in the manner in which it was taken up by Philolaus, Eurytas and Archytas, since its earlier adherents left nothing in a written form. 3. THE PYTHAGOREAN PRINCIPLE.—The ancients are united in affirming that the principle of the Pythagorean philosophy was number. But in what sense was this their principle—in a material or a formal sense? Did they hold number as the material of things, _i. e._ did they believe that things had their origin in numbers, or did they regard it as the archetype of things, _i. e._ did they believe that things were made as the copy or the representation of numbers? From this very point the accounts given by the ancients diverge, and even the expressions of Aristotle seem to contradict each other. At one time he speaks of Pythagoreanism in the former, and at another in the latter sense. From this circumstance modern scholars have concluded that the Pythagorean doctrine of numbers had different forms of development; that some of the Pythagoreans regarded numbers as the substances and others as the archetypes of things. Aristotle, however, gives an intimation how the two statements may be reconciled with each other. Originally, without doubt, the Pythagoreans regarded number as the material, as the inherent essence of things, and therefore Aristotle places them together with the Hylics (the Ionic natural philosophers), and says of them that “they held things for numbers” (_Metaph._ I., 5, 6). But as the Hylics did not identify their matter, _e. g._ water, immediately with the sensuous thing, but only gave it out as the fundamental element, as the original form of the individual thing, so, on the other side, numbers also might be regarded as similar fundamental types, and therefore Aristotle might say of the Pythagoreans, that “they held numbers to be the corresponding original forms of being, as water, air, &c.” But if there still remains a degree of uncertainty in the expressions of Aristotle respecting the sense of the Pythagorean doctrine of numbers, it can only have its ground in the fact that the Pythagoreans did not make any distinction between a formal and material principle, but contented themselves with the undeveloped view, that, “number is the essence of things, every thing is number.”