¹ Elea, a city in Lucania, was founded by that energetic seafaring people, the Phocaeans. Zeno's ‘floruit’ was probably about 460 B.C.

² E.g. Tannery, P.: Pour l'histoire de la science héllène (2 ^me éd. Paris 1930), pp. 255–70Google Scholar; Rev. Phil. xx, 1885, pp. 385–410 Google Scholar. Brochard, V.: Rev. de Mét. et Mor. i, 1893, pp. 209–15Google Scholar, etc. Noel, G.: Rev. de Mét. et Mor. i, 1893, pp. 108–25Google Scholar.

³ Russell, Bertrand: Principles of Mathematics (1937), pp. 347 ff.Google Scholar; Our Knowledge of the External World (1926), pp. 129 ff.Google Scholar; History of Western Philosophy (1948); Chapter xxviii; Monist, July 1912, pp. 337–41Google Scholar; and The Philosophy of Bergson (Bowes & Bowes, Cambridge, 1914)Google Scholar.

⁴ Aristotle's text of this argument contains one or two textual difficulties, and there are differences of opinion about some minor details of the arrangement of the A's, B's and C's; but the main sense is not in any doubt.

⁵ Plato apparently had a theory of ‘Indivisible Lines’. See Nicol, A. T., ‘Indivisible Lines’, C.Q. xxx, 1936 Google Scholar.

⁶ Another possible explanation is that Aristotle's ‘indivisible now’ was a vague expression, and did not necessarily imply a minimal period. In Physics 234822 Aristotle applies the term ‘indivisible’ to his own ‘now’ (which is a limit, and has no magnitude). If this is the correct explanation here, then Aristotle's remarks in Physics viii should be taken as applying particularly to the first two paradoxes, though they also help out his answer to the third.

⁷ For the theory of mathematical continuity, see Dedekind, R.: Stetigkeit und irrationale Zahlen, Braunschweig, 1872 Google Scholar; Was sind und was sollen die Zahlen, Braunschweig, 1888 Google Scholar. Cantor, Georg: Grundlagen einer allgemeinen Mannichfaltigkeilslehre, Leipzig, 1883 Google Scholar. Hobson, E. W.: ‘On the Infinite and the Infinitesimal in Mathematical Analysis’, Proceedings of the London Mathematical Society, Vol. 35, London, 1903, p. 117 Google Scholar.

Not being a trained mathamatician, I have had to rely too much on Earl Russell's accounts of mathematical continuity. Further discussion appears in the Aristotelian Society's supplementary Volume 4, Concepts of Continuity, 1924; the philosophers of this time evidently felt that relativity theory and quantum theory ought to be taken into account. There is also a most interesting discussion in Mind, 1946, pp. 151–65Google Scholar, written by Ushenko, Andrew (who refers to previous articles in Mind, pp. 58–73 and 310–11Google Scholar, and Mind, 1942, pp. 89–90 Google Scholar. Ryle, Gilbert in Dilemmas (1953)Google Scholar devotes a chapter to Zeno's paradoxes.

Earl Russell seems to have been curiously anxious, at one time or another of his career, to defend the idea of mathematical continuity against philosophies such as that of Bergson. Bergson seems to have had a ‘dynamic’ view of the external world, and to have thought that mathematics was a construction of the human brain, far too rigid and static to bear any real relation to the dynamic world outside. This physical theory was evidently accompanied by a rather ‘fluid’ attitude towards social and political matters. In the Philosophy of Bergson (1914), Earl Russell was at pains to defend not only mathematical physics (the imposition of order on the physical world), but also justice (the imposition of order on the social world) against Bergsonian fluidity. How like the ancient quarrel between Plato and the Sophists!

⁸ In one respect, Aristotle's theory is very different; his Potential-Actual distinction is bound up with an exciting teleological theory of nature, in which each thing strives to realise its own most perfect form. It is interesting to compare Andrew Ushenko's discussion in Mind, 1946, pp. 151–65Google Scholar.

⁹ See also Raven, J. E., Pythagoreans and Eleatics (1948)Google Scholar.

¹⁰ The fallacy may also be taken to be a confusion between ‘identical’ and ‘similar’.

¹¹ Aristotle, Physics 213b22; Stobaeus Ecl. 1, 18, Diels 58B30).

¹² Anaximander and Heraclitus.

¹³ E.g. Physics 234324 ff., where Aristotle argues that neither rest nor motion is possible in the now. Motion in the now is impossible, because if A moves faster than B, and B covers a certain distance in the now, then A could accomplish the same distance in less time than a now, so that the supposedly indivisible now would have to be divided. Rest in the now is impossible, because ‘we say a thing is at rest when it has not changed its position, either in respect to its totality, or in respect to its parts, between now and then; but there is no then in now, so there is no being at rest’. (Aristotle gives two other arguments.)

¹⁴ Elias, Diels 29A15. Uncertain evidence.

¹⁵ Cf. Plato, Sophist 240C–endGoogle Scholar. Plato refutes Parmenides by saying that, when we say A is not B, we mean A is other than B; and in this sense, not-being is possible. It has, however, been suggested to me that these arguments of Plato's were directed against Sophists who made capital out of Parmenides' thesis, rather than against Parmenides himself.

¹⁶ Parmenides was perhaps the first to deny the reality of the world of sense, and it is interesting to note that his for doing so was, probably, a confusion between the two senses of ‘be’. Plato relegated our awareness of the sensible world to mere opinion, because sensible objects are always changing; they ‘are’ and ‘are not’, and never securely ‘are’.

¹⁷ This does not sound like dialogue form; Tannery tried to make a dialogue out of Zeno's paradoxes. The suggestion is rather of a number of arguments, one after another, each perhaps containing one or more hypotheses, This proves nothing in regard to the suggested systematisations of the arguments on Motion.

¹⁸ By Raven, J. E., Pythagoreans and Eleatics (1948), p. 72 Google Scholar. For the anti-Pythagorean interpretation of Zeno's arguments, see P. Tannery, op cit. (see my note 2); F. M. Cornford, C.Q. xvi and xvii (1922 and 1923), and Plato and Parmenides (1939); J. E. Raven, op. cit.; H. D. P. Lee, op. cit. (the last two with considerable modifications). For criticisms of the theory, see G. Calogero and B. L. van der Waerden (works quoted by me in the introduction to my account of the motion arguments), and also Heidel, W. A. (A.J.P. 61, 1940, pp. 21–30)Google Scholar.

¹⁹ The argument about the point is ascribed to Zeno by Aristotle Metaph. 1001b7, and so is presumably genuine enough.

²⁰ Simplicius 140.27. Very definitely ascribed to Zeno by Simplicius.

²¹ Simplicius 140.34. Again, very definitely ascribed to Zeno.

²² Philoponus 80.23 ff. Ascription to Zeno less definite; Philoponus may be doing some analysis of his own.

²³ I cannot agree with G. Calogero's translation (op. cit., p. 99) ‘e necessario … che si distingua da essa, come da altra, quella che rispetto ad essa e altra.’ We have indeed to demand a ‘ricca pregnanza di senso’ if this is the sense of the Greek.

²⁴ That Simplicius took προύχοντος to imply the next term in the process of ‘dichotomy’ (or perhaps the prior one?) is proved by the following passage: (Simplicius 139.16–18). But Simplicius was perfectly capable of misunderstanding an argument. Other suggestions include the ideas that προ- implies the ‘prior’ term, and that ‘that-which-is-beyond’—the jutting out piece—implies the piece separating the first two units. The word προύχοντος makes the first suggestion difficult, and if the second explanation were the true one why should not Zeno say the ‘piece in-between’? The suggestion that Zeno is thinking of a series of geometrical points in a line does not convince me; such a rendering seems inconsistent with the actual language used by Zeno in this argument. Possibly his ‘one part separated from another’ implies the two limits at either end of an extension; but if so, they are (for the purposes of this argument) limits having magnitude.

²⁵ op. cit., p. 106.

²⁶ Whether Aristotle's answers are finally satisfactory, is a doubtful matter; but certainly they were a necessary advance in his own time.

²⁷ See Eudemus's remarks quoted in Simplicius 97.11 ff.

²⁸ See Aristotle, Metaph. 992a20 ff.

²⁹ An argument against ‘Place’ might help our arguments against Plurality and Motion in a variety of ways. Place might have seemed necessary so that a plurality of objects might be separated out (plurality), and also so that things might change place (motion). But compare J. E. Raven, op cit., pp. 81–2.

Addendum. In this article I am much indebted to Mr. F. P. Chambers, of the London School of Economics; also to Professor W. K. C. Guthrie, Mr. H. D. P. Lee, and Mr. J. E. Raven for criticism, comments and encouragement.