's conjecture (adopted in R. P.) seems to me arbitrary.
- ↑ 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.
- ↑ 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.
- ↑ See last note.
