Anaximander’s system was succeeded by a number of others which cannot be arranged according to any order of linear progression. Such arrangements are, indeed, false in principle. Intellectual life, like every other life, is a product of manifold conditions, and their varied combinations are certain to issue in a corresponding multiplicity of effects. Anaximenes, a fellow-townsman of Anaximander, followed most closely in the footsteps of the master. Attempting, as it would appear, to mediate between his two predecessors, he chose air for a primal element. Air is more omnipresent than water, which, as well as earth, is enclosed within its plastic sphere. On the other hand, it is more tangible and concrete than the Infinite, or may even be substituted for that conception by supposing it to extend as far as thought can reach. As before, cosmogony grows out of cosmography; the enclosing element is the parent of those embraced within it. Speculation now leaves its Asiatic cradle and travels with the Greek colonists to new homes in Italy and Sicily, where new modes of thought were fostered by a new environment. A name, round which mythical accretions have gathered so thickly that the original nucleus of fact almost defies definition, first claims our attention. Aristotle, as is well known, avoids mentioning Pythagoras, and always speaks of the Pythagoreans when he is discussing the opinions held by a certain Italian school. Their doctrine, whoever originated it, was that all things are made out of number. Brandis regards Pythagoreanism as an entirely original effort of speculation, standing apart from the main current of Hellenic thought, and to be studied without reference to Ionian philosophy. Zeller, with more plausibility, treats it as an outgrowth of Anaximander’s system. In that system the finite and the infinite remained opposed to one another as unreconciled moments of thought. Number, according to the Greek arithmeticians, was a synthesis of the two, and therefore superior to either. To a Pythagorean the finite and the infinite were only one among several antithetical couples, such as odd and even, light and darkness, male and female, and, above all, the one and the many whence every number after unity is formed. The tendency to search for antitheses everywhere, and to manufacture them where they do not exist, became ere long an actual disease of the Greek mind. A Thucydides could no more have dispensed with this cumbrous mechanism than a rope-dancer could get on without his balancing pole; and many a schoolboy has been sorely puzzled by the fantastic contortions which Italiote reflection imposed for a time on Athenian oratory.