ON THE FIRST PAET OF PLATO S PARMENIDES. 9 stantly recurring to definite examples of the class of scientific problems upon which the whole Platonic theory is demon- strably based. If we will only take this trouble, the purport of the dialogue at once becomes positively perspicuous. Let us begin then by assigning a precise meaning to the aporia of Zeno with which the discussion starts. As briefly sum- marised by Plato, Zeno's argument runs thus ; things cannot be a Many, for, if they are they will be both similar and dissimilar, and this is absurd (127 e). To realise the probable meaning of this we need to remind ourselves that the polemic of Zeno was directed against the Pythagorean view of the composition of geometrical figures out of points, and that its special object was to establish indirectly the continuity of extension. Probably then we ought to interpret the antinomy in some such way as this. If figured extension is made up of points having magnitude as is held by the Pythagoreans, (a) all lines will be straight, and there will be no qualitatively dissimilar curves. For, if the straight line itself is made up of these points, of course the " point " itself will also be a straight line of unit length, and what we commonly call curves of various kinds and orders must therefore be, not approximately but in reality, so many open or closed rectilinear polygons. (b) But again, since lines are made up of points, which are really unit lines, it will also be open to us to argue that there must be as many different kinds of unit lines as there are kinds of curves ; there will have to be not only one unit for the circle and another for the ellipse, but, since the curvature of one circle or the eccentricity of one ellipse is not the same as that of another, there must be a different unit for each circle and for each ellipse ; thus "if things are a Many," they must be at once composed of repetitions of one and the same identical element and of as many qualitatively unique elements as there are " things " in the Many. Thus they are at once "like and unlike". And, I may add, there is a grave objection to taking what at first might seem the simplest way out of the difficulty by adopting the second of these alternatives. That, so long as you regard the line as actually made up of a sum of points having magnitude, you cannot meet the difficulty by regarding the unit of each kind of line as unique is proved by the fact that a curve and a straight line may coincide at one or more points (they may cut or touch). Hence until we find some other explanation of the relation of the line to the points which, to use modern terminology, satisfy its equation, than that which treats the points as constituent parts of the line, Zeno's antinomy is insoluble. It is on the
