Complete Works (264 page)

That is how at that time the four kinds were being shaken by the receiver, which was itself agitating like a shaking machine, separating the kinds most unlike each other furthest apart and pushing those most like each other closest together into the same region. This, of course, explains how these different kinds came to occupy different regions of space, even before the universe was set in order and constituted from them at its coming to be. Indeed, it is a fact that before this took place the four kinds all lacked [b] proportion and measure, and at the time the ordering of the universe was undertaken, fire, water, earth and air initially possessed certain traces of what they are now. They were indeed in the condition one would expect thoroughly god-forsaken things to be in. So, finding them in this natural condition, the first thing the god then did was to give them their distinctive shapes, using forms and numbers.

Here is a proposition we shall always affirm above all else: The god fashioned these four kinds to be as perfect and excellent as possible, when they were not so before. It will now be my task to explain to you what [c] structure each of them acquired, and how each came to be. My account will be an unusual one, but since you are well schooled in the fields of learning in terms of which I must of necessity proceed with my exposition, I’m sure you’ll follow me.

First of all, everyone knows, I’m sure, that fire, earth, water and air are bodies. Now everything that has bodily form also has depth. Depth, moreover, is of necessity comprehended within surface, and any surface bounded by straight lines is composed of triangles. Every triangle, moreover, [d] derives from two triangles, each of which has one right angle and two acute angles. Of these two triangles, one [the isosceles right-angled triangle] has at each of the other two vertices an equal part of a right angle, determined by its division by equal sides; while the other [the scalene right-angled triangle] has unequal parts of a right angle at its other two vertices, determined by the division of the right angle by unequal sides. This, then, we presume to be the originating principle of fire and of the other bodies, as we pursue our likely account in terms of Necessity. Principles yet more ultimate than these are known only to the god, and to any man he may hold dear.

[e] We should now say which are the most excellent four bodies that can come to be. They are quite unlike each other, though some of them are capable of breaking up and turning into others and vice-versa. If our account is on the mark, we shall have the truth about how earth and fire and their proportionate intermediates [water and air] came to be. For we shall never concede to anyone that there are any visible bodies more excellent than these, each conforming to a single kind. So we must wholeheartedly proceed to fit together the four kinds of bodies of surpassing excellence, and to declare that we have come to grasp their natures well enough.

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Of the two [right-angled] triangles, the isosceles has but one nature, while the scalene has infinitely many. Now we have to select the most excellent one from among the infinitely many, if we are to get a proper start. So if anyone can say that he has picked out another one that is more excellent for the construction of these bodies, his victory will be that of a friend, not an enemy. Of the many [scalene right-angled] triangles, then, we posit as the one most excellent, surpassing the others, that one from [a pair of] which the equilateral triangle is constructed as a third figure. Why this is so is too long a story to tell now. But if anyone puts this [b] claim to the test and discovers that it isn’t so, his be the prize, with our congratulations. So much, then, for the selection of the two triangles out of which the bodies of fire and the other bodies are constructed—the [right-angled] isosceles, and [the right-angled] scalene whose longer side squared is always triple its shorter side squared [i.e., the half-equilateral].

At this point we need to formulate more precisely something that was not stated clearly earlier. For then it appeared that all four kinds of bodies could turn into one another by successive stages.
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But the appearance is wrong. While there are indeed four kinds of bodies that come to be from [c] the [right-angled] triangles we have selected, three of them come from triangles that have unequal sides, whereas the fourth alone is fashioned out of isosceles triangles. Thus not all of them have the capacity of breaking up and turning into one another, with a large number of small bodies turning into a small number of large ones and vice-versa. There are three that can do this. For all three are made up of a single type of triangle, so that when once the larger bodies are broken up, the same triangles can go to make up a large number of small bodies, assuming shapes appropriate to them. And likewise, when numerous small bodies are fragmented into [d] their triangles, these triangles may well combine to make up some single massive body belonging to another kind.

So much, then, for our account of how these bodies turn into one another. Let us next discuss the form that each of them has come to have, and the various numbers that have combined to make them up.

Leading the way will be the primary form [the tetrahedron], the tiniest structure, whose elementary triangle is the one whose hypotenuse is twice the length of its shorter side. Now when a pair of such triangles are juxtaposed along the diagonal [i.e., their hypotenuses] and this is done three times, and their diagonals and short sides converge upon a single [e] point as center, the result is a single equilateral triangle, composed of six such triangles. When four of these equilateral triangles are combined, a single solid angle is produced at the junction of three plane angles. This,
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it turns out, is the angle which comes right after the most obtuse of the plane angles.
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And once four such solid angles have been completed, we get the primary solid form, which is one that divides the entire circumference [sc. of the sphere in which it is inscribed] into equal and similar parts.

The second solid form [the octahedron] is constructed out of the same triangles which, however, are now arranged in eight equilateral triangles and produce a single solid angle out of four plane angles. And when six such solid angles have been produced, the second body has reached its completion.

Now the third body [the icosahedron] is made up of a combination of one hundred and twenty of the elementary triangles, and of twelve solid [b] angles, each enclosed by five plane equilateral triangles. This body turns out to have twenty equilateral triangular faces. And let us take our leave of this one of the elementary triangles, the one that has begotten the above three kinds of bodies and turn to the other one, the isosceles [right-angled] triangle, which has begotten the fourth [the cube]. Arranged in sets of four whose right angles come together at the center, the isosceles triangle produced a single equilateral quadrangle [i.e., a square]. And when six of [c] these quadrangles were combined together, they produced eight solid angles, each of which was constituted by three plane right angles. The shape of the resulting body so constructed is a cube, and it has six quadrangular equilateral faces.

One other construction, a fifth, still remained, and this one the god used for the whole universe, embroidering figures on it.
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Anyone following this whole line of reasoning might very well be puzzled about whether we should say that there are infinitely many worlds [d] or a finite number of them. If so, he would have to conclude that to answer, “infinitely many,” is to take the view of one who is really “unfinished” in things he ought to be “finished” in. He would do better to stop with the question whether we should say that there’s really just one world or five and be puzzled about that. Well, our “probable account” answer declares there to be but one world, a god—though someone else, taking other things into consideration, will come to a different opinion. We must set him aside, however.

Let us now assign to fire, earth, water and air the structures which have just been given their formations in our speech. To earth let us give the [e] cube, because of the four kinds of bodies earth is the most immobile and the most pliable—which is what the solid whose faces are the most secure must of necessity turn out to be, more so than the others. Now of the [right-angled] triangles we originally postulated, the face belonging to those that have equal sides has a greater natural stability than that belonging to triangles that have unequal sides, and the surface that is composed of the two triangles, the equilateral quadrangle [the square], holds its position with greater stability than does the equilateral triangle, both in
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their parts and as wholes. Hence, if we assign this solid figure to earth, we are preserving our “likely account.” And of the solid figures that are left, we shall next assign the least mobile of them to water, to fire the most mobile, and to air the one in between. This means that the tiniest body belongs to fire, the largest to water, and the intermediate one to air—and also that the body with the sharpest edges belongs to fire, the next sharpest to air, and the third sharpest to water. Now in all these cases the body that has the fewest faces is of necessity the most mobile, in that it, more [b] than any other, has edges that are the sharpest and best fit for cutting in every direction. It is also the lightest, in that it is made up of the least number of identical parts. The second body ranks second in having these same properties, and the third ranks third. So let us follow our account, which is not only likely but also correct, and take the solid form of the pyramid that we saw constructed as the element or the seed of fire. And let us say that the second form in order of generation is that of air, and the third that of water.

Now we must think of all these bodies as being so small that due to their small size none of them, whatever their kind, is visible to us individually. [c] When, however, a large number of them are clustered together, we do see them in bulk. And in particular, as to the proportions among their numbers, their motions and their other properties, we must think that when the god had brought them to complete and exact perfection (to the degree that Necessity was willing to comply obediently), he arranged them together proportionately.

Given all we have said so far about the kinds of elemental bodies, the following account [of their transformations] is the most likely: When earth [d] encounters fire and is broken up by fire’s sharpness, it will drift about—whether the breaking up occurred within fire itself, or within a mass of air or water—until its parts meet again somewhere, refit themselves together and become earth again. The reason is that the parts of earth will never pass into another form. But when water is broken up into parts by fire or even by air, it could happen that the parts recombine to form one corpuscle of fire and two of air. And the fragments of air could produce, [e] from any single particle that is broken up, two fire corpuscles. And conversely, whenever a small amount of fire is enveloped by a large quantity of air or water or perhaps earth and is agitated inside them as they move, and in spite of its resistance is beaten and shattered to bits, then any two fire corpuscles may combine to constitute a single form of air. And when air is overpowered and broken down, then two and one half entire forms of air will be consolidated into a single, entire form of water.

Let us recapitulate and formulate our account of these transformations as follows: Whenever one of the other kinds is caught inside fire and gets
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cut up by the sharpness of fire’s angles and edges, then if it is reconstituted as fire, it will stop getting cut. The reason is that a thing of any kind that is alike and uniform is incapable of effecting any change in, or being affected by, anything that is similar to it. But as long as something involved in a transformation has something stronger than it to contend with, the process of its dissolution will continue non-stop. And likewise, when a few of the smaller corpuscles are surrounded by a greater number of bigger [b] ones, they will be shattered and quenched. The quenching will stop when these smaller bodies are willing to be reconstituted into the form of the kind that prevailed over them, and so from fire will come air, and from air, water. But if these smaller corpuscles are in process of turning into these and one of the other kinds encounters them and engages them in battle, their dissolution will go on non-stop until they are either completely squeezed and broken apart and escape to their own likes, or else are defeated, and, melding from many into one, they are assimilated to the kind that prevailed over them, and come to share its abode from then on. [c] And, what is more, as they undergo these processes, they all exchange their territories: for as a result of the Receptacle’s agitation the masses of each of the kinds are separated from one another, with each occupying its own region, but because some parts of a particular kind do from time to time become unlike their former selves and like the other kinds, they are carried by the shaking towards the region occupied by whatever masses they are becoming like to.

These, then, are the sorts of causes by which the unalloyed primary bodies have come to be. Now the fact that different varieties are found within their respective forms is to be attributed to the constructions of [d] each of the elementary triangles. Each of these two constructions did not originally yield a triangle that had just one size, but triangles that were both smaller and larger, numerically as many as there are varieties within a given form. That is why when they are mixed with themselves and with each other they display an infinite variety, which those who are to employ a likely account in their study of nature ought to take note of.

Now as for motion and rest, unless there is agreement on the manner and the conditions in which these two come to be, we will have many [e] obstacles to face in our subsequent course of reasoning. Although we have already said something about them, we need to say this as well: there will be no motion in a state of uniformity. For it is difficult, or rather impossible, for something to be moved without something to set it in motion, or something to set a thing in motion without something to be moved by it. When either is absent, there is no motion, but [when they are present] it is quite impossible for them to be uniform. And so let us always presume that rest is found in a state of uniformity and to attribute motion to non-uniformity.
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The latter, moreover, is caused by inequality, the origin of which we have already discussed.
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We have not explained, however, how it is that the various corpuscles have not reached the point of being thoroughly separated from each other kind by kind, so that their transformations into each other and their movement [toward their own regions] would have come to a halt. So let us return to say this about it: Once the circumference of the universe has comprehended the [four] kinds, then, because it is round and has a natural tendency to gather in upon itself, it constricts them all and allows no empty [b] space to be left over. This is why fire, more than the other three, has come to infiltrate all of the others, with air in second place, since it is second in degree of subtlety, and so on for the rest. For the bodies that are generated from the largest parts will have the largest gaps left over in their construction, whereas the smallest bodies will have the tiniest. Now this gathering, contracting process squeezes the small parts into the gaps inside the big ones. So now, as the small parts are placed among the large ones and the smaller ones tend to break up the larger ones while the larger tend to cause the smaller to coalesce, they all shift, up and down, into their own respective regions. For as each changes in quantity, it also changes the [c] position of its region. This, then, is how and why the occurrence of non-uniformity is perpetually preserved, and so sets these bodies in perpetual motion, both now and in the future without interruption.