[41] Aristotle's words are: “The first is the one on the non-existence of motion on the ground that what is moved must always attain the middle point sooner than the end-point, on which we gave our opinion in the earlier part of our discourse.” Phys., vi. 9. 939B (R.P. 136). Aristotle seems to refer to Phys., vi. 2. 223AB [R.P. 136A]: “All space is continuous, for time and space are divided into the same and equal divisions…. Wherefore also Zeno's argument is fallacious, that it is impossible to go through an infinite collection or to touch an infinite collection one by one in a finite time. For there are two senses in which the term ‘infinite’ is applied both to length and to time, and in fact to all continuous things, either in regard to divisibility, or in regard to the ends. Now it is not possible to touch things infinite in regard to number in a finite time, but it is possible to touch things infinite in regard to divisibility: for time itself also is infinite in this sense. So that in fact we go through an infinite, [space] in an infinite [time] and not in a finite [time], and we touch infinite things with infinite things, not with finite things.” Philoponus, a sixth-century commentator (R.P. 136A, Exc. Paris Philop. in Arist. Phys., 803, 2. Vit.), gives the following illustration: “For if a thing were moved the space of a cubit in one hour, since in every space there are an infinite number of points, the thing moved must needs touch all the points of the space: it will then go through an infinite collection in a finite time, which is impossible.”
[42] Cf. Mr C. D. Broad, “Note on Achilles and the Tortoise,” Mind, N.S., vol. xxii. pp. 318–9.
[43] Op. cit.
[44] Aristotle's words are: “The second is the so-called Achilles. It consists in this, that the slower will never be overtaken in its course by the quickest, for the pursuer must always come first to the point from which the pursued has just departed, so that the slower must necessarily be always still more or less in advance.” Phys., vi. 9. 239B (R.P. 137).
[45] Phys., vi. 9. 239B (R.P. 138).
[46] Phys., vi. 9. 239B (R.P. 139).
[47] Loc. cit.
[48] Loc. cit., p. 105.
[49] Phil. Werke, Gerhardt's edition, vol. i. p. 338.
[50] Mathematical Discourses concerning two new sciences relating to mechanics and local motion, in four dialogues. By Galileo Galilei, Chief Philosopher and Mathematician to the Grand Duke of Tuscany. Done into English from the Italian, by Tho. Weston, late Master, and now published by John Weston, present Master, of the Academy at Greenwich. See pp. 46 ff.