[17] The assumptions made concerning time-relations in the above are as follows:� [18] The above paradox is essentially the same as Zeno's argument of the stadium which will be considered in our next lecture. [19] See next lecture. [20] Monist, July 1912, pp. 337�341. [21] �Le continu math�matique,� Revue de M�taphysique et de Morale, vol. i. p. 29. [22] In what concerns the early Greek philosophers, my knowledge is largely derived from Burnet's valuable work, Early Greek Philosophy (2nd ed., London, 1908). I have also been greatly assisted by Mr D. S. Robertson of Trinity College, who has supplied the deficiencies of my knowledge of Greek, and brought important references to my notice. [23] Cf. Aristotle, Metaphysics, M. 6, 1080b, 18 sqq., and 1083b, 8 sqq. [24] There is some reason to think that the Pythagoreans distinguished between discrete and continuous quantity. G. J. Allman, in his Greek Geometry from Thales to Euclid, says (p. 23): �The Pythagoreans made a fourfold division of mathematical science, attributing one of its parts to the how many, t? p?s??, and the other to the how much, t? p??????; and they assigned to each of these parts a twofold division. For they said that discrete quantity, or the how many, either subsists by itself or must be considered with relation to some other; but that continued quantity, or the how much, is either stable or in motion. Hence they affirmed that arithmetic contemplates that discrete quantity which subsists by itself, but music that which is related to another; and that geometry considers continued quantity so far as it is immovable; but astronomy (t?? sfa??????) contemplates continued quantity so far as it is of a self-motive nature. (Proclus, ed. Friedlein, p. 35. As to the distinction between t? p??????, continuous, and t? p?s??, discrete quantity, see Iambl., in Nicomachi Geraseni Arithmeticam introductionem, ed. Tennulius, p. 148.)� Cf. p. 48. [25] Referred to by Burnet, op. cit., p. 120. [26] iv., 6. 213b, 22; H. Ritter and L. Preller, Historia Philosophi� Gr�c�, 8th ed., Gotha, 1898, p. 75 (this work will be referred to in future as �R. P.�). [27] The Pythagorean proof is roughly as follows. If possible, let the ratio of the diagonal to the side of a square be m/n, where m and n are whole numbers having no common factor. Then we must have m2 = 2n2. Now the square of an odd number is odd, but m2, being equal to 2n2, is even. Hence m must be even. But the square of an even number divides by 4, therefore n2, which is half of m2, must be even. Therefore n must be even. But, since m is even, and m and n have no common factor, n must be odd. Thus n must be both odd and even, which is impossible; and therefore the diagonal and the side cannot have a rational ratio. [28] In regard to Zeno and the Pythagoreans, I have derived much valuable information and criticism from Mr P. E. B. Jourdain.