large numbers and complex relations of numbers, which otherwise we could only know indirectly by fixing them in concepts. Therefore by the union of these two very different but correct views of music we may arrive at a conception of the possibility of a philosophy of number, such as that of Pythagoras and of the Chinese in Y-King, and then interpret in this sense the saying of the Pythagoreans which Sextus Empiricus quotes (adv. Math., L. vii.): τῳ αριθμῳ δε τα παντ᾿ επεοικεν (numero cuncta assimilantur). And if, finally, we apply this view to the interpretation of harmony and melody given above, we shall find that a mere moral philosophy without an explanation of Nature, such as Socrates wanted to introduce, is precisely analogous to a mere melody without harmony, which Rousseau exclusively desired; and, in opposition to this mere physics and metaphysics without ethics, will correspond to mere harmony without melody. Allow me to add to these cursory observations a few more remarks concerning the analogy of music with the phenomenal world. We found in the second book that the highest grade of the objectification of will, man, could not appear alone and isolated, but presupposed the grades below him, as these again presupposed the grades lower still In the same way music, which directly objectifies the will, just as the world does, is complete only in full harmony. In order to achieve its full effect, the high leading voice of the melody requires the accompaniment of all the other voices, even to the lowest bass, which is to be regarded as the origin of all. The melody itself enters as an integral part into the harmony, as the harmony enters into it, and only thus, in the full harmonious whole, music expresses what it aims at expressing. Thus also the one will outside of time finds its full objectification only in the complete union of all the steps which reveal its nature in the innumerable ascending grades of distinctness. The following analogy is also very remarkable. We have seen in the preceding book that not-
