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1. notices of the Pythagorean philosophy which have been transmitted to us, whether in its earlier or in its later manifestations, are scanty and extremely obscure. With the later manifestations we need not trouble ourselves. They are founded on spurious data, or at least on data which are not sufficiently authenticated. They are mystical in the extreme, and their symbolism is utterly incomprehensible. The earlier form of the philosophy, in so far as it is extant, is preserved in the fragments of Philolaus, and in a few notices by Aristotle. Philolaus was a contemporary of Socrates, and flourished about 420 B.C. Aristotle was a good deal later: so that there was an interval of nearly a hundred years between Pythagoras, who was in his prime about the year 540, and the earliest expositor of his opinions with whom we are acquainted. These two, Philolaus and Aristotle, are the principal sources of our knowledge of the Pythagorean philosophy in its earlier form. For