Ancilla to the Pre-Socratic Philosophers, by Kathleen Freeman, , at sacred-texts.com
Zênô of Elea was in his prime about 450 B.C.
He wrote a book of Epicheirêmata (Attacks) in defence of Parmenides theory of Being as One and Indivisible; his method was to take the opposite proposition that Things are Many, and derive two contradictory conclusions therefrom.
1. (Second half of the Epicheirêma showing that if Things are Many, they must be (a) infinitely small, (b) infinitely great).
If Being had no size, it could not Be either.
If anything Is, it follows that each (part) must have a certain size and bulk, and distance one from the other. And the same reasoning applies also to the part preceding it; for that too will have size and there will be another part preceding it. The same reasoning, in fact, applies always: no part of the Whole will be such as to be outermost, nor will any part be unrelated to another part. Therefore, if Things are Many, they must be both small and great: so small as to have no size, so large as to be infinite.
2. (First half of the Epicheirêma showing that if Things are Many they must be (a) infinitely small, (b) infinitely great).
If it (a unit without magnitude) be added to another existing thing, it would not make the latter at all larger. For if a thing without magnitude is added (to another thing) the latter cannot gain anything in magnitude. And thus (it follows) at once that the thing added is nothing. And if when a unit is subtracted the other will not become at all less, and will not, on the other hand, increase when (this unit) is added, it is clear that the unit added or subtracted was nothing.
3. (Epicheirêma showing that if Things are Many, they must be (a) finite, (b) infinite in number).
If Things are Many, they must be as many as they are and neither more nor less than this. But if they are as many as they are, they must be finite (in number).
If Things are Many, they are infinite in number. For there are always other things between those that are, and again others between those. And thus things are infinite (in number).
4. (From an Epicheirêma showing the impossibility of motion).
That which moves, moves neither in the place in which it is, nor in that in which it is not.