while at the same time it remains eternal in the system of the universe; so that things not only have their origin and essence, their place and time according to numerical causes, but each is in effect a number as far as its individual properties and the universal process of cosmic life are concerned.  The reason of the number must depend upon the substance, by the configurations of which it is defined, divided, added, and multiplied, and to this geometry is added, which measures all things in relation to themselves and others.  This eternal cause makes it intelligible that if immaterial principles precede and govern the whole material world, it is also by means of these that the classification of science is in intrinsic agreement with that of nature.  Numbers have their value in music, in gymnastics, in medicine, in morals, in politics, in all branches of science.  The Pythagorean arithmetic is the bond and universal logic of the knowable.  But at the same time Pythagoras and his school peopled the world with demons and genii, which were the causes of disease; they did not abandon the old mythical ideas of the incarnation of spirits and the transmigration of souls—­theories and beliefs which recur in nearly all primitive and savage peoples.

In this vast Pythagorean scheme, which contrasts with that of the Ionic school of physics, thought is more explicitly freed from the ruder mythical ideas, and rises to a more intelligent and rational conception of the world, but the ancient popular traditions still persist, and there is an evident entification of number.  The primitive monad, numbers, their genesis and relations, are not regarded as abstract conceptions, necessary for understanding the order of nature, and a merely logical function of the mind; they are the substantial essence which underlies all mythical representations.  Although the essential life of the world is considered from a more abstract point of view, yet the mythical analogy of animal life evidently finds a place in the breath of the void and of time, assumed to be independent entities.  The subsequent train of beliefs in spirits, of their incarnations and transmigrations, are closely connected with the phantasmagoria of the past, and display their mythical genesis; yet by their deeper and more explicit thought they may be said to infuse intellectual life into the world and into science which relates to it.  In this first rational classification of science by the Greeks, both on its physical and its ideal side, thought sometimes issues in the simple contemplation of manifold nature, while it still continues mythical in its fundamental conceptions and spiritual corollaries; myth, however, instead of being altogether anthropomorphic, begins to become scientific.

I must here be allowed to quote a hymn in the Rig-Veda, which was historically earlier than the primitive philosophy of Greece, but which reveals the same tendency, the same mythical and scientific teaching in its interpretation of the world.  In this hymn, which has been translated and explained by Max Mueller, we see how boldly the problem of the origin of the world is stated (hymn 129, book x.)—­