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  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 untraversable. In point of fact they

  do not need the infinite and do not use it. They postulate only that

  the finite straight line may be produced as far as they wish. It is

  possible to have divided in 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 to have such an infinite

  instead, while its existence will be in the sphere of real magnitudes.

  In the fourfold scheme of causes, it is plain that the infinite is a

  cause in the 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.

  8

  It remains to dispose of 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 valid.

  (1) In order that coming to be should not fail, it is not

  necessary that there should be a sensible body which is actually

  infinite. The passing away of one thing may be the coming to be of

  another, the All being limited.

  (2) 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 necessarily possible between any two things taken at

  random.

  (3) To rely on mere 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.

  (a) Time indeed and movement are infinite, and also thinking, in the

  sense that each part that is taken passes in succession out of

  existence.

  (b) 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 somewhere (the non-existent is nowhere--where

  is the goat-stag or the sphinx?), and because 'motion' in its most

  general and primary 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 divergent

  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

  replacement. Where water now is, there in turn, when the water has

  gone out as from a vessel, air is present. When therefore another body

  occupies this same place, the place is thought to be different from

  all the bodies which come to be in it 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 typical 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 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, c.) hold only in relation to us. To us they are not

  always the same but change with the direction in which we are

  turned: that is why the same thing may be 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 weight and what is made of

  earth are carried-the implication being that these places do not

  differ merely in relative position, but also as possessing distinct

  potencies. This is made plain also by the objects studied by

  mathematics. Though they have no real place, they nevertheless, in

  respect of their position relatively to us, have a right and left as

  attributes ascribed to them only in consequence of their relative

  position, not having by nature these various characteristics. Again,

  the theory that the void exists involves the existence 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 broad-breasted

  earth,'

  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 potency of place must be a marvellous thing, and

  take precedence of all other things. For that without which nothing

  else can exist, while it can exist without the others, must needs be

  first; for 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 its nature presents difficulty-whether it is some sort of 'bulk' of

  body or some entity other than that, for we must first determine its

  genus.

  (1) Now it has three dimensions, length, breadth, depth, the

  dimensions by which all body also is bounded. But the place cannot

  be body; for if it were there would be two bodies in the same place.

  (2) Further, if body has a place and space, clearly so too have

  surface and the other limits of body; for the same statement will

  apply to them: where the bounding planes 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
<
br />   each of them.

  (3) What in the world then are we to suppose place to be? If it

  has the sort of 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.

  (4) 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 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.

  (5) Further, too, if it is itself an existent, where will it be?

  Zeno's difficulty demands an explanation: for if everything that

  exists has a place, place too will have a place, and so on ad

  infinitum.

  (6) 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 place-not only as to what it is, but even whether there is

  such a thing.

  2

  We may distinguish generally between predicating B of A because it

  (A) is itself, and because it is something else; and particularly

  between place which is common and in which all bodies are, and the

  special place occupied primarily by each. I mean, for instance, that

  you are now in the heavens because you are in the air and it is in the

  heavens; 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.

  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 by

  which the magnitude or the matter of the magnitude is defined: for

  this is the limit of each body.

  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; when the

  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, 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 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 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: qua containing, 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 called it in writing in

  the 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 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 resulting body is not in the same place. What

  sort of destruction then is that?

  This concludes my statement of the reasons why space must be

  something, and again of the difficulties that may be raised about

  its essential nature.

  3

  The next step we must take is to see in how many senses one thing is

  said to be 'in' another.

  (1) As the finger is 'in' the hand and generally the part 'in' the

  whole.

  (2) As the whole is 'in' the parts: for there is no whole over and

  above the parts.

  (3) As man is 'in' animal and generally species 'in' genus.

  (4) As the genus is 'in' the species and generally the part of the

  specific form 'in' the definition of the specific form.

  (5) As health is 'in' the hot and the cold and generally the form

  'in' the matter.

  (6) As the affairs of Greece centre 'in' the king, and generally

  events centre 'in' their primary motive agent.

  (7) As the existence of a thing centres 'in its good and generally

  'in' its end, i.e. in 'that for the sake of which' it exists.

  (8) In the strictest sense of all, as a thing is 'in' a vessel,

  and generally 'in' place.

  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 is said to be

  white because the visible surface of him is white, or to be scientific

  because his thinking faculty has been trained. 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 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, as

  'white' is in man because it is in body, and in body because it

  resides in the visible surface. We cannot go further and say that it

  is in surface in virtue of something other than itself. (Yet it is not

  in itself: though these are in a way the same thing,) they differ in

  essence, each having a special nature and capacity, 'surface' and

  'white'.

  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 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 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 their essence 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 incidentally:

  for two things would at the same time in the same thing. The jar would

  be in itself-if a thing whose nature it is to receive can be in

  itself; and that which it receives, namely (if wine) wine, will be