(a) Simplicius’s account of Zeno’s arguments, including the translation of the Fragments.

30 r 138, 30. For Eudemos says in his Physics, ‘Then does not this exist, and is there any one? This was the problem. He reports Zeno as saying that if any one explains to him the one, what it is, he can tell him what things are. But he is puzzled, it seems, because each of the senses declares that there are many things, both absolutely, and as the result of division, but no one establishes the mathematical point. He thinks that what is not increased by receiving additions, or decreased as parts are taken away, is not one of the things that are.’ It was natural that Zeno, who, as if for the sake of exercise, argued both sides of a case (so that he is called double-tongued), should utter such statements raising difficulties about the one; but in his book which has many arguments in regard to each point, he shows that a man who affirms multiplicity naturally falls into contradictions. Among these arguments is one by which he shows that if there are many things, these are both small and great—great enough to be infinite in size, and small enough to be nothing in size. By this he shows that what has neither greatness nor thickness nor bulk could not even be. (Fr. 1)⁠[66] ‘For if, he says, anything were added to another being, it could not make it any greater; for since greatness does not exist, it is impossible to increase the greatness of a thing by adding to it. So that which is added would be nothing. If when something is taken away that which is left is no less, and if it becomes no greater by receiving additions, evidently that which has been added or taken away is nothing.’ These things Zeno says, not denying the one, but holding that each thing has the greatness of [115]many and infinite things, since there is always something before that which is apprehended, by reason of its infinite divisibility; and this he proves by first showing that nothing has any greatness because each thing of the many is identical with itself and is one.

Ibid. 30 v 140, 27. And why is it necessary to say that there is a multiplicity of things when it is set forth in Zeno’s own book? For again in showing that, if there is a multiplicity of things, the same things are both finite and infinite, Zeno writes as follows, to use his own words: (Fr. 2) ‘If there is a multiplicity of things, it is necessary that these should be just as many as exist, and not more nor fewer. If there are just as many as there are, then the number would be finite. If there is a multiplicity at all, the number is infinite, for there are always others between any two, and yet others between each pair of these. So the number of things is infinite.’ So by the process of division he shows that their number is infinite. And as to magnitude, he begins with this same argument. For first showing that (Fr. 3) ‘if being did not have magnitude, it would not exist at all,’ he goes on, ‘if anything exists, it is necessary that each thing should have some magnitude and thickness, and that one part of it should be separated from another. The same argument applies to the thing that precedes this. That also will have magnitude and will have something before it. The same may be said of each thing once for all, for there will be no such thing as last, nor will one thing differ from another. So if there is a multiplicity of things, it is necessary that these should be great and small—small enough not to have any magnitude, and great enough to be infinite.’⁠[67]⁠

Ibid. 130 v 562, 3. Zeno’s argument seems to deny that place exists, putting the question as follows: (Fr. 4) [116]‘If there is such a thing as place, it will be in something, for all being is in something, and that which is in something is in some place. Then this place will be in a place, and so on indefinitely. Accordingly there is no such thing as place.’

Ibid. 131 r 563, 17. Eudemos’ account of Zeno’s opinion runs as follows:—‘Zeno’s problem seems to come to the same thing. For it is natural that all being should be somewhere, and if there is a place for things, where would this place be? In some other place, and that in another, and so on indefinitely.’

Ibid. 236 v. Zeno’s argument that when anything is in a space equal to itself, it is either in motion or at rest, and that nothing is moved in the present moment, and that the moving body is always in a space equal to itself at each present moment, may, I think, be put in a syllogism as follows: The arrow which is moving forward is at every present moment in a space equal to itself, accordingly it is <in a space equal to itself> in all time; but that which is in a space equal to itself in the present moment is not in motion. Accordingly it is in a state of rest, since it is not moved in the present moment, and that which is not moving is at rest, since everything is either in motion or at rest. So the arrow which is moving forward is at rest while it is moving forward, in every moment of its motion.

237 r. The Achilles argument is so named because Achilles is named in it as the example, and the argument shows that if he pursued a tortoise it would be impossible for him to overtake it.

255 r. Aristotle accordingly solves the problem of Zeno the Eleatic, which he propounded to Protagoras the Sophist.⁠[68] Tell me, Protagoras, said he, does one grain of millet make a noise when it falls, or does the [117]ten-thousandth part of a grain? On receiving the answer that it does not, he went on: Does a measure of millet grains make a noise when it falls, or not? He answered, it does make a noise. Well, said Zeno, does not the statement about the measure of millet apply to the one grain and the ten-thousandth part of a grain? He assented, and Zeno continued, Are not the statements as to the noise the same in regard to each? For as are the things that make a noise, so are the noises. Since this is the case, if the measure of millet makes a noise, the one grain and the ten-thousandth part of a grain make a noise.