[35] _Cf._ R. K. Gaye, "On Aristotle, _Physics_, Z ix." _Journal of Philology_, vol. xxxi., esp. p. 111. Also Moritz Cantor, _Vorlesungen über Geschichte der Mathematik_, 1st ed., vol. i., 1880, p. 168, who, however, subsequently adopted Paul Tannery's opinion, _Vorlesungen_, 3rd ed. (vol. i. p. 200). [36] "Le mouvement et les partisans des indivisibles," _Revue de Métaphysique et de Morale_, vol. i. pp. 382-395. [37] "Le mouvement et les arguments de Zénon d'Élée," _Revue de Métaphysique et de Morale_, vol. i. pp. 107-125. The historical questions raised by the above-mentioned discussions are no doubt largely insoluble, owing to the very scanty material from which our evidence is derived. The points which seem fairly clear are the following: (1) That, in spite of MM. Milhaud and Paul Tannery, Zeno is anxious to prove that motion is really impossible, and that he desires to prove this because he follows Parmenides in denying plurality;[38] (2) that the third and fourth arguments proceed on the hypothesis of indivisibles, a hypothesis which, whether adopted by the Pythagoreans or not, was certainly much advocated, as may be seen from the treatise _On Indivisible Lines_ attributed to Aristotle. As regards the first two arguments, they would seem to be valid on the hypothesis of indivisibles, and also, without this hypothesis, to be such as would be valid if the traditional contradictions in infinite numbers were insoluble, which they are not. [38] _Cf._ M. Brochard, "Les prétendus sophismes de Zénon d'Élée," _Revue de Métaphysique et de Morale_, vol. i. pp. 209-215. We may conclude, therefore, that Zeno's polemic is directed against the view that space and time consist of points and instants; and that as against the view that a finite stretch of space or time consists of a finite number of points and instants, his arguments are not sophisms, but perfectly valid. The conclusion which Zeno wishes us to draw is that plurality is a delusion, and spaces and times are really indivisible. The other conclusion which is possible, namely, that the number of points and instants is infinite, was not tenable so long as the infinite was infected with contradictions. In a fragment which is not one of the four famous arguments against motion, Zeno says: "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."[39] [39] Simplicius, _Phys._, 140, 28 D (R.P. 133); Burnet, _op. cit._, pp. 364-365.