by Aristotle
Our account does not rob the mathematicians of their science, by disproving the actual existence of the infinite in the direction of increase, in the sense of the [30] untraversable. In point of fact they do not need the infinite and do not use it. They postulate only that a finite straight line may be produced as far as they wish. It is possible to have divided into the same ratio as the largest quantity another magnitude of any size you like. Hence, for the purposes of proof, it will make no difference to them whether the infinite is found among existent magnitudes.
In the four-fold scheme of causes, it is plain that the infinite is a cause in the [208a1] sense of matter, and that its essence is privation, the subject as such being what is continuous and sensible. All the other thinkers, too, evidently treat the infinite as matter—that is why it is inconsistent in them to make it what contains, and not what is contained.
[5] 8 · It remains to go through the arguments which are supposed to support the view that the infinite exists not only potentially but as a separate thing. Some have no cogency; others can be met by fresh objections that are true.
In order that coming to be should not fail, it is not necessary that there should [10] be a sensible body which is actually infinite. The passing away of one thing may be the coming to be of another, the universe being limited.
There is a difference between touching and being limited. The former is relative to something and is the touching of something (for everything that touches touches something), and further is an attribute of some one of the things which are limited. On the other hand, what is limited is not limited in relation to anything. Again, contact is not possible between any two things taken at random.
[15] To rely on thinking is absurd; for then the excess or defect is not in the thing but in the thought. One might think that one of us is bigger than he is and magnify him ad infinitum. But it does not follow that he is bigger than the size we are, just because some one thinks he is, but only because he is the size he is. The thought is an accident.
[20] Time indeed and movement are infinite, and also thinking; but the parts that are taken do not persist.
Magnitude is not infinite either in the way of reduction or of magnification in thought.
This concludes my account of the way in which the infinite exists, and of the way in which it does not exist, and of what it is.
BOOK IV
1 · The physicist must have a knowledge of place, too, as well as of the infinite—namely, whether there is such a thing or not, and the manner of its existence and what it is—both because all suppose that things which exist are [30] somewhere (the non-existent is nowhere—where is the goat-stag or the sphinx?), and because motion in its most general and proper sense is change of place, which we call ‘locomotion’.
The question, what is place? presents many difficulties. An examination of all the relevant facts seems to lead to different conclusions. Moreover, we have inherited nothing from previous thinkers, whether in the way of a statement of difficulties or of a solution.
The existence of place is held to be obvious from the fact of mutual [208b1] replacement. Where water now is, there in turn, when the water has gone out as from a vessel, air is present; and at another time another body occupies this same place. The place is thought to be different from all the bodies which come to be in it [5] and replace one another. What now contains air formerly contained water, so that clearly the place or space into which and out of which they passed was something different from both.
Further, the locomotions of the elementary natural bodies—namely, fire, earth, and the like—show not only that place is something, but also that it exerts a [10] certain influence. Each is carried to its own place, if it is not hindered, the one up, the other down. Now these are regions or kinds of place—up and down and the rest of the six directions. Nor do such distinctions (up and down and right and left) hold only in relation to us. To us they are not always the same but change with the [15] direction in which we are turned: that is why the same thing is often both right and left, up and down, before and behind. But in nature each is distinct, taken apart by itself. It is not every chance direction which is up, but where fire and what is light are carried; similarly, too, down is not any chance direction but where what has [20] weight and what is made of earth are carried—the implication being that these places do not differ merely in position, but also as possessing distinct powers. This is made plain also by the objects studied by mathematics. Though they have no place, they nevertheless, in respect of their position relatively to us, have a right and left as these are spoken of merely in respect of relative position, not having by nature these various characteristics. Again, the theory that the void exists involves the existence [25] of place; for one would define void as place bereft of body.
These considerations then would lead us to suppose that place is something distinct from bodies, and that every sensible body is in place. Hesiod too might be held to have given a correct account of it when he made chaos first. At least he says:
First of all things came chaos to being, then broadbreasted earth,25 [30]
implying that things need to have space first, because he thought, with most people, that everything is somewhere and in place. If this is its nature, the power of place must be a marvellous thing, and be prior to all other things. For that without which nothing else can exist, while it can exist without the others, must needs be first; for [209a1] place does not pass out of existence when the things in it are annihilated.
True, but even if we suppose its existence settled, the question of what it is presents difficulty—whether it is some sort of ‘bulk’ of body or some entity other than that; for we must first determine its genus.
[5] Now it has three dimensions, length, breadth, depth, the dimensions by which all body is bounded. But the place cannot be body; for if it were there would be two bodies in the same place.
Further, if body has a place and space, clearly so too have surface and the other limits of body; for the same argument will apply to them: where the bounding planes [10] of the water were, there in turn will be those of the air. But when we come to a point we cannot make a distinction between it and its place. Hence if the place of a point is not different from the point, no more will that of any of the others be different, and place will not be something different from each of them.
What in the world, then, are we to suppose place to be? If it has the sort of [15] nature described, it cannot be an element or composed of elements, whether these be corporeal or incorporeal; for while it has size, it has not body. But the elements of sensible bodies are bodies, while nothing that has size results from a combination of intelligible elements.
[20] Also we may ask: of what in things is space the cause? None of the four modes of causation can be ascribed to it. It is neither cause in the sense of the matter of existents (for nothing is composed of it), nor as the form and definition of things, nor as end, nor does it move existents.
Further, too, if it is itself an existent, it will be somewhere. Zeno’s difficulty demands an explanation; for if everything that exists has a place, place too will have [25] a place, and so on ad infinitum.
Again, just as every body is in place, so, too, every place has a body in it. What then shall we say about growing things? It follows from these premisses that their place must grow with them, if their place is neither less nor greater than they are.
By asking these questions, then, we must raise the whole problem about [30] place—not only as to what it is, but even whether there is such a thing.
2 · Something can be said of a subject either in virtue of itself or in virtue of something else; and there is place which is common and in which all bodies are, and which is the proper and primary location of each body. I mean, for instance, that you are now in the world because you are in the air and it is in the world; and you are in the air because you are on the earth; and similarly on the earth because you are in this place which contains no more than you.
[2
09b1] Now if place is what primarily contains each body, it would be a limit, so that the place would be the form or shape of each body which the magnitude or the matter of the magnitude is defined; for this is the limit of each body.
[5] If, then, we look at the question in this way the place of a thing is its form. But, if we regard the place as the extension of the magnitude, it is the matter. For this is different from the magnitude: it is what is contained and defined by the form, as by a bounding plane. Matter or the indeterminate is of this nature; for when the [10] boundary and attributes of a sphere are taken away, nothing but the matter is left.
This is why Plato in the Timaeus says that matter and space are the same; for the ‘participant’ and space are identical. (It is true, indeed, that the account he gives there of the ‘participant’ is different from what he says in his so-called unwritten teaching. Nevertheless, he did identify place and space.) I mention Plato because, [15] while all hold place to be something, he alone tried to say what it is.
In view of these facts we should naturally expect to find difficulty in determining what place is, if indeed it is one of these two things, matter or form. They demand a very close scrutiny, especially as it is not easy to recognize them [20] apart.
But it is at any rate not difficult to see that place cannot be either of them. The form and the matter are not separate from the thing, whereas the place can be separated. As we pointed out, where air was, water in turn comes to be, the one [25] replacing the other; and similarly with other bodies. Hence the place of a thing is neither a part nor a state of it, but is separable from it. For place is supposed to be something like a vessel—the vessel being a transportable place. But the vessel is no part of the thing.
In so far then as it is separable from the thing, it is not the form; and in so far as [30] it contains it, it is different from the matter.
Also it is held that what is anywhere is both itself something and that there is a different thing outside it. (Plato of course, if we may digress, ought to tell us why the form and the numbers are not in place, if ‘what participates’ is place—whether what participates is the Great and the Small or the matter, as he has written in the [210a1] Timaeus.)
Further, how could a body be carried to its own place, if place was the matter or the form? It is impossible that what has no reference to motion or the distinction of up and down can be place. So place must be looked for among things which have these characteristics.
If the place is in the thing (it must be if it is either shape or matter) place will [5] have a place; for both the form and the indeterminate undergo change and motion along with the thing, and are not always in the same place, but are where the thing is. Hence the place will have a place.
Further, when water is produced from air, the place has been destroyed, for the [10] resulting body is not in the same place. What sort of destruction then is that?
This concludes my statement of the reasons why place must be something, and again of the difficulties that may be raised about is essential nature.
3 · The next step we must take is to see in how many ways one thing is said to be in another. In one way, as a finger is in a hand, and generally a part in a whole. In [15] another way, as a whole is in its parts; for there is no whole over and above the parts. Again, as man is in animal, and in general a species in a genus. Again, as the genus is in the species, and in general a part of the species in its definition. Again, as [20] health is in the hot and the cold, and in general the form in the matter. Again, as the affairs of Greece are in the King, and generally events are in their primary motive agent. Again, as a thing is in its good, and generally in its end, i.e. in that for the sake of which. And most properly of all, as something is in a vessel, and generally in a place.26
[25] One might raise the question whether a thing can be in itself, or whether nothing can be in itself—everything being either nowhere or in something else. The question is ambiguous; we may mean the thing qua itself or qua something else.
When there are parts of a whole—the one that in which a thing is, the other the thing which is in it—the whole will be described as being in itself. For a thing is described in terms of its parts, as well as in terms of the thing as a whole, e.g. a man [30] is said to be white because the visible surface of him is white, or to be scientific because his thinking faculty is. The jar then will not be in itself and the wine will not be in itself. But the jar of wine will; for the contents and the container are both parts of the same whole.
In this sense then, but not primarily, a thing can be in itself, namely, as white is [210b1] in body (for the visible surface is in body), and science is in the mind.
It is from these, which are parts (in the sense at least of being in the man), that the man is called white, &c. (But the jar and the wine in separation are not parts of a whole, though together they are.) So when there are parts, a thing will be in itself, [5] as white is in man because it is in body, and in body because it resides in the visible surface. But it is not in surface in virtue of something else. And these things—the surface and the white—differ in form, and each has a different nature and power.
Thus if we look at the matter inductively we do not find anything to be in itself in any of the senses that have been distinguished; and it can be seen by argument [10] that it is impossible. For each of two things will have to be both, e.g. the jar will have to be both vessel and wine, and the wine both wine and jar, if it is possible for a thing to be in itself; so that, however true it might be that they were in each other, the jar will receive the wine in virtue not of its being wine but of the wine’s being wine, and [15] the wine will be in the jar in virtue not of its being a jar but of the jar’s being a jar. Now that they are different in respect of what they are is evident; for that in which something is and that which is in it would be differently defined.
Nor is it possible for a thing to be in itself even accidentally; for two things would be at the same time in the same thing. The jar would be in itself—if a thing [20] whose nature it is to receive can be in itself; and that which it receives, namely (if wine) wine, will be in it.
Obviously then a thing cannot be in itself primarily.
Zeno’s problem—that if place is something it must be in something—is not difficult to solve. There is nothing to prevent the first place from being in something [25] else—not indeed in that as in a place, but as health is in the hot as a state of it or as the hot is in body as an affection. So we escape the infinite regress.
Another thing is plain: since the vessel is no part of what is in it (what contains something primarily is different from what is contained), place could not be either the matter or the form of the thing contained, but must be different—for the latter, [30] both the matter and the shape, are parts of what is contained.
This then may serve as a critical statement of the difficulties involved.
4 · What then after all is place? The answer to this question may be elucidated as follows.
Let us take for granted about it the various characteristics which are supposed correctly to belong to it essentially. We assume first that place is what contains that of which it is the place, and is no part of the thing; again, that the primary place of a [211a1] thing is neither less nor greater than the thing; again, that place can be left behind by the thing and is separable; and in addition that all place admits of the distinction of up and down, and each of the bodies is naturally carried to its appropriate place and rests there, and this makes the place either up or down. [5] Having laid these foundations, we must complete the theory. We ought to try to conduct our inquiry into what place is in such a way as not only to solve the difficulties connected with it, but also to show that the attributes supposed to belong to it do really belong to it, and further to make clear the cause of the trouble and of [10] the difficulties about it. In that way, each point will be proved in the most satisfactory manner.
First then we must understand that place would not have been inqu
ired into, if there had not been motion with respect to place. It is chiefly for this reason that we suppose the heaven also to be in place, because it is in constant movement. Of this kind of motion there are two species—locomotion on the one hand and, on the other, increase and diminution. For these too involve change: what was then in this place [15] has now in turn changed to what is larger or smaller.
Again, things are moved either in themselves, actually, or accidentally. In the latter case it may be either something which by its own nature is capable of being moved, e.g. the parts of the body or the nail in the ship, or something which is not in [20] itself capable of being moved, but is always moved accidentally, as whiteness or science. These have changed their place only because the subjects to which they belong do so.
We say that a thing is in the world, in the sense of in place, because it is in the air, and the air is in the world; and when we say it is in the air, we do not mean it is in [25] every part of the air, but that it is in the air because of the surface of the air which surrounds it; for if all the air were its place, the place of a thing would not be equal to the thing—which it is supposed to be, and which the primary place in which a thing is actually is.
When what surrounds, then, is not separate from the thing, but is in continuity with it, the thing is said to be in what surrounds it, not in the sense of in place, but as [30] a part in a whole. But when the thing is separate and in contact, it is primarily in the inner surface of the surrounding body, and this surface is neither a part of what is in it nor yet greater than its extension, but equal to it; for the extremities of things which touch are coincident.
Further, if one body is in continuity with another, it is not moved in that but [35] with that. On the other hand it is moved in that if it is separate. It makes no difference whether what contains is moved or not.
[Again, when it is not separate it is described as a part in a whole, as the pupil [211b1] in the eye or the hand in the body: when it is separate, as the water in the cask or the wine in the jar. For the hand is moved with the body and the water in the cask.]27