Aristotle · a new plain-English translation from the original language
1| It is clear, then, that in the things of nature, the necessary is what is called matter, along with the motions belonging to it. And both causes must be stated by the student of nature, but especially that for the sake of which; for this is the cause of the matter, while the matter is not the cause of the end. And the end is that for the sake of which, and the starting-point proceeds from the definition and the account, just as in the products of craft: since the house is of such-and-such a kind, these things must of necessity come to be and be present; and since health is this, these things must of necessity come to be and be present. So too, if a human being is this, then these things must be present; and if these, then these. Perhaps the necessary is present in the account as well. For once one has defined the function of sawing as being a division of such-and-such a kind, this will not be possible unless it has teeth of such-and-such a kind, and these will not be possible unless they are of iron. For even in the account there are certain parts that function as the matter of the account. Γ.
2| Since nature is a principle of motion and change, and our inquiry concerns nature, we must not let it escape us what motion is; for if it remains unknown, nature too must remain unknown. Once we have determined the question of motion, we must try to proceed in the same way concerning what comes next. Motion is held to belong among continuous things, and the infinite makes its first appearance in the continuous — which is why those who define the continuous often find themselves making further use of the account of the infinite, on the ground that the continuous is what is divisible without limit. Besides this, it is impossible for there to be motion without place, void, and time. It is clear, then, that for these reasons — and because these things are common and universal to all cases — we must examine each of them in turn, taking them up first (for the study of what is particular comes after the study of what is common); and first, as we said, we must examine motion. Now, one thing exists only in actuality, another both potentially and in actuality — one a this, another a so-much, another a such, and likewise for the other categories of being. Of the relative, one kind is spoken of in terms of excess and deficiency, another in terms of the active and the passive, and in general as what can cause motion and what can be moved;
3| for what can cause motion causes motion in what can be moved, and what can be moved is moved by what can cause motion; and there is no motion apart from things. For what changes always changes either in respect of substance, or of quantity, or of quality, or of place, and there is nothing common to be grasped over and above these, as we say — nothing that is neither a this, nor a quantity, nor a quality, nor any of the other predicates. So there will be no motion or change of anything over and above what has been mentioned, since nothing exists over and above what has been mentioned. Each of these belongs to everything in two ways — for example the this (one is its form, the other its privation), and likewise for quality (one is white, the other black), and for quantity, one is complete and the other incomplete. Similarly too for locomotion, one is up and the other down, or one light and the other heavy. So there are as many species of motion and change as there are of being. Now that a distinction has been drawn
4| within each genus between what is in actuality and what is in potentiality, the actuality of what exists potentially, qua such, is motion — for example, of what is alterable, qua alterable, alteration; of what admits of growth, and of its opposite, what admits of diminution (for there is no name common to both), growth and diminution; of what admits of coming-to-be and passing-away, coming-to-be and passing-away; of what admits of being carried along, locomotion. That this is what motion is, is clear from the following. When the buildable, in so far as we call it just that, is in actuality, it is being built, and this is building; and likewise learning, doctoring, rolling, leaping, ripening, and aging. But since some things are the same thing both potentially and in actuality, though not at the same time, or not in the same respect — for instance, a thing may be hot in actuality but cold in potentiality — many things will then act on one another and be acted on by one another; for everything will be at once active and passive. So too what causes motion is, in the natural world, itself movable; for everything of this sort causes motion by being itself in motion. Some, then, think that whatever causes motion is itself in motion; but how this actually stands will become clear from other considerations (for there is something that causes motion while itself unmoved) — and the actuality of what exists potentially, when, existing in actuality, it is active not qua itself but qua movable, is motion.
5| I mean it this way. Bronze is potentially a statue, but all the same the realization of the bronze, as bronze, is not motion. For being bronze and being potentially something are not the same thing, since if they were the same without qualification and by definition, then the realization of the bronze, as bronze, would be motion. But they are not the same, as has been said (this is clear in the case of contraries: being able to be healthy and being able to be sick are different — for otherwise being sick and being healthy would be the same — while the substrate that is healthy and that is diseased, whether it is fluid or blood, is the same and one). Since they are not the same, just as color and the visible are not the same, it is evident that the realization of the potential, as potential, is motion. That this is what motion is, then, and that a thing is in motion precisely when this realization occurs, and neither before nor after, is clear. For each thing admits of being at one time in activity and at another time not — take, for example, the buildable: the activity of the buildable, as buildable, is building (for the activity is either the building or the house; but once there is a house, the buildable no longer exists —
6| what is being built is the buildable; so the activity must be the building) — and building is a kind of motion. But indeed the same account will fit the other kinds of motion as well.
7| That this has been well stated is clear also from what others say about motion, and from the fact that it is not easy to define it in any other way. For one could not place motion and change in any other genus — this is clear if you consider how some people classify it, calling motion otherness, inequality, and not-being; but none of these is necessarily in motion, whether things are other, or unequal, or not-being; nor, again, is change any more into these or out of these than it is out of their opposites. The reason they classify it this way is that motion seems to be something indeterminate, while the principles of the other column are indeterminate because they are privative — for none of them is a this or a such, nor does it fall under any of the other categories. The reason motion seems indeterminate is that it cannot be placed either under the potentiality of things or under their actuality: what is potentially of a certain quantity is not necessarily in motion, nor is what is actually of a certain quantity, and motion seems to be a kind of activity, but an incomplete one — and the reason is that the potential thing, of which it is the activity, is incomplete. And this is exactly why it is hard to grasp what motion is: it seems necessary to place it either under privation, or under potentiality, or under unqualified actuality, and none of these appears to admit it. What remains, then, is the way we have stated: that it is a certain activity, an activity of the kind we have described — hard to see, but possible for it to be so. And what moves is also moved, as has been said, everything that is
8| potentially moveable, and whose immobility is rest (for that to which motion belongs, the immobility of this same thing is rest). For to be in activity toward this, as such, is precisely to move it; and it does this by contact, so that at the same time it is also acted on. Hence motion is the realization of the moveable, as moveable, and this comes about by contact with what can move it, so that at the same time it is also acted on. And what moves will always bring some form — either a this, or a such, or a so-much — which will be the starting point and cause of the motion, when it moves the thing, as for instance a man in actuality makes a man out of what is potentially a man. And the puzzle raised is also clear: that motion is in the moveable thing — for it is the realization of this thing, brought about by what can move it. And the activity of what can move is not a different thing; for there must be a realization belonging to both: a thing is capable of moving by having a capacity, and it moves by being in activity, but it is active upon the moveable — so that the activity of both alike is one, just as the same interval is one whether from one to two or from two to one, and just as the uphill and the downhill are one; for these are one, though their definition is not one; and likewise too in the case of what moves and what is moved.
9| But there is a puzzle of logic here. For it is presumably necessary that there be some activity belonging to what acts and to what is acted on: the one is a doing, the other an undergoing, and the work and end of the one is a product, of the other an affection. Since, then, both are motions, if they are different motions, in what are they? Either both are in the thing acted on and moved, or the doing is in the thing doing it while the undergoing is in the thing acted on (and if this too must be called a doing, it would be so only by an equivocation). But if this is so, the motion will be in the thing that moves it (for the same account applies to the mover and the moved), so that either everything that moves will itself be in motion, or a thing that has motion will not be in motion. And if both — the doing and the undergoing — are in the thing moved and acted on, and if teaching and learning, being two, are both in the learner, then, first of all, the activity of each will not belong to each, and next it is absurd for two motions to be going on at once — for what will the two alterations be, belonging to one thing and toward one form? But that is impossible. So the activity will be one. But it is unreasonable for the activity of two things differing in form to be the same, single activity — and yet it will be so, if teaching and learning are the same, and doing and undergoing are the same, and teaching is the same as learning and acting is the same as being acted on, so that it will be necessary for the teacher to be learning everything he teaches, and for the agent to be undergoing what he does.
10| Or rather, there is nothing absurd in the activity of one thing being in another (for teaching is an activity of the person capable of teaching, but it is in something — it is not detached, but belongs to this person and is in that other person), and nothing prevents one and the same activity from belonging to two things — not as though its being were the same in each, but rather as the potentially existing thing stands to what is actually operating — nor is it necessary for the teacher to be learning, even if acting and being acted on are the same, provided this does not mean that the account stating what it is to be each of them is one, as with a cloak and a garment, but rather as with the road from Thebes to Athens and the road from Athens to Thebes, as has also been said before. For it is not the case that all the same things belong to things that are the same in just any way, but only to things whose being is the same. So it does not follow, even if teaching is the same as learning, that learning is the same as teaching, any more than, if the distance between two separated things is one, the moving apart from here to there and from there to here is one and the same. To put it generally, neither teaching and learning, nor doing and undergoing, are, strictly speaking, the same thing — rather, what is the same is that to which these belong, namely the motion. For the being-in-activity of this thing in that one, and of this thing by that one, is different in account. What motion is, then, has been stated, both in general and in particular — for it is not unclear how each of its species will be defined: alteration is the realization of the alterable, as alterable. And, to put it in more familiar terms, it is the realization of what is potentially productive and potentially affectable, as such, both without qualification and again in each particular case — for instance, building or healing. The same account will apply, in the same way, to each of the other kinds of motion as well.
11| Since natural science is concerned with magnitudes and motion and time, each of which must be either infinite or finite, even if not everything is infinite or finite (an affection, for instance, or a point — for perhaps none of such things need be in either of these), it would be fitting for the one who studies nature to examine the infinite: whether it exists or not, and if it exists, what it is. A sign that the inquiry into it belongs to this science is this: all those who seem to have engaged worthily with this kind of philosophy have given an account of the infinite, and all posit it as a kind of principle of the things that are — some, like the Pythagoreans and Plato, positing it in its own right, not as an accident belonging to something else but as the infinite itself being a substance. Except that the Pythagoreans place it among the perceptible things (for they do not make number separable), and hold that what is outside the heaven is infinite; whereas Plato holds that there is no body outside, nor are the forms outside either, since they are not even anywhere, but that the infinite is present both in the perceptible things and in those forms. And some hold that the infinite is the even (for this, they say, when enclosed and bounded by the odd, provides things with infinitude; a sign of this is what happens in the case of numbers: for when the gnomons are placed around the one, and separately, at one time the resulting figure is always different, at another time it is one); Plato, however, holds that there are two infinites, the great and the small.
12| The natural philosophers, on the other hand, all of them posit for the infinite some further, particular nature drawn from the so-called elements — water, for instance, or air, or what is between these. Of those who make the elements finite, none makes them infinite; but all who make the elements infinite, such as Anaxagoras and Democritus — the one out of the things with like parts, the other out of the seed-mixture of shapes — say that the infinite is continuous by contact. And Anaxagoras holds that any part whatever is a mixture similar to the whole, because he sees anything whatever coming to be out of anything whatever; for it is from this, it seems, that he also says that all things were once together, such as this flesh and this bone, and so with anything whatever — and therefore all things were together; and therefore also all at once. For there is a starting point of separating-out not only within each thing but of all things together. For since what comes to be comes to be from a body of this kind, and there is coming-to-be of all things, though not all at once, and there must be some principle of the coming-to-be, and this is one — what he calls Intellect — and Intellect, in thinking, works from some starting point; so it is necessary that all things were at some time together and began at some time to be in motion. Democritus, however, says that none of the primary bodies comes to be from another;
13| but nonetheless, for him, the common body underlying all things is a principle, differing part by part in magnitude and shape. That the inquiry, then, belongs to the natural philosophers is clear from these considerations. And reasonably all posit it as a principle; for it cannot exist in vain, nor can any other power belong to it except that of a principle — for everything is either a principle or from a principle. But there is no principle of the infinite; for then there would be a limit of it. Further, it is also ungenerated and indestructible, as being a kind of principle; for what has come to be must reach an end, and there is a termination to every process of perishing. Therefore, as we say, there is no principle of it, but it itself seems to be the principle of the other things, and to encompass all things and govern all things, as those say who posit no causes beyond the infinite, such as Intellect or Love; and that this is the divine — for it is immortal and indestructible, as Anaximander says, and most of the natural philosophers say. The conviction that there is something infinite would arise, on examination, especially from five considerations
14| namely, from time (for this is infinite), and from the division of magnitudes (for the mathematicians too make use of the infinite); further, from the fact that coming-to-be and perishing would fail to give out only in this way, if that from which what comes to be is drawn off were infinite; further, from the fact that the finite always reaches its limit against something else, so that, if one thing must always be bounded against another, there must be no limit at all. But most of all, and most decisively, there is the consideration that creates the common difficulty for everyone: because thought does not give out, number too seems to be infinite, and so do the mathematical magnitudes, and what is outside the heaven. And since what is outside is infinite, body too seems to be infinite, and worlds as well; for why should there be void here rather than there? So that, if bulk exists in one place, it must exist everywhere. And at the same time, if there is void and infinite place, then body too must exist; for in the case of eternal things, being possible and actually being make no difference.
15| But the inquiry concerning the infinite involves difficulty; for many impossibilities result both for those who posit that it does not exist and for those who posit that it does. Further, in which way does it exist — as a substance, or as an accident in its own right belonging to some particular nature? Or in neither way, but nonetheless there is something infinite, or infinitely many things in number? But it belongs above all to the natural philosopher to examine whether there is an infinite perceptible magnitude. First, then, we must determine in how many senses "the infinite" is spoken of. In one sense, it is what cannot be gone through because it is not of such a nature as to be gone through, just as a sound is invisible; in another sense, it is what has a passage-through that is endless, or one that is scarcely possible, or what, though naturally fitted to have a passage-through or a limit, does not have one. Further, everything infinite is so either by addition or by division or in both ways.
16| That the infinite should exist separately from perceptible things, itself being some infinite thing, is not possible. For if it is neither a magnitude nor a multitude, but the infinite is itself a substance and not an accident, it will be indivisible (for what is divisible will be either a magnitude or a multitude); and if it is such, it is not infinite, except as a sound is invisible. But this is not how those who assert that the infinite exists say it exists, nor is it how we are inquiring about it, but rather as what cannot be traversed. But if the infinite exists by accident, it would not be an element of the things that are, insofar as it is infinite, just as the invisible is not an element of speech, even though sound is invisible. Further, how is it possible for something to be itself infinite, unless number and magnitude also are, of which the infinite is an attribute in its own right? Indeed it is even less necessary for the infinite itself to be a substance than it is for number or magnitude. And it is also clear that the infinite cannot exist as something actual, and as a substance and a principle; for any part of it whatever that is taken will be infinite, if it is divisible into parts (for to be infinite and to be an infinite thing are the same, if the infinite is a substance and not said of a subject), so that it will be either indivisible or divisible into infinitely many parts; but it is impossible for the same thing to be many infinites (and yet, just as a part of air is air, so too a part of the infinite would be infinite, if indeed it is a substance and a principle); therefore it is without parts and indivisible. But it is impossible for the infinite to exist in actuality; for it must be some quantity. Therefore the infinite belongs by accident. But if so, it has been said that it is not possible to call the infinite itself a principle, but rather that to which it is accidental — air, or the even. So that those who speak as the Pythagoreans do would be declaring something absurd; for at the same time they make the infinite a substance and divide it into parts.
17| But perhaps this inquiry is a universal one — whether it is possible for the infinite to exist among mathematical objects as well, and among objects of thought that have no magnitude at all. We, however, are examining perceptible things, and the things about which we are conducting our inquiry: whether or not there is among them a body infinite in the direction of increase. Now to those examining the question logically, it would seem from considerations of the following kind that there is not. For if the definition of body is that which is bounded by a surface, there could not be an infinite body, neither an intelligible one nor a perceptible one (nor, again, could number, taken as separate, be infinite; for number is what is countable, or what has a count, so that if what is countable can be counted, then it would also be possible to traverse the infinite completely). But to those who examine the question more in the manner of natural science, it appears from the following. It cannot be either composite or simple. It will not be a composite infinite body, then, if the elements are finite in number. For there must be more than one element, and the opposites must always be equal to one another, and none of them can be infinite (for if the power of one element in a given body falls short at all of that of the other — suppose fire is finite and air is infinite, and an equal quantity of fire exceeds an equal quantity of air in power by some multiple whatever, provided only that multiple is some definite number — it is nonetheless clear that the infinite will exceed and destroy the finite). But it is impossible for each element to be infinite, since a body is what has extension in every direction, and the infinite is what is extended without limit, so that an infinite body would be extended without limit in every direction.
18| But, moreover, it is not possible for the infinite body to be one and simple either — neither, as some say, something apart from the elements out of which they generate the elements, nor simply one of the elements. For there are some who make this the infinite, rather than air or water, so that the other elements are not destroyed by the one of them being infinite; for the elements stand in opposition to one another — air, for example, is cold, water is wet, fire is hot — and if one of these were infinite, the others would already have been destroyed. As it is, they say that the thing from which these come is something different from them. But it is impossible for such a thing to exist — not because it is infinite (for on that point something must be said that applies alike to every case, to air and water and anything else), but because there is no such perceptible body apart from the so-called elements. For everything is resolved into that from which it comes, and dissolves into it, so that this thing would have to appear here alongside air and fire and earth and water; but nothing of the sort appears. Nor, then, can fire or any other of the elements be infinite. For quite apart from the question of any one of them being infinite, it is impossible for the universe, even if finite, either to be or to become any single one of them, as Heraclitus says everything at some time becomes fire (and the same argument applies also to the one that the natural philosophers posit apart from the elements); for everything changes from opposite into opposite, as, for example, from hot into cold.
19| We must examine the question for every case in general on the following basis, asking whether or not it is possible for such a thing to exist. That it is altogether impossible for there to be an infinite perceptible body is clear from what follows. Every perceptible thing naturally exists somewhere, and there is some place for each thing, and the same place holds for the part as for the whole — for the whole earth and for a single clod alike, and for fire and for a spark alike. So that if the infinite body is uniform in kind, it will be either unmoved or always in motion; yet this is impossible (for why should it move down rather than up, or in any direction whatever? I mean, for instance, if it were a clod, where would it move to, or where would it remain? For the place proper to the body of the same kind as it is infinite. Will it, then, occupy the whole of that place? And how? What, then, or where, is its remaining at rest and its motion? Will it remain everywhere? Then it will not move at all. Or will it move everywhere? Then it will never come to rest). But if the whole is not uniform, then the places are not uniform either; and, first of all, the body of the universe will not be one except by contact; and next, these constituents will be either finite or infinite in kind. They cannot be finite in kind (for then some will be infinite and others not, if the whole is infinite — fire, say, or water
20| — and such a state of affairs means destruction for things that are opposites). But if they are infinite in kind, and simple, then the places too will be infinite, and the elements will be infinite in number; and if that is impossible, and the places are finite, then the whole must be finite too. For it is impossible for place and body not to match each other exactly: the whole place is not greater than the body could possibly be (and in that case the body will no longer be infinite either), nor is the body greater than the place; for otherwise there would be either some void, or a body that by nature exists nowhere. And it is for this reason that none of the natural philosophers made the one infinite body fire or earth, but rather water or air, or what is intermediate between them, because the place proper to each of the first two was clearly marked out, whereas water and air are ambiguous as between up and down. But Anaxagoras —
21| — speaks absurdly about the infinite's remaining at rest. For he says that the infinite fixes itself in place, and this because it is in itself — since nothing else contains it, on the assumption that wherever a thing is, it is its nature to be there. But this is not true; for a thing might be somewhere by force and not where it naturally belongs. So even granting, as far as possible, that the whole does not move (for what is fixed by itself and is in itself must necessarily be unmoved), one must still state why it is not its nature to move. For it is not enough to say this and consider the matter settled; a thing might also fail to move simply because it has nowhere else to move to, while nothing prevents its being naturally disposed to move all the same. Consider the earth: it does not move, even supposing it were infinite, being held fast by the center; yet it would not remain at rest because there is no other place for it to move to, but because that is its nature. And yet one could equally well say that it fixes itself. If, then, this is not the cause even in the case of the earth, supposing it infinite, but rather the fact that it has weight, and what is heavy remains at the center, while the earth is at the center — then likewise the infinite, too, would remain within itself on account of some other cause, and not because it is infinite and fixes itself. At the same time it is clear that any part of it whatever would equally have to remain at rest; for just as the infinite remains in itself, fixing itself, so too any part of it, once taken, will remain in itself. For the places of the whole and of the part are the same in kind — the place of the whole earth and of a clod alike is down, and that of all fire and of a spark alike is up. So that if the place of the infinite is to be within itself, the place of the part is the same too. It will therefore remain within itself.
22| In general, it is clear that it is impossible to say at once that body is infinite and that there is some place for bodies, if every perceptible body has either weight or lightness, and if, being heavy, its natural motion is toward the center, while if light, upward. For this must hold of the infinite body too; but it is impossible for it to be affected in either way, either wholly one way or the other, or half one way and half the other. For how will you divide it? Or how will one part of the infinite be up and another down, or how will it have an outermost part and a middle? Further, every perceptible body is in a place, and the kinds and differentiae of place are up and down, front and back, right and left; and these are marked out not merely relative to us and by our own position, but within the whole universe itself. But it is impossible for these distinctions to exist within the infinite. And, simply put, if it is impossible for place to be infinite, and every body is in a place, then it is impossible for body to be infinite. But surely what is somewhere is in a place, and what is in a place is somewhere. If, then, the infinite cannot even be a determinate quantity — for it would then be some definite quantity, say two cubits or three cubits, since that is what quantity signifies — so too being in a place means being somewhere, and this means being either up or down or in some other of the six directions, and each of these is itself a kind of limit. That there is, then, no body infinite in actuality is clear from these considerations.
23| It is clear that if there is no infinite at all, many impossible consequences follow. For time will have a beginning and an end, magnitudes will not be divisible into magnitudes, and number will not be infinite. Now when, matters having been distinguished in this way, neither alternative appears possible, an arbiter is needed, and it is clear that in one sense the infinite exists and in another it does not. Being, then, is spoken of in one sense as potential and in another as actual, and the infinite exists both by addition and by division. Now it has been said that a magnitude is not actually infinite, but it is infinite by division — for it is not difficult to refute the theory of indivisible lines. It remains, then, that the infinite exists potentially. But we must not take "what exists potentially" the way we do when we say this may possibly become a statue, meaning that this will also actually be a statue, and suppose the infinite is likewise something that will actually be. Rather, since being is spoken of in many ways — just as the day is, and the games are, by one thing after another continually coming to be — so too is the infinite (for in these cases too there is both potentiality and actuality: the Olympic games exist both in the sense that the contest is capable of occurring and in the sense that it is occurring). This is clear in another way too, both in time and in the case of human beings, and in the division of magnitudes. For in general the infinite exists in this way: by one thing after another continually being taken, and what is taken is always finite, but always different in turn —
24| but in the case of magnitudes what has been taken persists, while in the case of time and of human beings, they perish in such a way that the series does not run out. And what proceeds by addition is in a way the same as
25| what proceeds by division. For in a finite whole, addition happens in a way that is the inverse of division: in the way a thing is seen to be divisible without limit, in that same way what is added will appear to tend toward a determinate whole. For if, within a finite magnitude, one takes a determinate amount and keeps taking more in the same ratio, without taking in each time the same fraction of the whole magnitude, one will not use up the finite magnitude; but if one increases the ratio so as always to take in the same magnitude, one will use it up, because every finite thing is exhausted by any determinate amount whatever. So the infinite does not exist in any other way, but it does exist in this way — potentially, and by way of reduction (and it also exists in actuality, in the sense in which we say the day exists, and the games) — and it exists potentially the way matter does, not in its own right, the way the finite does. And so too by way of addition the infinite exists potentially, which we say is in a certain way the same as the infinite by division: for there will always be something to take from outside, but it will never exceed every magnitude — whereas in division what is taken does exceed every determinate amount, and will always be less. But that it should exceed every magnitude by addition is not even possible potentially, unless there is an accidental actual infinite of the kind the natural philosophers speak of — the body outside the universe, whose substance, whether air or something else of that sort, they say is infinite. But if it is not possible for a perceptible body to be actually infinite in that way, then clearly it could not be potentially infinite by addition either, except in the way stated — inversely to division. This, indeed, is why Plato made two infinites: because both in growth and in diminution a thing seems to exceed every limit and go on without end. Yet having made two, he does not use them: for in numbers the infinite by diminution does not occur, since the unit is the smallest, nor does the infinite by growth, since he makes number go only as far as the decad.
26| But it turns out that the infinite is the opposite of what they say: for the infinite is not that outside of which there is nothing, but that outside of which there is always something. Here is a sign of this: people call rings that have no bezel "infinite," because it is always possible to take something further outside — speaking by way of a certain resemblance, but not strictly, since strictly this must also hold: that the same point is never reached again; whereas in a circle this does not happen, but each successive part is simply other than the last. The infinite, then, is that outside of which, however much quantity is taken, there is always something further to take. But that outside of which there is nothing is complete and whole; for this is how we define the whole — that from which nothing is absent, a whole man, say, or a whole box. And just as this holds for a particular case, so it holds strictly in general: the whole is that outside of which there is nothing, while that from which something is absent is not a whole, whatever it is that is missing. Whole and complete are either exactly the same in nature or close to it. And nothing is complete that has no end, and the end is a limit. This is why one should think Parmenides spoke better than Melissus: for the one says the infinite is whole, while the other says the whole is limited, "equally balanced from the middle."
27| For the infinite is not, so to speak, thread joined to thread with the all and the whole — even though it is precisely from this that people derive their reverence for the infinite, the idea that it contains all things and holds the universe within itself, because it bears a certain resemblance to the whole. For the infinite is the matter belonging to the completeness of magnitude, and is the whole potentially, not actually; it is divisible both by reduction and by the addition that is the inverse of that; and it is whole and finite not in its own right but in virtue of something else. And it does not contain but is contained, insofar as it is infinite. This is also why it is unknowable insofar as it is infinite: for matter has no form. So it is clear that the infinite falls under the account of a part rather than of a whole — for matter is a part of the whole, as bronze is a part of the bronze statue. Since, if the infinite really did contain things in the case of perceptible objects, then in the case of intelligible objects the great and the small would have to contain the intelligible objects too. But it is absurd and impossible for what is unknowable and indeterminate to contain and to define what is determinate.
28| It also follows, reasonably enough, that the infinite by addition does not seem to exist in such a way as to exceed every magnitude, while the infinite by division does exist (for matter and the infinite are contained within, while form contains). And it is also reasonable that in the case of number there is a limit toward the least, while toward the greater it always exceeds every plurality, whereas with magnitudes the opposite holds: toward the smaller it exceeds every magnitude, but toward the larger there is no infinite magnitude. The cause is that the one is indivisible, whatever the thing is that is one — one man, for instance, is one man and not many — whereas number is a plurality of ones, some determinate number of them, so that it is necessary to stop at the indivisible (for "three" and "two" are derivative terms, and likewise each of the other numbers), while toward the greater it is always possible to think further, since the bisections of a magnitude are infinite. So it exists potentially, not actually, but what is taken always exceeds every determinate plurality. But this number is not separable, nor does the infinity remain fixed — rather it comes to be, just as time also does, and the number belonging to time.
29| But with magnitudes it is the opposite: the continuous is divided into infinite parts, but toward the greater there is no infinite. For whatever amount is possible potentially, that same amount is also possible actually. So since no perceptible magnitude is infinite, it is not possible for there to be an excess over every determinate magnitude — for then there would be something greater than the heavens. But the infinite is not the same thing in magnitude, in motion, and in time, as though it were a single nature; rather the later is spoken of in accordance with the earlier — motion is called infinite, for instance, because the magnitude over which something moves, or alters, or grows is infinite, and time is called infinite on account of motion. For now, then, we are simply using these terms; later we will also say what each of them is, and why every magnitude is divisible into magnitudes. This account does not take away the mathematicians' study either, by doing away with the infinite in the sense that would make it actually exist so as to be inexhaustible in growth. For they have no need of the infinite even now — they make no use of it — but only need there to be a finite magnitude as large as they wish; and any other magnitude whatever, however large, can be cut in the same ratio as the greatest magnitude. So, as far as proving their results goes, whether the infinite actually occurs among real magnitudes changes nothing for them.
30| Since the causes have been divided four ways, it is clear that the infinite is a cause as matter, and that while its being consists in privation, its substrate in its own right is the continuous and the perceptible. All the others, too, evidently treat the infinite as matter; that is why it is also strange to make it the container rather than something contained.
31| It remains to go through the arguments by which the infinite seems to exist not only potentially but as something determinate. Some of them are not compelling, while others admit certain other, true answers. For it is not necessary, in order that coming-to-be should not run short, that there be a perceptible body infinite in actuality; for it is possible for the destruction of one thing to be the coming-to-be of another, the whole being finite. Again, being in contact and being finite are different things. For contact is relative to something and of something (for everything that is in contact is in contact with something), and belongs accidentally to some finite things, whereas the finite is not itself relative to something; nor can just anything be in contact with just anything. And it is strange to put one's trust in thinking here; for the excess and the deficiency lie not in the thing but in the thinking. For one might conceive of each of us as many times larger than we are, increasing us to infinity; but it does not follow from this that anyone is outside the size he actually has because someone thinks it, but because he actually is that size — the thinking of it is merely accidental. But time and motion are infinite, and so is the process of thinking, without what is taken persisting. Magnitude, however, is not infinite either by division or by increase in thought. As for the infinite, then — how it exists, how it does not exist, and what it is — this has been stated.