ON THE FIRST PAET OF PLATO'S PARMENIDES. 19 that there is any true science, and thus it is by no unfortunate leaning to Pythagorean symbolism but as a necessary logical consequence of his central doctrine that he finally identified the Ideas with the principles of number. How completely he did so can easily be seen from his writings if the mathe- matical physics of the Timceus be studied side by side with the treatment of proportion and symmetry as the essence of goodness and beauty in the Philebus. From our own modern point of view this identification of science with the study of quantity is hardly likely to be judged satisfactory; on the contrary, many circumstances, especially the growth of psychology into a great independent scientific discipline, have led to a growing conviction that there are, or may be, branches of scientific knowledge which are non-quantitative, but the quantitative ideal in science still retains sufficient attractiveness for many minds to enable us to realise how much it meant to the philosopher who formally prescribed the study of geometry as the one propaedeutic to philosophy. But what then in the end, it may be asked, is on this interpretation the answer to the question with which we started, the problem of the relation of Idea to thing ? Simply this ; the Idea as such is not of the same order as the sensible thing, but is connected with it in a peculiar way which can enly be understood by bearing in mind its character as an ideal number. The Idea of the circle, or as we should now say, the circle as defined by its equation in the general form, is not itself properly speaking a curve, that is, is not a unique qualitative form of perceptual extension ; it is a general rule for the construction of curves of a certain kind by the mental synthesis of positions fulfilling a certain relation, and these positions themselves have no existence as parts of per- ceptual actuality ; they do not exist in the perceived circle, but are derived from it by a conceptual analysis. Precisely the same is true of the equation to a particular circle which is got by giving the coefficients of the general equation numer- ical values. Such an equation, like the ideal number, is at once many, as synthesising an indefinite plurality of positions, and one, as synthesising them in accord with a definite law. Let the circle corresponding to such an equation be actually described from any point as centre, and in its unique quality as perceived you have the particular which according to Plato's phrase is what it is in virtue of the " presence " of the Idea. This language about " presence " and " participation " and "likeness" is indeed Plato's way of saying that the qualitatively unique character of the perceived curve, when you come mentally to analyse it, may be replaced by and
