There are, then, according to Pythagoras, two worlds: one intelligible, which has the monad for an originating principle; and the other sensible. But of this (latter) is the quaternion having the iota, the one tittle, a perfect number. And there likewise is, according to the Pythagoreans, the i, the one tittle, which is chief and most dominant, and enables us to apprehend the substance of those intelligible entities which are capable of being understood through the medium of intellect and of sense. (And in this substance inhere) the nine incorporeal accidents which cannot exist without substance, viz., “quality,” and “quantity,” and “relation,” and “where,” and “when,” and “position,” and “possession,” and “action,” and “passion.” These, then, are the nine accidents (inhering in) substance, and when reckoned with these (substances), contains the perfect number, the i. Wherefore, the universe being divided, as we said, into the intelligible and sensible world, we have also reason from the intelligible (world), in order that by reason we may behold the substance of things that are cognised by intellect, and are incorporeal and divine. But we have, he says, five senses—smelling, seeing, hearing, taste, and touch. Now, by these we arrive at a knowledge of things that are discerned by sense; and so, he says, the sensible is divided from the intelligible world. And that we have for each of these an instrument for attaining knowledge, we perceive from the following consideration. Nothing, he says, of intelligibles can be known to us from sense. For he says neither eye has seen, nor ear heard, nor any whatsoever of the other senses known that (which is cognised by mind). Neither, again, by reason is it possible to arrive at a knowledge of any of the things discernible by sense. But one must see that a thing is white, and taste that it is sweet, and know by hearing that it is musical or out of tune. And whether any odour is fragrant or disagreeable, is the function of smell, not of reason. It is the same with objects of touch; for anything rough, or soft, or warm, or cold, it is not possible to know by hearing, but (far from it), for touch is the judge of such (sensations). Things being thus constituted, the arrangement of things that have been made and are being made is observed to happen in conformity with numerical (combinations). For in the same manner as, commencing from monad, by an addition of monads or triads, and a collection of the succeeding numbers, we make some one very large complex whole of number; (and) then, again, from an amassed number thus formed by addition, we accomplish, by means of a certain subtraction and re-calculation, a solution of the totality of the aggregate numbers; so likewise he asserts that the world, bound by a certain arithmetical and musical chain, was, by its tension and relaxation, and by addition and subtraction, always and for ever preserved incorrupt.