Tim. This being so, we must agree that One Kind is the self-identical Form, ungenerated and indestructible, neither receiving into itself any other from any quarter nor itself passing anywhither into another, invisible and in all ways imperceptible by sense, it being the object which it is the province of Reason to contemplate; and a second Kind is that which is named after the former and similar thereto, an object perceptible by sense, generated, ever carried about, becoming in a place and out of it again perishing, apprehensible by Opinion with the aid of Sensation; and a third Kind is ever-existing Place, which admits not of destruction, and provides room for all things that have birth, itself being apprehensible by a kind of bastard reasoning by the aid of non-sensation, barely an object of belief; for when we regard this we dimly dream and affirm that it is somehow necessary that all that exists should exist in some spot and occupying some place , and that that which is neither on earth nor anywhere in the Heaven is nothing. So because of all these and other kindred notions, we are unable also on waking up to distinguish clearly the unsleeping and truly subsisting substance, owing to our dreamy condition, or to state the truth—how that it belongs to a copy—seeing that it has not for its own even that substance for which it came into being, but fleets ever as a phantom of something else—to come into existence in some other thing, clinging to existence as best it may, on pain of being nothing at all; whereas to the aid of the really existent there comes the accurately true argument, that so long as one thing is one thing, and another something different, neither of the two will ever come to exist in the other so that the same thing becomes simultaneously both one and two.

 Let this, then, be, according to my verdict, a reasoned account of the matter summarily stated,—that Being and Place and Becoming were existing, three distinct things, even before the Heaven came into existence; and that the Nurse of Becoming, being liquefied and ignified and receiving also the forms of earth and of air, and submitting to all the other affections which accompany these, exhibits every variety of appearance; but owing to being filled with potencies that are neither similar nor balanced, in no part of herself is she equally balanced, but sways unevenly in every part, and is herself shaken by these forms and shakes them in turn as she is moved. Tim. And the forms, as they are moved, fly continually in various directions and are dissipated; just as the particles that are shaken and winnowed by the sieves and other instruments used for the cleansing of corn fall in one place if they are solid and heavy, but fly off and settle elsewhere if they are spongy and light. So it was also with the Four Kinds when shaken by the Recipient: her motion, like an instrument which causes shaking, was separating farthest from one another the dissimilar, and pushing most closely together the similar; wherefore also these Kinds occupied different places even before that the Universe was organized and generated out of them.

 Before that time, in truth, all these things were in a state devoid of reason or measure, but when the work of setting in order this Universe was being undertaken, fire and water and earth and air, although possessing some traces of their own nature, were yet so disposed as everything is likely to be in the absence of God; and inasmuch as this was then their natural condition, God began by first marking them out into shapes by means of forms and numbers. And that God constructed them, so far as He could, to be as fair and good as possible, whereas they had been otherwise,—this above all else must always be postulated in our account. Now, however, it is the disposition and origin of each of these Kinds which I must endeavor to explain to you in an exposition of an unusual type; yet, inasmuch as you have some acquaintance with the technical method which I must necessarily employ in my exposition, you will follow me.

 In the first place, then, it is plain I presume to everyone that fire and earth and water and air are solid bodies; and the form of a body, in every case, possesses depth also. Further, it is absolutely necessary that depth should be bounded by a plane surface; and the rectilinear plane is composed of triangles. Now all triangles derive their origin from two triangles, each having one angle right and the others acute i.e., the rectangular isosceles triangle and the rectangular scalene; all other triangles can be built up from these two (e.g. see 54 E N.). ; and the one of these triangles has on each side half a right angle marked off by equal sides, while the other has the right angle divided into unequal parts by unequal sides. These we lay down as the principles of fire and all the other bodies, proceeding according to a method in which the probable is combined with the necessary; but the principles which are still higher than these are known only to God and the man who is dear to God. We must now declare what will be the four fairest bodies, dissimilar to one another, but capable in part of being produced out of one another by means of dissolution; for if we succeed herein we shall grasp the truth concerning the generation of earth and fire and the mean proportionals. For to no one will we concede that fairer bodies than these, each distinct of its kind, are anywhere to be seen. Wherefore we must earnestly endeavor to frame together these four kinds of bodies which excel in beauty, and to maintain that we have apprehended their nature adequately. Tim. Now of the two triangles, the isosceles possesses one single nature, but the scalene an infinite number; and of these infinite natures we must select the fairest, if we mean to make a suitable beginning. If, then, anyone can claim that he has chosen one that is fairer for the construction of these bodies, he, as friend rather than foe, is the victor. We, however, shall pass over all the rest and postulate as the fairest of the triangles that triangle out of which, when two are conjoined, the equilateral triangle is constructed as a third. i.e., the half of an equilateral triangle; e.g. if the triangle ABC is bisected by the line AD, we have two such triangles in ADB and ADC. The reason why is a longer story; but should anyone refute us and discover that it is not so, we begrudge him not the prize. Accordingly, let these two triangles be selected as those wherefrom are contrived the bodies of fire and of the other elements,— one being the isosceles, and the other that which always has the square on its greater side three times the square on the lesser side. i.e., in the triangle ADB (see last note) AB = 2Bsquared, and (AB)squared = (BD)squared + (AD)squared; therefore 4(BD)squared = (BD)squared + (AD)squared, and so 3(BD)squared = (AD)squared. 

 Moreover, a point about which our previous statement was obscure must now be defined more clearly. It appeared as if the four Kinds, in being generated, all passed through one another into one another, but this appearance was deceptive. For out of the triangles which we have selected four Kinds are generated, three of them out of that one triangle which has its sides unequal, and the fourth Kind alone composed of the isosceles triangle. Consequently, they are not all capable of being dissolved into one another so as to form a few large bodies composed of many small ones, or the converse; but three of them do admit of this process. For these three are all naturally compounded of one triangle, so that when the larger bodies are dissolved many small ones will form themselves from these same bodies, receiving the shapes that befit them; and conversely, when many small bodies are resolved into their triangles they will produce, when unified, one single large mass of another Kind. So let thus much be declared concerning their generation into one another.

 In the next place we have to explain the form in which each Kind has come to exist and the numbers from which it is compounded. First will come that form which is primary and has the smallest components, and the element thereof is that triangle which has its hypotenuse twice as long as its lesser side. And when a pair of such triangles are joined along the line of the hypotenuse, and this is done thrice, by drawing the hypotenuses and the short sides together as to a center, there is produced from those triangles, six in number, one equilateral triangle. As in the figure the equilateral triangle ABC is divided into 6 triangles of unequal sides by joining the vertical points A, B, C to the points of bisection of the opposite sides, viz. D, E, F. Then the hypotenuse in each such triangle is double the shortest side (e.g. AO = 2FO). And FAO = 1/3 right angle; while FOA = 2/3 right angle. The three plane edges are thus of 60 degrees each = 180 degrees = the most obtuse plane angle; so that the solid angle is one degree less, i.e., 179 degrees. Tim. And when four equilateral triangles are combined so that three plane angles meet in a point, they form one solid angle, which comes next in order to the most obtuse of the plane angles. And when four such angles are produced, the first solid figure i.e., the tetrahedron or pyramid (molecule of fire). is constructed, which divides the whole of the circumscribed sphere into equal and similar parts. And the second solid i.e., the octahedron (molecule of air). is formed from the same triangles, but constructed out of eight equilateral triangles, which produce one solid angle out of four planes; and when six such solid angles have been formed, the second body in turn is completed. And the third solid i.e., the icosahedron (molecule of water). is composed of twice sixty of the elemental triangles conjoined, and of twelve solid angles, each contained by five plane equilateral triangles, and it has, by its production, twenty equilateral triangular bases.

 Now the first of the elemental triangles ceased acting when it had generated these three solids, the substance of the fourth Kind i.e., the cube, composed of 6x4 rectangular isosceles triangles (molecule of earth). being generated by the isosceles triangle. Four of these combined, with their right angles drawn together to the center, produced one equilateral quadrangle; and six such quadrangles, when joined together, formed eight solid angles, each composed of three plane right angles; and the shape of the body thus constructed was cubic, having six plane equilateral quadrangular bases. And seeing that there still remained one other compound figure, the fifth, i.e., the dodecahedron. How God used it up is obscure: the reference may be to the 12 signs of the Zodiac. God used it up for the Universe in his decoration thereof.

 Now in reasoning about all these things, a man might question whether he ought to affirm the existence of an infinite diversity of Universes or a limited number; and if he questioned aright he would conclude that the doctrine of an infinite diversity is that of a man unversed There is a play here on the two senses of ἄπειρος , unlimited and unskilled ; Cf. Phileb . 17 E. The doctrine of an infinite number of worlds was held by the Atomists. in matters wherein he ought to be versed; but the question whether they ought really to be described as one Universe or five is one which might with more reason give us pause. Now our view declares the Universe to be essentially one, in accordance with the probable account; but another man, considering other facts, will hold a different opinion. Him, however, we must let pass. But as for the Kinds which have now been generated by our argument, let us assign them severally to fire and earth and water and air. To earth let us give the cubic form; for of the four Kinds earth is the most immobile and the most plastic body, and of necessity the body which has the most stable bases must be pre-eminently of this character. Now of the triangles we originally assumed, the basis formed by equal sides is of its nature more stable than that formed by unequal sides; and of the plane surfaces which are compounded of these several triangles, the equilateral quadrangle, both in its parts and as a whole, has a more stable base than the equilateral triangle. Tim. Wherefore, we are preserving the probable account when we assign this figure to earth, and of the remaining figures the least mobile to water, and the most mobile to fire, and the intermediate figure to air; and, further, when we assign the smallest body to fire, and the greatest to water, and the intermediate to air; and again, the first in point of sharpness to fire, the second to air, and the third to water. As regards all these forms, that which has the fewest bases must necessarily be the most mobile, since it is in all ways the sharpest and most acute of all; and it must also be the lightest, since it is composed of the fewest identical parts; and the second comes second in point of these same qualities, and the third third.

 Thus, in accordance with the right account and the probable, that solid which has taken the form of a pyramid shall be the element and seed of fire; the second in order of generation we shall affirm to be air, and the third water. Now one must conceive all these to be so small that none of them, when taken singly each in its several kind, is seen by us, but when many are collected together their masses are seen. And, moreover, as regards the numerical proportions which govern their masses and motions and their other qualities, we must conceive that God realized these everywhere with exactness, in so far as the nature of Necessity submitted voluntarily or under persuasion, and thus ordered all in harmonious proportion.

 From all that we have hitherto said about these Kinds, they will, in all likelihood, behave themselves as follows. Earth will keep moving when it happens to meet with fire and has been dissolved by its acuteness, whether this dissolution takes place in pure fire or in a mass of air or of water; and this motion will continue until the particles of earth happen to meet together somewhere and reunite one with another, when they become earth again; for assuredly earth will never change into another form. But water, when broken up by fire or even by air, is capable of becoming a compound of one corpuscle of fire with two of air; and the fractions of air which come from the dissolving of one particle will form two corpuscles of fire. And again, when a small quantity of fire is enclosed by a large quantity of air and water, or of earth, and moves within them as they rush along, and is defeated in its struggle and broken up, then two corpuscles of fire unite to make one form of air. And when air is defeated and disintegrated, from two whole forms of air and a half, one whole form of water will be compounded.