(5) Pleasures are of two kinds, the mixed and unmixed. Of mixed pleasures there are three classes--(a) those in which both the pleasures and pains are corporeal, as in eating and hunger; (b) those in which there is a pain of the body and pleasure of the mind, as when you are hungry and are looking forward to a feast; (c) those in which the pleasure and pain are both mental. Of unmixed pleasures there are four kinds: those of sight, hearing, smell, knowledge. (6) The sciences are likewise divided into two classes, theoretical and productive: of the latter, one part is pure, the other impure. The pure part consists of arithmetic, mensuration, and weighing. Arts like carpentering, which have an exact measure, are to be regarded as higher than music, which for the most part is mere guess-work. But there is also a higher arithmetic, and a higher mensuration, which is exclusively theoretical; and a dialectical science, which is higher still and the truest and purest knowledge. (7) We are now able to determine the composition of the perfect life. First, we admit the pure pleasures and the pure sciences; secondly, the impure sciences, but not the impure pleasures. We have next to discover what element of goodness is contained in this mixture. There are three criteria of goodness--beauty, symmetry, truth. These are clearly more akin to reason than to pleasure, and will enable us to fix the places of both of them in the scale of good. First in the scale is measure; the second place is assigned to symmetry; the third, to reason and wisdom; the fourth, to knowledge and true opinion; the fifth, to pure pleasures; and here the Muse says 'Enough.' 'Bidding farewell to Philebus and Socrates,' we may now consider the metaphysical conceptions which are presented to us. These are (I) the paradox of unity and plurality; (II) the table of categories or elements; (III) the kinds of pleasure; (IV) the kinds of knowledge; (V) the conception of the good. We may then proceed to examine (VI) the relation of the Philebus to the Republic, and to other dialogues. I. The paradox of the one and many originated in the restless dialectic of Zeno, who sought to prove the absolute existence of the one by showing the contradictions that are involved in admitting the existence of the many (compare Parm.). Zeno illustrated the contradiction by well-known examples taken from outward objects. But Socrates seems to intimate that the time had arrived for discarding these hackneyed illustrations; such difficulties had long been solved by common sense ('solvitur ambulando'); the fact of the co-existence of opposites was a sufficient answer to them. He will leave them to Cynics and Eristics; the youth of Athens may discourse of them to their parents. To no rational man could the circumstance that the body is one, but has many members, be any longer a stumbling-block.