Aristotle/Doctrine of Change

The Doctrine of Change, by Aristotle.

The translation used is of Aristotle's Physics, by R. P. Hardie and R. K. Gaye. Glimmerguard has selected materials and has made adjustments to punctuation and grammar to ensure smooth flow of reading in this context.

The full translation by Hardie and Gaye is at MIT classics.

I
Nature has been defined as a 'principle of motion and change', and it is the subject of our inquiry.

We must therefore see that we understand the meaning of 'motion'; for if it were unknown, the meaning of 'nature' too would be unknown. When we have determined the nature of motion, our next task will be to attack in the same way the terms which are involved in it. Now motion is supposed to belong to the class of things which are continuous; and the infinite presents itself first in the continuous.

That is how it comes about that 'infinite' is often used in definitions of the continuous ('what is infinitely divisible is continuous').

Besides these, place, void, and time are thought to be necessary conditions of motion. Clearly, then, for these reasons and also because the attributes mentioned are common to, and coextensive with, all the objects of our science, we must first take each of them in hand and discuss it. For the investigation of special attributes comes after that of the common attributes.

II
To begin then, as we said, with motion. We may start by distinguishing

1. What exists in a state of fulfilment only,

2. What exists as potential,

3. What exists as potential and also in fulfilment. One being a 'this', another 'so much', a third 'such', and similarly in each of the other modes of the predication of being.

III
Further, the word 'relative' is used with reference to

1. excess and defect,

2. agent and patient and generally what can move and what can be moved. For 'what can cause movement' is relative to 'what can be moved', and vice versa. Again, there is no such thing as motion over and above the things. It is always with respect to substance or to quantity or to quality or to place that what changes changes. But it is impossible, as we assert, to find anything common to these which is neither 'this' nor number nor quality nor any of the other predicates. Hence neither will motion and change have reference to something over and above the things mentioned, for there is nothing over and above them.

IV
Now each of these belongs to all its subjects in either of two ways:

1. substance: the one is positive form, the other privation.

2. in quality, white and black.

3. in quantity, complete and incomplete.

4. in respect of locomotion, upwards and downwards or light and heavy.

V
Hence there are as many types of motion or change as there are meanings of the word 'is'. We have now before us the distinctions in the various classes of being between what is full real and what is potential. Definition: The fulfilment of what exists potentially, in so far as it exists potentially, is motion.

Namely,

of what is alterable by virtue of being alterable, alteration.

of what can be increased and its opposite what can be decreased (there is no common name), increase and decrease.

of what can come to be and can pass away, coming to be and passing away.

of what can be carried along, locomotion.

VI
Examples will elucidate this definition of motion. When the buildable, in so far as it is just that, is fully real, it is being built, and this is building. Similarly, learning, doctoring, rolling, leaping, ripening, ageing. The same thing, if it is of a certain kind, can be both potential and fully real, not indeed at the same time or not in the same respect, but that is, potentially hot and actually cold.

Hence at once such things will act and be acted on by one another in many ways: each of them will be capable at the same time of causing alteration and of being altered.

Hence, too, what effects motion as a physical agent can be moved: when a thing of this kind causes motion, it is itself also moved. This, indeed, has led some people to suppose that every mover is moved. But this question depends on another set of arguments, and the truth will be made clear later. is possible for a thing to cause motion, though it is itself incapable of being moved.

VII
It is the fulfilment of what is potential when it is already fully real and operates not as itself but as movable, that is motion.

What I mean by 'as' is this: Bronze is potentially a statue. But it is not the fulfilment of bronze as bronze which is motion. For 'to be bronze' and 'to be a certain potentiality' are not the same. If they were identical without qualification, that is, in definition, then the fulfilment of bronze as bronze would have been motion.

But they are not the same, as has been said. This is obvious in contraries. 'To be capable of health' and 'to be capable of illness' are not the same, for if they were there would be no difference between being ill and being well. Yet the subject both of health and of sickness, whether it is humour or blood, is one and the same. We can distinguish, then, between the two. Just as, to give another example, 'colour' and visible' are different. And clearly it is the fulfilment of what is potential as potential that is motion. So this, precisely, is motion. Further it is evident that motion is an attribute of a thing just when it is fully real in this way, and neither before nor after. For each thing of this kind is capable of being at one time actual, at another not.

Take for instance the buildable as buildable.

The actuality of the buildable as buildable is the process of building. For the actuality of the buildable must be either this or the house. But when there is a house, the buildable is no longer buildable. On the other hand, it is the buildable which is being built. The process then of being built must be the kind of actuality required.

But building is a kind of motion, and the same account will apply to the other kinds also.

VIII
The soundness of this definition is evident both when we consider the accounts of motion that the others have given, and also from the difficulty of defining it otherwise. One could not easily put motion and change in another genus.

This is plain if we consider where some people put it: they identify motion with or 'inequality' or 'not being'. But such things are not necessarily moved, whether they are 'different' or 'unequal' or 'non-existent'.

Nor is change either to or from these rather than to or from their opposites. The reason why they put motion into these genera is that it is thought to be something indefinite, and the principles in the second column are indefinite because they are privative: none of them is either 'this' or 'such' or comes under any of the other modes of predication.

The reason in turn why motion is thought to be indefinite is that it cannot be classed simply as a potentiality or as an actuality.

A thing that is merely capable of having a certain size is not undergoing change, nor yet a thing that is actually of a certain size, and motion is thought to be a sort of actuality, but incomplete, the reason for this view being that the potential whose actuality it is is incomplete.

This is why it is hard to grasp what motion is.

IX
It is necessary to class it with privation or with potentiality or with sheer actuality, yet none of these seems possible.

There remains then the suggested mode of definition, namely that it is a sort of actuality, or actuality of the kind described, hard to grasp, but not incapable of existing. The mover too is moved, as has been said. Every mover, that is, which is capable of motion, and whose immobility is rest.

When a thing is subject to motion its immobility is rest.

For to act on the movable as such is just to move it. But this it does by contact, so that at the same time it is also acted on.

Hence we can define motion as the fulfilment of the movable by virtue of being movable, the cause of the attribute being contact with what can move so that the mover is also acted on.

The mover or agent will always be the vehicle of a form, either a 'this' or 'such', which, when it acts, will be the source and cause of the change, that is, the full-formed man becomes man from what is potentially man.

X
The solution of the difficulty that is raised about the motion, whether it is in the movable, is plain.

It is the fulfilment of this potentiality, and by the action of that which has the power of causing motion.

And the actuality of that which has the power of causing motion is not other than the actuality of the movable, for it must be the fulfilment of both.

A thing is capable of causing motion because it can do this, it is a mover because it actually does it. But it is on the movable that it is capable of acting. Hence there is a single actuality of both alike, just as one to two and two to one are the same interval, and the steep ascent and the steep descent are one.

For these are one and the same, although they can be described in different ways.

So it is with the mover and the moved.

XI
This view has a dialectical difficulty. Perhaps it is necessary that the actuality of the agent and that of the patient should not be the same.

The one is 'agency' and the other 'patiency'; and the outcome and completion of the one is an 'action', that of the other a 'passion'.

Since then they are both motions, we may ask: in what are they, if they are different?

Either

a. both are in what is acted on and moved,

or

b. the agency is in the agent and the patiency in the patient. Now, in alternative (b), the motion will be in the mover, for the same statement will hold of 'mover' and 'moved'. Hence either every mover will be moved, or, though having motion, it will not be moved. If on the other hand (a) both are in what is moved and acted on, both the agency and the patiency. That is, both teaching and learning, though they are two, in the learner, then, first, the actuality of each will not be present in each, and, a second absurdity, a thing will have two motions at the same time. How will there be two alterations of quality in one subject towards one definite quality? The thing is impossible: the actualization will be one. But, some one will say, it is contrary to reason to suppose that there should be one identical actualization of two things which are different in kind.

Yet there will be, if teaching and learning are the same, and agency and patiency.

To teach will be the same as to learn, and to act the same as to be acted on, the teacher will necessarily be learning everything that he teaches, and the agent will be acted on.

One may reply: 1. It is not absurd that the actualization of one thing should be in another. Teaching is the activity of a person who can teach, yet the operation is performed on some patient. It is not cut adrift from a subject, but is of A on B. 2. There is nothing to prevent two things having one and the same actualization, provided the actualizations are not described in the same way, but are related as what can act to what is acting. 3. Nor is it necessary that the teacher should learn, even if to act and to be acted on are one and the same, provided they are not the same in definition (as 'raiment' and 'dress'), but are the same merely in the sense in which the road from Thebes to Athens and the road from Athens to Thebes are the same, as has been explained above.

For it is not things which are in a way the same that have all their attributes the same, but only such as have the same definition.

But indeed it by no means follows from the fact that teaching is the same as learning, that to learn is the same as to teach, any more than it follows from the fact that there is one distance between two things which are at a distance from each other, that the two vectors AB and Ba, are one and the same.

XII
To generalize, teaching is not the same as learning, or agency as patiency, in the full sense, though they belong to the same subject, the motion.

For the 'actualization of X in Y' and the 'actualization of Y through the action of X' differ in definition. What then Motion is, has been stated both generally and particularly.

It is not difficult to see how each of its types will be defined:

Alteration is the fulfillment of the alterable by virtue of being alterable.

Or, more scientifically:

The fulfilment of what can act and what can be acted on, as such, generally and again in each particular case, building, healing, and suchlike.

A similar definition will apply to each of the other kinds of motion.

Book Three Part Four.

XIII
The science of nature is concerned with spatial magnitudes and motion and time, and each of these at least is necessarily infinite or finite, even if some things dealt with by the science are not, that is, a quality or a point.

It is not necessary perhaps that such things should be put under either head.

Hence it is a duty of the person who specializes in physics to discuss the infinite and to inquire whether there is such a thing or not, and, if there is, to describe what it is. The appropriateness to the science of this problem is clearly indicated.

All who have touched on this kind of science in a way worth considering have formulated views about the infinite, and indeed, to a man, make it a principle of things.

XIV
Some, as the Pythagoreans and Plato, make the infinite a principle in the sense of a self-subsistent substance, and not as a mere attribute of some other thing.

Only the Pythagoreans place the infinite among the objects of sense since they do not regard number as separable from these, and assert that what is outside the heaven is infinite.

Plato, on the other hand, holds that there is no body outside, the Forms are not outside because they are nowhere, yet that the infinite is present not only in the objects of sense but in the Forms also. Further, the Pythagoreans identify the infinite with the even. For this, they say, when it is cut off and shut in by the odd, provides things with the element of infinity.

XV
An indication of this is what happens with numbers.

If the gnomons are placed round the one, and without the one, in the one construction the figure that results is always different, in the other it is always the same.

XVI
But Plato has two infinites, the Great and the Small. The physicists, on the other hand, all of them, always regard the infinite as an attribute of a substance which is different from it and belongs to the class of the so-called elements, water or air or what is intermediate between them.

Those who make them limited in number never make them infinite in amount.

But those who make the elements infinite in number, as Anaxagoras and Democritus do, say that the infinite is continuous by contact. Compounded of the homogeneous parts according to the one, of the seed-mass of the atomic shapes according to the other. Further, Anaxagoras held that any part is a mixture in the same way as the All, on the ground of the observed fact that anything comes out of anything. For it is probably for this reason that he maintains that once upon a time all things were together. That is, this flesh and this bone were together, and so of any thing. Therefore all things, and at the same time too.

For there is a beginning of separation, not only for each thing, but for all.

Each thing that comes to be comes from a similar body, and there is a coming to be of all things, though not, it is true, at the same time.

Hence there must also be an origin of coming to be.

One such source there is which he calls Mind, and Mind begins its work of thinking from some starting-point.

So necessarily all things must have been together at a certain time, and must have begun to be moved at a certain time.

XVII
Democritus, for his part, asserts the contrary, namely that no element arises from another element. Nevertheless for him the common body is a source of all things, differing from part to part in size and in shape. It is clear then from these considerations that the inquiry concerns the physicist.

Nor is it without reason that they all make it a principle or source. We cannot say that the infinite has no effect, and the only effectiveness which we can ascribe to it is that of a principle.

Everything is either a source or derived from a source.

But there cannot be a source of the infinite or limitless, for that would be a limit of it. Further, as it is a beginning, it is both uncreatable and indestructible. For there must be a point at which what has come to be reaches completion, and also a termination of all passing away.

That is why, as we say, there is no principle of this, but it is this which is held to be the principle of other things, and to encompass all and to steer all, as those assert who do not recognize, alongside the infinite, other causes, such as Mind or Friendship.

Further they identify it with the Divine, for it is 'deathless and imperishable' as Anaximander says, with the majority of the physicists.

XVIII
Belief in the existence of the infinite comes mainly from five considerations: 1. From the nature of time, for it is infinite.

2. From the division of magnitudes, for the mathematicians also use the notion of the infinite. 3. If coming to be and passing away do not give out, it is only because that from which things come to be is infinite. 4. Because the limited always finds its limit in something, so that there must be no limit, if everything is always limited by something different from itself. 5. Most of all, a reason which is peculiarly appropriate and presents the difficulty that is felt by everybody, not only number but also mathematical magnitudes and what is outside the heaven are supposed to be infinite because they never give out in our thought.

XIX
The last fact, that what is outside is infinite, leads people to suppose that body also is infinite, and that there is an infinite number of worlds.

Why should there be body in one part of the void rather than in another?

Grant only that mass is anywhere and it follows that it must be everywhere.

Also, if void and place are infinite, there must be infinite body too, for in the case of eternal things what may be must be.

But the problem of the infinite is difficult. Many contradictions result whether we suppose it to exist or not to exist. If it exists, we have still to ask how it exists, as a substance or as the essential attribute of some entity? Or in neither way, yet none the less is there something which is infinite or some things which are infinitely many?

XX
The problem, however, which specially belongs to the physicist is to investigate whether there is a sensible magnitude which is infinite. We must begin by distinguishing the various senses in which the term 'infinite' is used. 1. What is incapable of being gone through, because it is not in its nature to be gone through (the sense in which the voice is 'invisible'). 2. What admits of being gone through, the process however having no termination, or what scarcely admits of being gone through. 3. What naturally admits of being gone through, but is not actually gone through or does not actually reach an end. Further, everything that is infinite may be so in respect of addition or division or both.

XXI
Everything which changes does so in one of three senses. It may change:

1. accidentally, as for instance when we say that something musical walks, that which walks being something in which aptitude for music is an accident.

2. a thing is said without qualification to change because something belonging to it changes, i.e. in statements which refer to part of the thing in question: thus the body is restored to health because the eye or the chest, that is to say a part of the whole body, is restored to health.

3. the case of a thing which is in motion neither accidentally nor in respect of something else belonging to it, but in virtue of being itself directly in motion. Here we have a thing which is essentially movable: and that which is so is a different thing according to the particular variety of motion: for instance it may be a thing capable of alteration: and within the sphere of alteration it is again a different thing according as it is capable of being restored to health or capable of being heated.

XXII
And there are the same distinctions in the case of the mover:

1. one thing causes motion accidentally,

2. another partially (because something belonging to it causes motion),

3. another of itself directly, as, for instance, the physician heals, the hand strikes.

XXIII
We have, then, the following factors:

a. on the one hand that which directly causes motion,

b. on the other hand that which is in motion.

c. that in which motion takes place, namely time, and (distinct from these three)

d. that from which it proceeds.

e) that to which it proceeds.

XXIV
For every motion proceeds from something and to something, that which is directly in motion being distinct from that to which it is in motion and that from which it is in motion.

For instance, we may take the three things 'wood', 'hot', and 'cold', of which the first is that which is in motion, the second is that to which the motion proceeds, and the third is that from which it proceeds.

This being so, it is clear that the motion is in the wood, not in its form: for the motion is neither caused nor experienced by the form or the place or the quantity.

So we are left with a mover, a moved, and a goal of motion.

XXV
I do not include the starting-point of motion: for it is the goal rather than the starting-point of motion that gives its name to a particular process of change.

Thus 'perishing' is change to not-being, though it is also true that that that which perishes changes from being: and 'becoming' is change to being, though it is also change from not-being.

Now a definition of motion has been given above, from which it will be seen that every goal of motion, whether it be a form, an affection, or a place, is immovable, as, for instance, knowledge and heat. Here, however, a difficulty may be raised. Affections, it may be said, are motions, and whiteness is an affection: thus there may be change to a motion.

To this we may reply that it is not whiteness but whitening that is a motion.

XXVI
Here also the same distinctions are to be observed: a goal of motion may be so accidentally, or partially and with reference to something other than itself, or directly and with no reference to anything else: for instance, a thing which is becoming white changes accidentally to an object of thought, the colour being only accidentally the object of thought; it changes to colour, because white is a part of colour, or to Europe, because Athens is a part of Europe; but it changes essentially to white colour.

It is now clear in what sense a thing is in motion essentially, accidentally, or in respect of something other than itself, and in what sense the phrase 'itself directly' is used in the case both of the mover and of the moved: and it is also clear that the motion is not in the form but in that which is in motion, that is to say 'the movable in activity'.

Now accidental change we may leave out of account: for it is to be found in everything, at any time, and in any respect.

XXVII
Change which is not accidental on the other hand is not to be found in everything, but only in contraries, in things intermediate contraries, and in contradictories, as may be proved by induction. An intermediate may be a starting-point of change, since for the purposes of the change it serves as contrary to either of two contraries: for the intermediate is in a sense the extremes.

Hence we speak of the intermediate as in a sense a contrary relatively to the extremes and of either extreme as a contrary relatively to the intermediate: for instance, the central note is low relatively-to the highest and high relatively to the lowest, and grey is light relatively to black and dark relatively to white.

XXVIII
And since every change is from something to something-as the word itself (metabole) indicates, implying something 'after' (meta) something else, that is to say something earlier and something later-that which changes must change in one of four ways:

1. from subject to subject,

2. from subject to nonsubject,

3. from non-subject to subject,

4. from non-subject to non-subject,

where by 'subject' I mean what is affirmatively expressed.

XXIX
So it follows necessarily from what has been said above that there are only three kinds of change, that from subject to subject, that from subject to non-subject, and that from non-subject to subject: for the fourth conceivable kind, that from non-subject to nonsubject, is not change, as in that case there is no opposition either of contraries or of contradictories.

Now change from non-subject to subject, the relation being that of contradiction, is 'coming to be'-'unqualified coming to be' when the change takes place in an unqualified way, 'particular coming to be' when the change is change in a particular character: for instance, a change from not-white to white is a coming to be of the particular thing, white, while change from unqualified not-being to being is coming to be in an unqualified way, in respect of which we say that a thing 'comes to be' without qualification, not that it 'comes to be' some particular thing.

Change from subject to non-subject is 'perishing'-'unqualified perishing' when the change is from being to not-being, 'particular perishing' when the change is to the opposite negation, the distinction being the same as that made in the case of coming to be.

Now the expression 'not-being' is used in several senses: and there can be motion neither of that which 'is not' in respect of the affirmation or negation of a predicate, nor of that which 'is not' in the sense that it only potentially 'is', that is to say the opposite of that which actually 'is' in an unqualified sense:

for although that which is 'not-white' or 'not-good' may nevertheless he in motion accidentally (for example that which is 'not-white' might be a man), yet that which is without qualification 'not-so-and-so' cannot in any sense be in motion:

therefore it is impossible for that which is not to be in motion.

XXX
This being so, it follows that 'becoming' cannot be a motion: for it is that which 'is not' that 'becomes'. For however true it may be that it accidentally 'becomes', it is nevertheless correct to say that it is that which 'is not' that in an unqualified sense 'becomes'. And similarly it is impossible for that which 'is not' to be at rest.

There are these difficulties, then, in the way of the assumption that that which 'is not' can be in motion: and it may be further objected that, whereas everything which is in motion is in space, that which 'is not' is not in space: for then it would be somewhere.

So, too, 'perishing' is not a motion: for a motion has for its contrary either another motion or rest, whereas 'perishing' is the contrary of 'becoming'.

XXXI
Since, then, every motion is a kind of change, and there are only the three kinds of change mentioned above, and since of these three those which take the form of 'becoming' and 'perishing', that is to say those which imply a relation of contradiction, are not motions: it necessarily follows that only change from subject to subject is motion.

And every such subject is either a contrary or an intermediate (for a privation may be allowed to rank as a contrary) and can be affirmatively expressed, as naked, toothless, or black.

If, then, the categories are severally distinguished as Being, Quality, Place, Time, Relation, Quantity, and Activity or Passivity, it necessarily follows that there are three kinds of motion-

1. qualitative,

2. quantitative,

3. local.

XXXII
In respect of Substance there is no motion, because Substance has no contrary among things that are.

Nor is there motion in respect of Relation: for it may happen that when one correlative changes, the other, although this does not itself change, is no longer applicable, so that in these cases the motion is accidental.

Nor is there motion in respect of Agent and Patient-in fact there can never be motion of mover and moved, because there cannot be motion of motion or becoming of becoming or in general change of change.

XXXIII
For in the first place there are two senses in which motion of motion is conceivable.

1. The motion of which there is motion might be conceived as subject; e.g. a man is in motion because he changes from fair to dark. Can it be that in this sense motion grows hot or cold, or changes place, or increases or decreases? Impossible: for change is not a subject.

2) can there be motion of motion in the sense that some other subject changes from a change to another mode of being, as e.g. a man changes from falling ill to getting well? Even this is possible only in an accidental sense. For, whatever the subject may be, movement is change from one form to another. (And the same holds good of becoming and perishing, except that in these processes we have a change to a particular kind of opposite, while the other, motion, is a change to a different kind.)

XXXIV
So, if there is to be motion of motion, that which is changing from health to sickness must simultaneously be changing from this very change to another.

It is clear, then, that by the time that it has become sick, it must also have changed to whatever may be the other change concerned (for that it should be at rest, though logically possible, is excluded by the theory).

Moreover this other can never be any casual change, but must be a change from something definite to some other definite thing. So in this case it must be the opposite change, viz. convalescence.

It is only accidentally that there can be change of change, e.g. there is a change from remembering to forgetting only because the subject of this change changes at one time to knowledge, at another to ignorance.

XXXV
In the second place, if there is to be change of change and becoming of becoming, we shall have an infinite regress.

Thus if one of a series of changes is to be a change of change, the preceding change must also be so: e.g. if simple becoming was ever in process of becoming, then that which was becoming simple becoming was also in process of becoming, so that we should not yet have arrived at what was in process of simple becoming but only at what was already in process of becoming in process of becoming.

And this again was sometime in process of becoming, so that even then we should not have arrived at what was in process of simple becoming.

And since in an infinite series there is no first term, here there will be no first stage and therefore no following stage either.

On this hypothesis, then, nothing can become or be moved or change.

XXXVI
Thirdly, if a thing is capable of any particular motion, it is also capable of the corresponding contrary motion or the corresponding coming to rest, and a thing that is capable of becoming is also capable of perishing:

consequently, if there be becoming of becoming, that which is in process of becoming is in process of perishing at the very moment when it has reached the stage of becoming:

since it cannot be in process of perishing when it is just beginning to become or after it has ceased to become: for that which is in process of perishing must be in existence.

XXXVII
Fourthly, there must be a substrate underlying all processes of becoming and changing.

What can this be in the present case?

It is either the body or the soul that undergoes alteration:

what is it that correspondingly becomes motion or becoming?

And again what is the goal of their motion?

It must be the motion or becoming of something from something to something else.

But in what sense can this be so?

For the becoming of learning cannot be learning:

so neither can the becoming of becoming be becoming, nor can the becoming of any process be that process.

XXXVIII
Finally, since there are three kinds of motion, the substratum and the goal of motion must be one or other of these, e.g. locomotion will have to be altered or to be locally moved.

XXXIX
To sum up, then, since everything that is moved is moved in one of three ways, either accidentally, or partially, or essentially, change can change only accidentally, as e.g. when a man who is being restored to health runs or learns: and accidental change we have long ago decided to leave out of account.

Since, then, motion can belong neither to Being nor to Relation nor to Agent and Patient, it remains that there can be motion only in respect of Quality, Quantity, and Place:

for with each of these we have a pair of contraries.

Motion in respect of Quality let us call alteration, a general designation that is used to include both contraries: and by Quality I do not here mean a property of substance (in that sense that which constitutes a specific distinction is a quality) but a passive quality in virtue of which a thing is said to be acted on or to be incapable of being acted on.

Motion in respect of Quantity has no name that includes both contraries, but it is called increase or decrease according as one or the other is designated: that is to say motion in the direction of complete magnitude is increase, motion in the contrary direction is decrease.

Motion in respect of Place has no name either general or particular: but we may designate it by the general name of locomotion, though strictly the term 'locomotion' is applicable to things that change their place only when they have not the power to come to a stand, and to things that do not move themselves locally.

XL
Change within the same kind from a lesser to a greater or from a greater to a lesser degree is alteration:

for it is motion either from a contrary or to a contrary, whether in an unqualified or in a qualified sense:

for change to a lesser degree of a quality will be called change to the contrary of that quality, and change to a greater degree of a quality will be regarded as change from the contrary of that quality to the quality itself.

It makes no difference whether the change be qualified or unqualified, except that in the former case the contraries will have to be contrary to one another only in a qualified sense:

and a thing's possessing a quality in a greater or in a lesser degree means the presence or absence in it of more or less of the opposite quality.

It is now clear, then, that there are only these three kinds of motion.

XLI
The term 'immovable' we apply in the first place to that which is absolutely incapable of being moved (just as we correspondingly apply the term invisible to sound);

in the second place to that which is moved with difficulty after a long time or whose movement is slow at the start-in fact, what we describe as hard to move;

and in the third place to that which is naturally designed for and capable of motion, but is not in motion when, where, and as it naturally would be so. This last is the only kind of immovable thing of which I use the term 'being at rest': for rest is contrary to motion, so that rest will be negation of motion in that which is capable of admitting motion.

XLII
The foregoing remarks are sufficient to explain the essential nature of motion and rest, the number of kinds of change, and the different varieties of motion.

XLIII
Further, everything that changes must be divisible.

For since every change is from something to something, and when a thing is at the goal of its change it is no longer changing, and when both it itself and all its parts are at the starting-point of its change it is not changing, for that which is in whole and in part in an unvarying condition is not in a state of change.

It follows, therefore, that part of that which is changing must be at the starting-point and part at the goal: for as a whole it cannot be in both or in neither.

Here by 'goal of change' I mean that which comes first in the process of change: e.g. in a process of change from white the goal in question will be grey, not black: for it is not necessary that that that which is changing should be at either of the extremes. It is evident, therefore, that everything that changes must be divisible.

XLIV
Now motion is divisible in two senses.

In the first place it is divisible in virtue of the time that it occupies.

In the second place it is divisible according to the motions of the several parts of that which is in motion.

That is, if the whole AG is in motion, there will be a motion of AB and a motion of BG. That being so, let DE be the motion of the part AB and EZ the motion of the part BG. Then the whole DZ must be the motion of AG: for DZ must constitute the motion of AG inasmuch as DE and EZ severally constitute the motions of each of its parts.

But the motion of a thing can never be constituted by the motion of something else: consequently the whole motion is the motion of the whole magnitude.

XLV
Again, since every motion is a motion of something, and the whole motion DZ is not the motion of either of the parts, for each of the parts DE, EZ is the motion of one of the parts AB, BG.

Or of anything else for, the whole motion being the motion of a whole, the parts of the motion are the motions of the parts of that whole: and the parts of DZ are the motions of AB, BG and of nothing else.

Since, as we saw, a motion that is one cannot be the motion of more things than one.

And the whole motion will be the motion of the magnitude ABG.

XLVI
Again, if there is a motion of the whole other than DZ, say the the of each of the arts may be subtracted from it:

and these motions will be equal to DE, EZ respectively: for the motion of that which is one must be one.

So if the whole motion OI may be divided into the motions of the parts, OI will be equal to DZ:

if on the other hand there is any remainder, say KI, this will be a motion of nothing:

for it can be the motion neither of the whole nor of the parts (as the motion of that which is one must be one) nor of anything else:

for a motion that is continuous must be the motion of things that are continuous.

And the same result follows if the division of OI reveals a surplus on the side of the motions of the parts.

Consequently, if this is impossible, the whole motion must be the same as and equal to DZ.

XLVII
This then is what is meant by the division of motion according to the motions of the parts: and it must be applicable to everything that is divisible into parts.

XLVIII
Motion is also susceptible of another kind of division, that according to time.

For since all motion is in time and all time is divisible, and in less time the motion is less, it follows that every motion must be divisible according to time.

And since everything that is in motion is in motion in a certain sphere and for a certain time and has a motion belonging to it, it follows that the time, the motion, the being-in-motion, the thing that is in motion, and the sphere of the motion must all be susceptible of the same divisions (though spheres of motion are not all divisible in a like manner: thus quantity is essentially, quality accidentally divisible).

For suppose that A is the time occupied by the motion B.

Then if all the time has been occupied by the whole motion, it will take less of the motion to occupy half the time, less again to occupy a further subdivision of the time, and so on to infinity.

Again, the time will be divisible similarly to the motion:

for if the whole motion occupies all the time half the motion will occupy half the time, and less of the motion again will occupy less of the time.

XLIX
In the same way the being-in-motion will also be divisible.

For let G be the whole being-in-motion.

Then the being-in-motion that corresponds to half the motion will be less than the whole being-in-motion, that which corresponds to a quarter of the motion will be less again, and so on to infinity.

Moreover by setting out successively the being-in-motion corresponding to each of the two motions DG (say) and GE,

we may argue that the whole being-in-motion will correspond to the whole motion

for if it were some other being-in-motion that corresponded to the whole motion, there would be more than one being-in motion corresponding to the same motion.

The argument being the same as that whereby we showed that the motion of a thing is divisible into the motions of the parts of the thing.

For if we take separately the being-in motion corresponding to each of the two motions, we shall see that the whole being-in motion is continuous.

L
The same reasoning will show the divisibility of the length, and in fact of everything that forms a sphere of change, though some of these are only accidentally divisible because that which changes is so.

For the division of one term will involve the division of all.

So, too, in the matter of their being finite or infinite, they will all alike be either the one or the other.

And we now see that in most cases the fact that all the terms are divisible or infinite is a direct consequence of the fact that the thing that changes is divisible or infinite.

For the attributes 'divisible' and 'infinite' belong in the first instance to the thing that changes. That divisibility does so we have already shown: that infinity does so will be made clear in what follows.

LI
Since everything that changes changes from something to something, that which has changed must at the moment when it has first changed be in that to which it has changed.

For that which changes retires from or leaves that from which it changes

And leaving, if not identical with changing, is at any rate a consequence of it.

And if leaving is a consequence of changing, having left is a consequence of having changed.

For there is a like relation between the two in each case.

LII
One kind of change, then, being change in a relation of contradiction, where a thing has changed from not-being to being it has left not-being.

Therefore it will be in being.

For everything must either be or not be.

It is evident, then, that in contradictory change that which has changed must be in that to which it has changed.

And if this is true in this kind of change, it will be true in all other kinds as well, for in this matter what holds good in the case of one will hold good likewise in the case of the rest.

LIII
Moreover, if we take each kind of change separately, the truth of our conclusion will be equally evident, on the ground that that that which has changed must be somewhere or in something.

For, since it has left that from which it has changed and must be somewhere, it must be either in that to which it has changed or in something else.

If, then, that which has changed to B is in something other than B, say G, it must again be changing from G to B: for it cannot be assumed that there is no interval between G and B, since change is continuous.

Thus we have the result that the thing that has changed, at the moment when it has changed, is changing to that to which it has changed, which is impossible: that which has changed, therefore, must be in that to which it has changed.

So it is evident likewise that that that which has come to be, at the moment when it has come to be, will be, and that which has ceased to be will not-be.

For what we have said applies universally to every kind of change, and its truth is most obvious in the case of contradictory change.

It is clear, then, that that which has changed, at the moment when it has first changed, is in that to which it has changed.

LIV
We will now show that the 'primary when' in which that which has changed effected the completion of its change must be indivisible, where by 'primary' I mean possessing the characteristics in question of itself and not in virtue of the possession of them by something else belonging to it.

For let AG be divisible, and let it be divided at B.

If then the completion of change has been effected in AB or again in BG, AG cannot be the primary thing in which the completion of change has been effected.

If, on the other hand, it has been changing in both AB and BG (for it must either have changed or be changing in each of them), it must have been changing in the whole AG.

But our assumption was that AG contains only the completion of the change.

It is equally impossible to suppose that one part of AG contains the process and the other the completion of the change: for then we shall have something prior to what is primary.

So that in which the completion of change has been effected must be indivisible.

It is also evident, therefore, that that that in which that which has ceased to be has ceased to be and that in which that which has come to be has come to be are indivisible.

LV
But there are two senses of the expression 'the primary when in which something has changed'.

On the one hand it may mean the primary when containing the completion of the process of change- the moment when it is correct to say 'it has changed':

on the other hand it may mean the primary when containing the beginning of the process of change.

Now the primary when that has reference to the end of the change is something really existent: for a change may really be completed, and there is such a thing as an end of change, which we have in fact shown to be indivisible because it is a limit.

But that which has reference to the beginning is not existent at all:

for there is no such thing as a beginning of a process of change, and the time occupied by the change does not contain any primary when in which the change began.

For suppose that AD is such a primary when.

Then it cannot be indivisible: for, if it were, the moment immediately preceding the change and the moment in which the change begins would be consecutive (and moments cannot be consecutive).

Again, if the changing thing is at rest in the whole preceding time GA, for we may suppose that it is at rest, it is at rest in A also: so if AD is without parts, it will simultaneously be at rest and have changed: for it is at rest in A and has changed in D.

Since then AD is not without parts, it must be divisible, and the changing thing must have changed in every part of it, for if it has changed in neither of the two parts into which AD is divided, it has not changed in the whole either.

If, on the other hand, it is in process of change in both parts, it is likewise in process of change in the whole.

And if, again, it has changed in one of the two parts, the whole is not the primary when in which it has changed.

It must therefore have changed in every part. It is evident, then, that with reference to the beginning of change there is no primary when in which change has been effected: for the divisions are infinite.

LVI
So, too, of that which has changed there is no primary part that has changed.

For suppose that of AE the primary part that has changed is AZ.

Everything that changes having been shown to be divisible.

And let OI be the time in which DZ has changed.

If, then, in the whole time DZ has changed, in half the time there will be a part that has changed, less than and therefore prior to DZ.

And again there will be another part prior to this, and yet another, and so on to infinity.

Thus of that which changes there cannot be any primary part that has changed.

It is evident, then, from what has been said, that neither of that which changes nor of the time in which it changes is there any primary part.

LVII
With regard, however, to the actual subject of change-that is to say that in respect of which a thing changes-there is a difference to be observed.

For in a process of change we may distinguish three terms-

1. that which changes,

2. that in which it changes,

3. the actual subject of change, e.g. the man, the time, and the fair complexion.

Of these the man and the time are divisible: but with the fair complexion it is otherwise (though they are all divisible accidentally, for that in which the fair complexion or any other quality is an accident is divisible).

For of actual subjects of change it will be seen that those which are classed as essentially, not accidentally, divisible have no primary part.

Take the case of magnitudes: let AB be a magnitude, and suppose that it has moved from B to a primary 'where' G. Then if BG is taken to be indivisible, two things without parts will have to be contiguous (which is impossible): if on the other hand it is taken to be divisible, there will be something prior to G to which the magnitude has changed, and something else again prior to that, and so on to infinity, because the process of division may be continued without end.

Thus there can be no primary 'where' to which a thing has changed.

And if we take the case of quantitative change, we shall get a like result, for here too the change is in something continuous.

It is evident, then, that only in qualitative motion can there be anything essentially indivisible.

LVIII
Now everything that changes changes time, and that in two senses:

for the time in which a thing is said to change may be the primary time, or on the other hand it may have an extended reference, as e.g. when we say that a thing changes in a particular year because it changes in a particular day.

That being so, that which changes must be changing in any part of the primary time in which it changes.

This is clear from our definition of 'primary', in which the word is said to express just this.

It may also, however, be made evident by the following argument.

Let ChRh be the primary time in which that which is in motion is in motion.

and (as all time is divisible) let it be divided at K.

Now in the time ChK it either is in motion or is not in motion, and the same is likewise true of the time KRh.

Then if it is in motion in neither of the two parts, it will be at rest in the whole:

for it is impossible that it should be in motion in a time in no part of which it is in motion.

If on the other hand it is in motion in only one of the two parts of the time, ChRh cannot be the primary time in which it is in motion.

For its motion will have reference to a time other than ChRh.

It must, then, have been in motion in any part of ChRh.

LIX
And now that this has been proved, it is evident that everything that is in motion must have been in motion before.

For if that which is in motion has traversed the distance KL in the primary time ChRh, in half the time a thing that is in motion with equal velocity and began its motion at the same time will have traversed half the distance.

But if this second thing whose velocity is equal has traversed a certain distance in a certain time, the original thing that is in motion must have traversed the same distance in the same time.

Hence that which is in motion must have been in motion before.

LX
Again, if by taking the extreme moment of the time-for it is the moment that defines the time, and time is that which is intermediate between moments.

We are enabled to say that motion has taken place in the whole time ChRh or in fact in any period of it, motion may likewise be said to have taken place in every other such period.

But half the time finds an extreme in the point of division.

Therefore motion will have taken place in half the time and in fact in any part of it.

For as soon as any division is made there is always a time defined by moments.

If, then, all time is divisible, and that which is intermediate between moments is time, everything that is changing must have completed an infinite number of changes.

LXI
Again, since a thing that changes continuously and has not perished or ceased from its change must either be changing or have changed in any part of the time of its change,

and since it cannot be changing in a moment, it follows that it must have changed at every moment in the time.

Consequently, since the moments are infinite in number, everything that is changing must have completed an infinite number of changes.

LXII
And not only must that which is changing have changed, but that which has changed must also previously have been changing, since everything that has changed from something to something has changed in a period of time.

For suppose that a thing has changed from A to B in a moment.

Now the moment in which it has changed cannot be the same as that in which it is at A (since in that case it would be in A and B at once).

For we have shown above that that that which has changed, when it has changed, is not in that from which it has changed.

If, on the other hand, it is a different moment, there will be a period of time intermediate between the two.

For, as we saw, moments are not consecutive.

Since, then, it has changed in a period of time, and all time is divisible, in half the time it will have completed another change, in a quarter another, and so on to infinity.

Consequently when it has changed, it must have previously been changing.

LXIII
Moreover, the truth of what has been said is more evident in the case of magnitude, because the magnitude over which what is changing changes is continuous.

For suppose that a thing has changed from G to D.

Then if GD is indivisible, two things without parts will be consecutive.

But since this is impossible, that which is intermediate between them must be a magnitude and divisible into an infinite number of segments.

Consequently, before the change is completed, the thing changes to those segments.

Everything that has changed, therefore, must previously have been changing.

For the same proof also holds good of change with respect to what is not continuous, changes, that is to say, between contraries and between contradictories.

In such cases we have only to take the time in which a thing has changed and again apply the same reasoning.

So that which has changed must have been changing and that which is changing must have changed, and a process of change is preceded by a completion of change and a completion by a process: and we can never take any stage and say that it is absolutely the first.

The reason of this is that no two things without parts can be contiguous, and therefore in change the process of division is infinite, just as lines may be infinitely divided so that one part is continually increasing and the other continually decreasing.

LXIV
So it is evident also that that that which has become must previously have been in process of becoming, and that which is in process of becoming must previously have become, everything (that is) that is divisible and continuous.

Though it is not always the actual thing that is in process of becoming of which this is true, sometimes it is something else, that is to say, some part of the thing in question, e.g. the foundation-stone of a house.

So, too, in the case of that which is perishing and that which has perished: for that which becomes and that which perishes must contain an element of infiniteness as an immediate consequence of the fact that they are continuous things.

And so a thing cannot be in process of becoming without having become or have become without having been in process of becoming.

So, too, in the case of perishing and having perished: perishing must be preceded by having perished, and having perished must be preceded by perishing.

It is evident, then, that that which has become must previously have been in process of becoming, and that which is in process of becoming must previously have become: for all magnitudes and all periods of time are infinitely divisible.

LXV
Consequently no absolutely first stage of change can be represented by any particular part of space or time which the changing thing may occupy.