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. [51] In his Grundlagen einer allgemeinen Mannichfaltigkeitslehre and in articles in Acta Mathematica, vol. ii. [52] The definition of number contained in this book, and elaborated in the Grundgesetze der Arithmetik (vol. i., 1893; vol. ii., 1903), was rediscovered by me in ignorance of Frege's work. I wish to state as emphatically as possible�what seems still often ignored�that his discovery antedated mine by eighteen years. [53] Giles, The Civilisation of China (Home University Library), p. 147. [54] Cf. Principia Mathematica, � 20, and Introduction, chapter iii. [55] In the above remarks I am making use of unpublished work by my friend Ludwig Wittgenstein. [56] Thus we are not using �thing� here in the sense of a class of correlated �aspects,� as we did in Lecture III. Each �aspect� will count separately in stating causal laws. [57] The above remarks, for purposes of illustration, adopt one of several possible opinions on each of several disputed points.