For whenever the middle term of any three numbers, cubic or square,Dealing first with square numbers, the proportion here indicated is—a2 : ab :: ab :: b2; conversely, b2 : ab :: ab :: a2; alternately, ab : a2 :: b2 : ab. is such that as the first term is to it, so is it to the last term, and again, conversely, as the last term is to the middle, so is the middle to the first,—then the middle term becomes in turn the first and the last, while the first and last become in turn middle terms, and the necessary consequence will be that all the terms are interchangeable, and being interchangeable they all form a unity. Now if the body of the All had had to come into existence as a plane surface, having no depth, one middle term would have sufficed to bind together both itself and its fellow-terms; but now it is otherwise: for it behoved it to be solid of shape, and what brings solids into unison is never one middle term alone but always two.Two mean terms are required for a continuous proportion of solid (or cubic) numbers, e.g. a3 :a2b :: a2b : ab2 :: ab2 : b3. Thus it was that in the midst between fire and earth God set water and air, and having bestowed upon them so far as possible a like ratio one towards another—air being to water as fire to air, and water being to earth as air to water, —he joined together and constructed a Heaven visible and tangible. For these reasons and out of these materials, such in kind and four in number, the body of the Cosmos was harmonized by proportion and brought into existence. These conditions secured for it Amity, so that being united in identity with itself it became indissoluble by any agent other than Him who had bound it together. Now of the four elements the construction of the Cosmos had taken up the whole of every one. For its Constructor had constructed it of all the fire and water and air and earth that existed, leaving over, outside it, no single particle or potency of any one of these elements. And these were his intentions: first, that it might be, so far as possible, a Living Creature, perfect and whole, with all its parts perfect;

and next, that it might be One, inasmuch as there was nothing left over out of which another like Creature might come into existence; and further, that it might be secure from age and ailment, since He perceived that when heat and cold, and all things which have violent potencies, surround a composite body from without and collide with it they dissolve it unduly and make it to waste away by bringing upon it ailments and age. Wherefore, because of this reasoning, He fashioned it to be One single Whole, compounded of all wholes, perfect and ageless and unailing. And he bestowed on it the shape which was befitting and akin. Now for that Living Creature which is designed to embrace within itself all living creatures the fitting shape will be that which comprises within itself all the shapes there are; wherefore He wrought it into a round, in the shape of a sphere, equidistant in all directions from the center to the extremities, which of all shapes is the most perfect and the most self-similar, since He deemed that the similar is infinitely fairer than the dissimilar. And on the outside round about, it was all made smooth with great exactness, and that for many reasons. For of eyes it had no need, since outside of it there was nothing visible left over; nor yet of hearing, since neither was there anything audible; nor was there any air surrounding it which called for respiration; nor, again, did it need any organ whereby it might receive the food that entered and evacuate what remained undigested. For nothing went out from it or came into it from any side, since nothing existed; for it was so designed as to supply its own wastage as food for itself, and to experience by its own agency and within itself all actions and passions, since He that had constructed it deemed that it would be better if it were self-sufficing rather than in need of other things. Hands, too, He thought He ought not to attach unto it uselessly, seeing they were not required either for grasping or for repelling anyone; nor yet feet, nor any instruments of locomotion whatsoever.

Midway between the Being which is indivisible and remains always the same and the Being which is transient and divisible in bodies, He blended a third form of Being compounded out of the twain, that is to say, out of the Same and the Other; and in like manner He compounded it midway between that one of them which is indivisible and that one which is divisible in bodies. And He took the three of them, and blent them all together into one form, by forcing the Other into union with the Same, in spite of its being naturally difficult to mix. And when with the aid of Being He had mixed them, and had made of them one out of three, straightway He began to distribute the whole thereof into so many portions as was meet; and each portion was a mixture of the Same, of the Other, and of Being.The choice of these three as constituents of the Soul is explained by the use of the same terms in the (244-245) to denote certain Greatest Kinds or main categories. As Professor Paul Shorey has aptly observed ( ix. p. 298), It is necessary that the Soul should recognize everywhere . . . the same, the other and essence, those three μέγιστα γένηof the . . .. Hence on the Greek principle that like is known by like, Plato makes real substances out of these three abstractions and puts them as plastic material into the hands of the Demiurgus for the formation of the Soul. And He began making the division thus: First He took one portion from the whole; then He took a portion double of this; then a third portion, half as much again as the second portion, that is, three times as much as the first; he fourth portion He took was twice as much as the second; the fifth three times as much as the third; the sixth eight times as much as the first; and the seventh twenty-seven times as much as the first.These seven numbers may be arranged in two branches, in order to show the two series of which Timaeus immediately goes on the speak:1 (the 1st), 2 (the 2nd), 4 (the 4th), 8 (the 6th); 3 (the 3rd), 9 (the 5th), 27 (the 7th). The former branch contains the double intervals, i.e., the powers of 2; the latter one the triple intervals, i.e., the powers of 3.