3. ANAXIMENES.—Anaximenes, who is called by some the scholar, and by others the companion of Anaximander, turned back more closely to the view of Thales, in that he made air as the principle of all things. The perception that air surrounds the whole world, and that breath conditions the activity of life, seems to have led him to his position. 4. RETROSPECT.—The whole philosophy of the three Ionic sages may be reduced to these three points, viz:—(1.) They sought for the universal essence of concrete being; (2.) They found this essence in a material substance or substratum; (3.) They gave some intimation respecting the derivation of the elements from this original matter. SECTION V. PYTHAGOREANISM. 1. ITS RELATIVE POSITION.—The development of the Ionic philosophy discloses the tendency to abstract matter from all else; though they directed this process solely to the determined _quality_ of matter. It is this abstraction carried to a higher step, when we look away from the sensible concretions of matter, and no more regard its _qualitative_ determinateness as water, air, &c., but only direct our attention to its _quantitative_ determinateness,—to its space-filling property. But the determinateness of quantity is number, and this is the principle and standpoint of Pythagoreanism. 2. HISTORICAL AND CHRONOLOGICAL.—The Pythagorean doctrine of numbers is referred to Pythagoras of Samos, who is said to have flourished between 540 and 500 B. C. He dwelt in the latter part of his life at Crotonia, in Magna Grecia, where he founded a society, or, more properly, an order, for the moral and political regeneration of the lower Italian cities. Through this society, this new direction of philosophy seems to have been introduced,—though more as a mode of life than in the form of a scientific theory. What is related concerning the life of Pythagoras, his journeys, the new order which he founded, his political influence upon the lower Italian cities, &c., is so thoroughly interwoven with traditions, legends, and palpable fabrications, that we can be certain at no point that we stand upon a historical basis. Not only the old Pythagoreans, who have spoken of him, delighted in the mysterious and esoteric, but even his new-Platonistic biographers, Porphyry and Jamblichus, have treated his life as a historico-philosophical romance. We have the same uncertainty in reference to his doctrines, _i. e._ in reference to his share in the number-theory. Aristotle, _e. g._ does not ascribe this to Pythagoras himself, but only to the Pythagoreans generally, _i. e._ to their school. The accounts which are given respecting his school have no certainty till the time of Socrates, a hundred years after Pythagoras. Among the few sources of light which we have upon this subject, are the mention made in Plato’s Phædon of the Pythagorean Philolaus and his doctrines, and the writings of Archytas, a cotemporary of Plato. We possess in fact the Pythagorean doctrine only