14 A. E. TAYLOR : remains but to say it must have been made less by the addition. In fact a quantity less than the unit, if there were such an indivisible minimum as the unit, would be, as we should now say, a negative quantity ; to add it to any- thing is to subtract from the thing. In other words, there can be no such thing as an absolute minimum of quantity which is not zero. If understood in this way, the argument together with its predecessors forms just such an antinomy as that by which the historical Zeno contended that, on the Pythagorean view, every magnitude must be either infinity or zero. This would agree well with our suggestion that one half of Plato's general argument is directed against Eleatic and the other against Pythagorean theories of the extension studied by the geometer. We now see why Parmenides proceeds straight from the considerations we have dealt with to the " third man " objec- tion to the Ideas (132 a ff.). For that objection simply states in a general form the principle of the Pythagorean error already exposed. In each of the three puzzles just exhibited the source of our difficulties was that by treating extension as made up of indivisible units we were driven to assume an infinite number of successively diminishing orders of these units. Thus our Idea turned out to be " not one but in- definitely numerous ". The passage in which this famous crux is brought forward then does not raise a new difficulty, but simply puts the old one in an abstract form. Hence if we wish to know what was Plato's answer to the "jthird man " argument, which Aristotle seems to have thought so irrefutable, we must find out his conception of the relation between the geometrical curve as defined by its characteristic property and the directly perceived curve accessible to sense. Or, what is the same thing, we must discover the theory of extension by which he hoped to escape at once from the Eleatic and from the Pythagorean side of the dilemma which arises when only one extension, the purely perceptual, is admitted, and it is then asked whether this perceptual ex- tension is continuous or discrete. On the "third man " difficulty itself I do not propose to add any observations. The method by which Socrates pro- poses to evade the difficulty will, however, repay a brief examination. He suggests, and some rash readers have held that Plato did wrong in rejecting the suggested solution, that the Idea is merely a concept in the mind (132 b). At first sight this no doubt seems a good answer to the question how it can be at once one and many, for common-sense finds
