But these remarks have been extended somewhat beyond what the occasion requires. However, it is clear that men make Five an attribute of the god, which at one time of itself creates itself, like fire, and at another time out of itself creates ten, like the universe. And in music, which is especially pleasing to him, do we imagine that this number plays no part? For the main application of harmony, so far as it can be put into words, is concerned with chords. That these are five, and no more, reason convinces anyone who wishes, by perception alone without employing reason, to pursue these matters on the strings and stops Cf. Plato, Republic , 530 d - 531 c. ; for they all have their origin in numerical ratios. The ratio of the fourth is four to three, Cf . Moralia , 1018 e. that of the fifth is three to two, and that of the octave two to one; that of the octave plus the fifth is three to one, Cf. 429 e, infra . and that of the double octave four to one. The extra chord which the writers on harmony introduce, naming it the octave and the fourth extra metrum , does not deserve acceptance, since we should be favouring the unreasoning element in our sense of hearing contrary to reason, which is as much as to say, contrary to law. Now if I may omit any discussion of the five stops of the tetrachord, Cf. 430 a, infra , and Moralia , 1021 e and 1029 a. and the first five tones or tropes or harmonies, whatever be their right name, from the changes in which, through a greater or a less tension, the remaining lower and higher notes are derived, I must ask whether, although the intervals are numerous, or rather of infinite number, yet the elements of melody are not five only, Cf. 430 a, infra . quarter tone, half tone, tone, a tone plus a half tone, and double tone; and there is, in the range of notes, no additional space, either smaller or greater within the limits set by the high and the low, which can yield melody. There are many other examples of this sort of thing, said I, which I shall pass over. I shall merely adduce Plato, Plato, Timaeus , 31 a. who, in speaking about a single world, says that if there are others besides ours, and ours is not the only one, then there are five altogether and no more. Cf . Moralia , 421 f, 422 f, 430 b, and 887 b. Nevertheless, even if this world of ours is the only one ever created, as Aristotle De Caelo , i. 8-9 (276 a 18). also thinks, even ours, he says, is in a way put together through the union of five worlds, of which one is of earth, another of water, a third of fire, a fourth of air; and the fifth, the heavens, others call light, and others aether, and others call this very thing a fifth substance (Quintessence), which alone of the bodies has by nature a circular motion that is not the result of any compelling powrer or any other incidental cause. Wherefore also Plato, apparently noting the five most beautiful and complete forms among those found in Nature, pyramid, cube, octahedron, icosahedron, and dodecahedron, appropriately assigned each to each. There are some who associate the senses also, since they are of the same number, with those primal elements, observing that touch functions against something resistant, and is earthly, and that taste, through moisture in the things tasted, absorbs their qualities. Air, when it is struck, becomes voice or sound in the hearing of it. Of the two remaining senses, odour, which the sense of smell has received as its portion, since it is an exhalation and is engendered by heat, bears a resemblance to fire; and in sight, which flashes to its goal owing to its kinship with aether and light, there occurs a combination and coalescence of the two, which behaves as they do. The living being possesses no other sense, nor has the world any other nature single and uncombined; but a marvellous distribution and apportionment each to each has, as it seems, been made of the five to the five. Therewith I checked myself and, after waiting a moment, said, What ails us, Eustrophus, that we all but passed over Homer Il. xv. 187. as if he were not the first to divide the world into five parts? For he duly assigned the three in the middle to the three gods, and the two extremes, the heaven and the earth, of which the one is the boundary of things below and the other of things above, he left to all in common, undistributed. But the discussion must be carried further back, as Euripides Nauck, Trag. Graec. Frag. , Euripides, no. 970; repeated in 431 a, infra . remarks. For those who exalt Four teach us a lesson that is not without value, that by reason of this number all solids have come into being. For since every such solid body exists through the acquisition of depth by length and breadth, and for length must be presupposed a single point assigned to unity, and length without breadth, which is called a line, is also duality, and the movement of the line breadthwise generates a plane in the third instance, and when depth is added, through the four factors the increase progresses to a solid — it is clear to everyone that four, when it has carried Nature forward to the point of completing a solid body and producing a volume that may be felt and that is resistant, has then left Nature lacking in the most important thing of all. For the inanimate thing is, to put it simply, orphaned, incomplete, and good for nothing, unless there be an animating soul to make use of it. The impulse or dispensation that creates the soul therein, a transformation brought about through five factors in all, gives to Nature its due completeness, and is as much more potent than four as the living being differs in worth from the inanimate thing. Moreover, the symmetry and power of five, rather than that of any other number, has prevailed and has not permitted the animate to progress to unlimited classes of beings, but has produced five forms of all living things. For there are, as we know, gods, demigods, and heroes, and after these the fourth class, man Cf. 415 b, infra . ; and fifth and last the class of unreasoning animals. If you should, moreover, make divisions of the soul itself to accord with Nature, the first and least clear part of it is the nutritive, second the perceptive, then the appetitive, and, next after this, the spirited; but when it had reached the power to reason, and had completed its nature, it came to rest there at the fifth element as at the highest point. Cf. 429 e, infra . Of this number, which has so many and such great powers, the origin also is fair and lovely; not that which we have expounded, that it is composed of two and three, but that which the beginning combined with the first square produces. For the beginning of all number is one, and the first square is four Cf. 429 e, infra . ; and from these, as though from perfected form and matter, comes five. And if certain authorities are right, who, as we know, posit one as the first square, since it is a power of itself and its product is itself, then five, the offspring of the first two squares, does not lack a surpassing nobility of lineage.