[Footnote 428: See Plato, "Timæus," ch. ix. p. 331 (Bohn's edition); Aristotle's "Metaphysics," bk. v. ch. iii.] Aristotle affirms that Pythagoras taught "that _numbers_ are the first principles of all entities," and, "as it were, a _material_ cause of things,"[429] or, in other words, "that numbers are substances that involve a separate subsistence, and are primary causes of entities."[430] [Footnote 429: Aristotle's "Metaphysics," bk. i. ch. v.] [Footnote 430: Id., ib., bk. xii. ch. vi.] Are we then required to accept the dictum of Aristotle as final and decisive? Did Pythagoras really teach that numbers are real entities--the _substance_ and cause of all other existences? The reader may be aware that this is a point upon which the historians of philosophy are not agreed. Ritter is decidedly of opinion that the Pythagorean formula "can only be taken symbolically."[431] Lewes insists it must be understood literally.[432] On a careful review of all the arguments, we are constrained to regard the conclusion of Ritter as most reasonable. The hypothesis "that numbers are real entities" does violence to every principle of common sense. This alone constitutes a strong _à priori_ presumption that Pythagoras did not entertain so glaring an absurdity. The man who contributed so much towards perfecting the mathematical sciences, who played so conspicuous a part in the development of ancient philosophy, and who exerted so powerful a determining influence on the entire current of speculative thought, did not obtain his ascendency over the intellectual manhood of Greece by the utterance of such enigmas. And further, in interpreting the philosophic opinions of the ancients, we must be guided by this fundamental canon--"The human mind has, under the necessary operation of its own laws, been compelled to entertain the same fundamental ideas, and the human heart to cherish the same feelings in all ages." Now if a careful philosophic criticism can not render the _reported_ opinions of an ancient teacher into the universal language of the reason and heart of humanity, we must conclude either that his opinions were misunderstood and misrepresented by some of his successors, or else that he stands in utter isolation, both from the present and the past. His doctrine has, then, no relation to the successions of thought, and no place in the history of philosophy. Nay, more, such a doctrine has in it no element of vitality, no germ of eternal truth, and must speedily perish. Now it is well known that the teaching of Pythagoras awakened the deepest intellectual sympathy of his age; that his doctrine exerted a powerful influence on the mind of Plato, and, through him, upon succeeding ages; and that, in some of its aspects, it now survives, and is more influential to-day than in any previous age; but this element of immutable and eternal truth was certainly not contained in the inane and empty formula, "that numbers are real existences, the causes of all