Page:Mind (New Series) Volume 12.djvu/25

 ON THE FIEST PAET OF PLATO'S PARMENIDES. 11 whole or only in part. Then in the first case the Idea loses its unity by multiplication, and there is a distinct Idea inherent in each particular belonging to the given class ; in the second, it loses its unity by indefinite subdivision, and the impossible consequences of this supposition are followed out at somewhat greater length. It will be worth our while to illustrate both sides of this new antinomy by refer- ence to the same body of mathematical conceptions with which we have all along been dealing. The first of the alternatives suggested by Parmenides would be realised if e.g., every accurately drawn geometrical circle were in every respect an exact facsimile of every other, just as every correctly struck exemplar of the same coin would be of every other coin struck with equal accuracy from the same die. This would be the case if in the general equation of the curve the coefficients and the independent term had fixed numerical values, so that the only remaining difference between two circles, two parabolas, etc., would be that of position of the origin of co-ordinates, as determined in turn by reference to some standard system of axes. On these terms and on no others would it be possible for the "Idea" of the curve, -i.e., the relation expressed by the general equation, to be entirely exhausted in the single exemplar. This way of regarding the relation of the curve to its "equation," it should be observed, would naturally follow from the Eleatic view of the extended as a mere continuum. For the moment you try to explain how e.g. two circles can be equally circles and yet have different curvature, you are driven to fall back upon the conception of the circle as a form of relation between a plurality of elements of some kind or another, and thus to admit the discontinuity, in some sense yet to be determined, of the extended. Whether you regard these elements in Pythagorean fashion as con- stituent "parts" or not makes no difference to this result. The second alternative again apparently corresponds to what we know to have been the Pythagorean view of the nature of extension ; the " Idea " is now supposed to be itself com- posed of an indefinite plurality of parts which are outside one another. Thus the circle, for instance, is taken as being simply formed by the repetition of an indivisible unit line or Pythagorean point, and it will follow that, as any one actually perceived circle only contains a limited number of these units, part of the " Idea " will constitute this particular curve, and another part, i.e., other similar units, some other curve of the same kind. Such at least was the Pythagorean view of the straight line, and it is reasonable to suppose that other lines