a. The Pythagoreans say that the first simple Notion is unity (_μονάς_); not the discrete, multifarious, arithmetic one, but identity as continuity and positivity, the entirely universal essence. They further say, according to Sextus (adv. Math. X. 260, 261): “All numbers come under the Notion of the one; for duality is one duality and triplicity is equally a ‘one,’ but the number ten is the one chief number. This moved Pythagoras to assert unity to be the principle of things, because, through partaking of it, each is called one.” That is to say, the pure contemplation of the implicit being of a thing is the one, the being like self; to all else it is not implicit, but a relationship to what is other. Things, however, are much more determined than being merely this dry “one.” The Pythagoreans have expressed this remarkable relationship of the entirely abstract one to the concrete existence of things through “simulation” (_μίμησις_). The same difficulty which they here encounter is also found in Plato’s Ideas; since they stand over against the concrete as species, the relation of concrete to universal is naturally an important point. Aristotle (Metaph. I. 6) ascribes the expression “participation” (_μέθεξις_) to Plato, who took it in place of the Pythagorean expression “simulation.” Simulation is a figurative, childish way of putting the relationship; participation is undoubtedly more definite. But Aristotle says, with justice, that both are insufficient; that Plato has not here arrived at any further development, but has only substituted another name. “To say that ideas are prototypes and that other things participate in them is empty talk and a poetic metaphor; for what is the active principle that looks upon the ideas?” (Metaph. I. 9). Simulation and participation are nothing more than other names for relation; to give names is easy, but it is another thing to comprehend. b. What comes next is the opposition, the duality (_δυάς_), the distinction, the particular; such determinations have value even now in Philosophy; Pythagoras merely brought them first to consciousness. Now, as this unity relates to multiplicity, or this being-like-self to being another, different applications are possible, and the Pythagoreans have expressed themselves variously as to the forms which this first opposition takes.