diff --git "a/data/aristotle_physics.txt" "b/data/aristotle_physics.txt"
new file mode 100644--- /dev/null
+++ "b/data/aristotle_physics.txt"
@@ -0,0 +1,7907 @@
+Provided by The Internet Classics Archive.
+See bottom for copyright. Available online at
+    http://classics.mit.edu//Aristotle/physics.html
+
+Physics
+By Aristotle
+
+Translated by R. P. Hardie and R. K. Gaye
+
+----------------------------------------------------------------------
+
+BOOK I
+
+Part 1 
+
+When the objects of an inquiry, in any department, have principles,
+conditions, or elements, it is through acquaintance with these that
+knowledge, that is to say scientific knowledge, is attained. For we
+do not think that we know a thing until we are acquainted with its
+primary conditions or first principles, and have carried our analysis
+as far as its simplest elements. Plainly therefore in the science
+of Nature, as in other branches of study, our first task will be to
+try to determine what relates to its principles. 
+
+The natural way of doing this is to start from the things which are
+more knowable and obvious to us and proceed towards those which are
+clearer and more knowable by nature; for the same things are not 'knowable
+relatively to us' and 'knowable' without qualification. So in the
+present inquiry we must follow this method and advance from what is
+more obscure by nature, but clearer to us, towards what is more clear
+and more knowable by nature. 
+
+Now what is to us plain and obvious at first is rather confused masses,
+the elements and principles of which become known to us later by analysis.
+Thus we must advance from generalities to particulars; for it is a
+whole that is best known to sense-perception, and a generality is
+a kind of whole, comprehending many things within it, like parts.
+Much the same thing happens in the relation of the name to the formula.
+A name, e.g. 'round', means vaguely a sort of whole: its definition
+analyses this into its particular senses. Similarly a child begins
+by calling all men 'father', and all women 'mother', but later on
+distinguishes each of them. 
+
+Part 2
+
+The principles in question must be either (a) one or (b) more than
+one. If (a) one, it must be either (i) motionless, as Parmenides and
+Melissus assert, or (ii) in motion, as the physicists hold, some declaring
+air to be the first principle, others water. If (b) more than one,
+then either (i) a finite or (ii) an infinite plurality. If (i) finite
+(but more than one), then either two or three or four or some other
+number. If (ii) infinite, then either as Democritus believed one in
+kind, but differing in shape or form; or different in kind and even
+contrary. 
+
+A similar inquiry is made by those who inquire into the number of
+existents: for they inquire whether the ultimate constituents of existing
+things are one or many, and if many, whether a finite or an infinite
+plurality. So they too are inquiring whether the principle or element
+is one or many. 
+
+Now to investigate whether Being is one and motionless is not a contribution
+to the science of Nature. For just as the geometer has nothing more
+to say to one who denies the principles of his science-this being
+a question for a different science or for or common to all-so a man
+investigating principles cannot argue with one who denies their existence.
+For if Being is just one, and one in the way mentioned, there is a
+principle no longer, since a principle must be the principle of some
+thing or things. 
+
+To inquire therefore whether Being is one in this sense would be like
+arguing against any other position maintained for the sake of argument
+(such as the Heraclitean thesis, or such a thesis as that Being is
+one man) or like refuting a merely contentious argument-a description
+which applies to the arguments both of Melissus and of Parmenides:
+their premisses are false and their conclusions do not follow. Or
+rather the argument of Melissus is gross and palpable and offers no
+difficulty at all: accept one ridiculous proposition and the rest
+follows-a simple enough proceeding. 
+
+We physicists, on the other hand, must take for granted that the things
+that exist by nature are, either all or some of them, in motion which
+is indeed made plain by induction. Moreover, no man of science is
+bound to solve every kind of difficulty that may be raised, but only
+as many as are drawn falsely from the principles of the science: it
+is not our business to refute those that do not arise in this way:
+just as it is the duty of the geometer to refute the squaring of the
+circle by means of segments, but it is not his duty to refute Antiphon's
+proof. At the same time the holders of the theory of which we are
+speaking do incidentally raise physical questions, though Nature is
+not their subject: so it will perhaps be as well to spend a few words
+on them, especially as the inquiry is not without scientific interest.
+
+The most pertinent question with which to begin will be this: In what
+sense is it asserted that all things are one? For 'is' is used in
+many senses. Do they mean that all things 'are' substance or quantities
+or qualities? And, further, are all things one substance-one man,
+one horse, or one soul-or quality and that one and the same-white
+or hot or something of the kind? These are all very different doctrines
+and all impossible to maintain. 
+
+For if both substance and quantity and quality are, then, whether
+these exist independently of each other or not, Being will be many.
+
+If on the other hand it is asserted that all things are quality or
+quantity, then, whether substance exists or not, an absurdity results,
+if the impossible can properly be called absurd. For none of the others
+can exist independently: substance alone is independent: for everything
+is predicated of substance as subject. Now Melissus says that Being
+is infinite. It is then a quantity. For the infinite is in the category
+of quantity, whereas substance or quality or affection cannot be infinite
+except through a concomitant attribute, that is, if at the same time
+they are also quantities. For to define the infinite you must use
+quantity in your formula, but not substance or quality. If then Being
+is both substance and quantity, it is two, not one: if only substance,
+it is not infinite and has no magnitude; for to have that it will
+have to be a quantity. 
+
+Again, 'one' itself, no less than 'being', is used in many senses,
+so we must consider in what sense the word is used when it is said
+that the All is one. 
+
+Now we say that (a) the continuous is one or that (b) the indivisible
+is one, or (c) things are said to be 'one', when their essence is
+one and the same, as 'liquor' and 'drink'. 
+
+If (a) their One is one in the sense of continuous, it is many, for
+the continuous is divisible ad infinitum. 
+
+There is, indeed, a difficulty about part and whole, perhaps not relevant
+to the present argument, yet deserving consideration on its own account-namely,
+whether the part and the whole are one or more than one, and how they
+can be one or many, and, if they are more than one, in what sense
+they are more than one. (Similarly with the parts of wholes which
+are not continuous.) Further, if each of the two parts is indivisibly
+one with the whole, the difficulty arises that they will be indivisibly
+one with each other also. 
+
+But to proceed: If (b) their One is one as indivisible, nothing will
+have quantity or quality, and so the one will not be infinite, as
+Melissus says-nor, indeed, limited, as Parmenides says, for though
+the limit is indivisible, the limited is not. 
+
+But if (c) all things are one in the sense of having the same definition,
+like 'raiment' and 'dress', then it turns out that they are maintaining
+the Heraclitean doctrine, for it will be the same thing 'to be good'
+and 'to be bad', and 'to be good' and 'to be not good', and so the
+same thing will be 'good' and 'not good', and man and horse; in fact,
+their view will be, not that all things are one, but that they are
+nothing; and that 'to be of such-and-such a quality' is the same as
+'to be of such-and-such a size'. 
+
+Even the more recent of the ancient thinkers were in a pother lest
+the same thing should turn out in their hands both one and many. So
+some, like Lycophron, were led to omit 'is', others to change the
+mode of expression and say 'the man has been whitened' instead of
+'is white', and 'walks' instead of 'is walking', for fear that if
+they added the word 'is' they should be making the one to be many-as
+if 'one' and 'being' were always used in one and the same sense. What
+'is' may be many either in definition (for example 'to be white' is
+one thing, 'to be musical' another, yet the same thing be both, so
+the one is many) or by division, as the whole and its parts. On this
+point, indeed, they were already getting into difficulties and admitted
+that the one was many-as if there was any difficulty about the same
+thing being both one and many, provided that these are not opposites;
+for 'one' may mean either 'potentially one' or 'actually one'.
+
+Part 3
+
+If, then, we approach the thesis in this way it seems impossible for
+all things to be one. Further, the arguments they use to prove their
+position are not difficult to expose. For both of them reason contentiously-I
+mean both Melissus and Parmenides. [Their premisses are false and
+their conclusions do not follow. Or rather the argument of Melissus
+is gross and palpable and offers no difficulty at all: admit one ridiculous
+proposition and the rest follows-a simple enough proceeding.] The
+fallacy of Melissus is obvious. For he supposes that the assumption
+'what has come into being always has a beginning' justifies the assumption
+'what has not come into being has no beginning'. Then this also is
+absurd, that in every case there should be a beginning of the thing-not
+of the time and not only in the case of coming to be in the full sense
+but also in the case of coming to have a quality-as if change never
+took place suddenly. Again, does it follow that Being, if one, is
+motionless? Why should it not move, the whole of it within itself,
+as parts of it do which are unities, e.g. this water? Again, why is
+qualitative change impossible? But, further, Being cannot be one in
+form, though it may be in what it is made of. (Even some of the physicists
+hold it to be one in the latter way, though not in the former.) Man
+obviously differs from horse in form, and contraries from each other.
+
+The same kind of argument holds good against Parmenides also, besides
+any that may apply specially to his view: the answer to him being
+that 'this is not true' and 'that does not follow'. His assumption
+that one is used in a single sense only is false, because it is used
+in several. His conclusion does not follow, because if we take only
+white things, and if 'white' has a single meaning, none the less what
+is white will be many and not one. For what is white will not be one
+either in the sense that it is continuous or in the sense that it
+must be defined in only one way. 'Whiteness' will be different from
+'what has whiteness'. Nor does this mean that there is anything that
+can exist separately, over and above what is white. For 'whiteness'
+and 'that which is white' differ in definition, not in the sense that
+they are things which can exist apart from each other. But Parmenides
+had not come in sight of this distinction. 
+
+It is necessary for him, then, to assume not only that 'being' has
+the same meaning, of whatever it is predicated, but further that it
+means (1) what just is and (2) what is just one. 
+
+It must be so, for (1) an attribute is predicated of some subject,
+so that the subject to which 'being' is attributed will not be, as
+it is something different from 'being'. Something, therefore, which
+is not will be. Hence 'substance' will not be a predicate of anything
+else. For the subject cannot be a being, unless 'being' means several
+things, in such a way that each is something. But ex hypothesi 'being'
+means only one thing. 
+
+If, then, 'substance' is not attributed to anything, but other things
+are attributed to it, how does 'substance' mean what is rather than
+what is not? For suppose that 'substance' is also 'white'. Since the
+definition of the latter is different (for being cannot even be attributed
+to white, as nothing is which is not 'substance'), it follows that
+'white' is not-being--and that not in the sense of a particular not-being,
+but in the sense that it is not at all. Hence 'substance' is not;
+for it is true to say that it is white, which we found to mean not-being.
+If to avoid this we say that even 'white' means substance, it follows
+that 'being' has more than one meaning. 
+
+In particular, then, Being will not have magnitude, if it is substance.
+For each of the two parts must he in a different sense. 
+
+(2) Substance is plainly divisible into other substances, if we consider
+the mere nature of a definition. For instance, if 'man' is a substance,
+'animal' and 'biped' must also be substances. For if not substances,
+they must be attributes-and if attributes, attributes either of (a)
+man or of (b) some other subject. But neither is possible.
+
+(a) An attribute is either that which may or may not belong to the
+subject or that in whose definition the subject of which it is an
+attribute is involved. Thus 'sitting' is an example of a separable
+attribute, while 'snubness' contains the definition of 'nose', to
+which we attribute snubness. Further, the definition of the whole
+is not contained in the definitions of the contents or elements of
+the definitory formula; that of 'man' for instance in 'biped', or
+that of 'white man' in 'white'. If then this is so, and if 'biped'
+is supposed to be an attribute of 'man', it must be either separable,
+so that 'man' might possibly not be 'biped', or the definition of
+'man' must come into the definition of 'biped'-which is impossible,
+as the converse is the case. 
+
+(b) If, on the other hand, we suppose that 'biped' and 'animal' are
+attributes not of man but of something else, and are not each of them
+a substance, then 'man' too will be an attribute of something else.
+But we must assume that substance is not the attribute of anything,
+that the subject of which both 'biped' and 'animal' and each separately
+are predicated is the subject also of the complex 'biped animal'.
+
+Are we then to say that the All is composed of indivisible substances?
+Some thinkers did, in point of fact, give way to both arguments. To
+the argument that all things are one if being means one thing, they
+conceded that not-being is; to that from bisection, they yielded by
+positing atomic magnitudes. But obviously it is not true that if being
+means one thing, and cannot at the same time mean the contradictory
+of this, there will be nothing which is not, for even if what is not
+cannot be without qualification, there is no reason why it should
+not be a particular not-being. To say that all things will be one,
+if there is nothing besides Being itself, is absurd. For who understands
+'being itself' to be anything but a particular substance? But if this
+is so, there is nothing to prevent there being many beings, as has
+been said. 
+
+It is, then, clearly impossible for Being to be one in this sense.
+
+Part 4
+
+The physicists on the other hand have two modes of explanation.
+
+The first set make the underlying body one either one of the three
+or something else which is denser than fire and rarer than air then
+generate everything else from this, and obtain multiplicity by condensation
+and rarefaction. Now these are contraries, which may be generalized
+into 'excess and defect'. (Compare Plato's 'Great and Small'-except
+that he make these his matter, the one his form, while the others
+treat the one which underlies as matter and the contraries as differentiae,
+i.e. forms). 
+
+The second set assert that the contrarieties are contained in the
+one and emerge from it by segregation, for example Anaximander and
+also all those who assert that 'what is' is one and many, like Empedocles
+and Anaxagoras; for they too produce other things from their mixture
+by segregation. These differ, however, from each other in that the
+former imagines a cycle of such changes, the latter a single series.
+Anaxagoras again made both his 'homceomerous' substances and his contraries
+infinite in multitude, whereas Empedocles posits only the so-called
+elements. 
+
+The theory of Anaxagoras that the principles are infinite in multitude
+was probably due to his acceptance of the common opinion of the physicists
+that nothing comes into being from not-being. For this is the reason
+why they use the phrase 'all things were together' and the coming
+into being of such and such a kind of thing is reduced to change of
+quality, while some spoke of combination and separation. Moreover,
+the fact that the contraries proceed from each other led them to the
+conclusion. The one, they reasoned, must have already existed in the
+other; for since everything that comes into being must arise either
+from what is or from what is not, and it is impossible for it to arise
+from what is not (on this point all the physicists agree), they thought
+that the truth of the alternative necessarily followed, namely that
+things come into being out of existent things, i.e. out of things
+already present, but imperceptible to our senses because of the smallness
+of their bulk. So they assert that everything has been mixed in every.
+thing, because they saw everything arising out of everything. But
+things, as they say, appear different from one another and receive
+different names according to the nature of the particles which are
+numerically predominant among the innumerable constituents of the
+mixture. For nothing, they say, is purely and entirely white or black
+or sweet, bone or flesh, but the nature of a thing is held to be that
+of which it contains the most. 
+
+Now (1) the infinite qua infinite is unknowable, so that what is infinite
+in multitude or size is unknowable in quantity, and what is infinite
+in variety of kind is unknowable in quality. But the principles in
+question are infinite both in multitude and in kind. Therefore it
+is impossible to know things which are composed of them; for it is
+when we know the nature and quantity of its components that we suppose
+we know a complex. 
+
+Further (2) if the parts of a whole may be of any size in the direction
+either of greatness or of smallness (by 'parts' I mean components
+into which a whole can be divided and which are actually present in
+it), it is necessary that the whole thing itself may be of any size.
+Clearly, therefore, since it is impossible for an animal or plant
+to be indefinitely big or small, neither can its parts be such, or
+the whole will be the same. But flesh, bone, and the like are the
+parts of animals, and the fruits are the parts of plants. Hence it
+is obvious that neither flesh, bone, nor any such thing can be of
+indefinite size in the direction either of the greater or of the less.
+
+Again (3) according to the theory all such things are already present
+in one another and do not come into being but are constituents which
+are separated out, and a thing receives its designation from its chief
+constituent. Further, anything may come out of anything-water by segregation
+from flesh and flesh from water. Hence, since every finite body is
+exhausted by the repeated abstraction of a finite body, it seems obviously
+to follow that everything cannot subsist in everything else. For let
+flesh be extracted from water and again more flesh be produced from
+the remainder by repeating the process of separation: then, even though
+the quantity separated out will continually decrease, still it will
+not fall below a certain magnitude. If, therefore, the process comes
+to an end, everything will not be in everything else (for there will
+be no flesh in the remaining water); if on the other hand it does
+not, and further extraction is always possible, there will be an infinite
+multitude of finite equal particles in a finite quantity-which is
+impossible. Another proof may be added: Since every body must diminish
+in size when something is taken from it, and flesh is quantitatively
+definite in respect both of greatness and smallness, it is clear that
+from the minimum quantity of flesh no body can be separated out; for
+the flesh left would be less than the minimum of flesh. 
+
+Lastly (4) in each of his infinite bodies there would be already present
+infinite flesh and blood and brain- having a distinct existence, however,
+from one another, and no less real than the infinite bodies, and each
+infinite: which is contrary to reason. 
+
+The statement that complete separation never will take place is correct
+enough, though Anaxagoras is not fully aware of what it means. For
+affections are indeed inseparable. If then colours and states had
+entered into the mixture, and if separation took place, there would
+be a 'white' or a 'healthy' which was nothing but white or healthy,
+i.e. was not the predicate of a subject. So his 'Mind' is an absurd
+person aiming at the impossible, if he is supposed to wish to separate
+them, and it is impossible to do so, both in respect of quantity and
+of quality- of quantity, because there is no minimum magnitude, and
+of quality, because affections are inseparable. 
+
+Nor is Anaxagoras right about the coming to be of homogeneous bodies.
+It is true there is a sense in which clay is divided into pieces of
+clay, but there is another in which it is not. Water and air are,
+and are generated 'from' each other, but not in the way in which bricks
+come 'from' a house and again a house 'from' bricks; and it is better
+to assume a smaller and finite number of principles, as Empedocles
+does. 
+
+Part 5
+
+All thinkers then agree in making the contraries principles, both
+those who describe the All as one and unmoved (for even Parmenides
+treats hot and cold as principles under the names of fire and earth)
+and those too who use the rare and the dense. The same is true of
+Democritus also, with his plenum and void, both of which exist, be
+says, the one as being, the other as not-being. Again he speaks of
+differences in position, shape, and order, and these are genera of
+which the species are contraries, namely, of position, above and below,
+before and behind; of shape, angular and angle-less, straight and
+round. 
+
+It is plain then that they all in one way or another identify the
+contraries with the principles. And with good reason. For first principles
+must not be derived from one another nor from anything else, while
+everything has to be derived from them. But these conditions are fulfilled
+by the primary contraries, which are not derived from anything else
+because they are primary, nor from each other because they are contraries.
+
+But we must see how this can be arrived at as a reasoned result, as
+well as in the way just indicated. 
+
+Our first presupposition must be that in nature nothing acts on, or
+is acted on by, any other thing at random, nor may anything come from
+anything else, unless we mean that it does so in virtue of a concomitant
+attribute. For how could 'white' come from 'musical', unless 'musical'
+happened to be an attribute of the not-white or of the black? No,
+'white' comes from 'not-white'-and not from any 'not-white', but from
+black or some intermediate colour. Similarly, 'musical' comes to be
+from 'not-musical', but not from any thing other than musical, but
+from 'unmusical' or any intermediate state there may be.
+
+Nor again do things pass into the first chance thing; 'white' does
+not pass into 'musical' (except, it may be, in virtue of a concomitant
+attribute), but into 'not-white'-and not into any chance thing which
+is not white, but into black or an intermediate colour; 'musical'
+passes into 'not-musical'-and not into any chance thing other than
+musical, but into 'unmusical' or any intermediate state there may
+be. 
+
+The same holds of other things also: even things which are not simple
+but complex follow the same principle, but the opposite state has
+not received a name, so we fail to notice the fact. What is in tune
+must come from what is not in tune, and vice versa; the tuned passes
+into untunedness-and not into any untunedness, but into the corresponding
+opposite. It does not matter whether we take attunement, order, or
+composition for our illustration; the principle is obviously the same
+in all, and in fact applies equally to the production of a house,
+a statue, or any other complex. A house comes from certain things
+in a certain state of separation instead of conjunction, a statue
+(or any other thing that has been shaped) from shapelessness-each
+of these objects being partly order and partly composition.
+
+If then this is true, everything that comes to be or passes away from,
+or passes into, its contrary or an intermediate state. But the intermediates
+are derived from the contraries-colours, for instance, from black
+and white. Everything, therefore, that comes to be by a natural process
+is either a contrary or a product of contraries. 
+
+Up to this point we have practically had most of the other writers
+on the subject with us, as I have said already: for all of them identify
+their elements, and what they call their principles, with the contraries,
+giving no reason indeed for the theory, but contrained as it were
+by the truth itself. They differ, however, from one another in that
+some assume contraries which are more primary, others contraries which
+are less so: some those more knowable in the order of explanation,
+others those more familiar to sense. For some make hot and cold, or
+again moist and dry, the conditions of becoming; while others make
+odd and even, or again Love and Strife; and these differ from each
+other in the way mentioned. 
+
+Hence their principles are in one sense the same, in another different;
+different certainly, as indeed most people think, but the same inasmuch
+as they are analogous; for all are taken from the same table of columns,
+some of the pairs being wider, others narrower in extent. In this
+way then their theories are both the same and different, some better,
+some worse; some, as I have said, take as their contraries what is
+more knowable in the order of explanation, others what is more familiar
+to sense. (The universal is more knowable in the order of explanation,
+the particular in the order of sense: for explanation has to do with
+the universal, sense with the particular.) 'The great and the small',
+for example, belong to the former class, 'the dense and the rare'
+to the latter. 
+
+It is clear then that our principles must be contraries.
+
+Part 6
+
+The next question is whether the principles are two or three or more
+in number. 
+
+One they cannot be, for there cannot be one contrary. Nor can they
+be innumerable, because, if so, Being will not be knowable: and in
+any one genus there is only one contrariety, and substance is one
+genus: also a finite number is sufficient, and a finite number, such
+as the principles of Empedocles, is better than an infinite multitude;
+for Empedocles professes to obtain from his principles all that Anaxagoras
+obtains from his innumerable principles. Lastly, some contraries are
+more primary than others, and some arise from others-for example sweet
+and bitter, white and black-whereas the principles must always remain
+principles. 
+
+This will suffice to show that the principles are neither one nor
+innumerable. 
+
+Granted, then, that they are a limited number, it is plausible to
+suppose them more than two. For it is difficult to see how either
+density should be of such a nature as to act in any way on rarity
+or rarity on density. The same is true of any other pair of contraries;
+for Love does not gather Strife together and make things out of it,
+nor does Strife make anything out of Love, but both act on a third
+thing different from both. Some indeed assume more than one such thing
+from which they construct the world of nature. 
+
+Other objections to the view that it is not necessary to assume a
+third principle as a substratum may be added. (1) We do not find that
+the contraries constitute the substance of any thing. But what is
+a first principle ought not to be the predicate of any subject. If
+it were, there would be a principle of the supposed principle: for
+the subject is a principle, and prior presumably to what is predicated
+of it. Again (2) we hold that a substance is not contrary to another
+substance. How then can substance be derived from what are not substances?
+Or how can non-substances be prior to substance? 
+
+If then we accept both the former argument and this one, we must,
+to preserve both, assume a third somewhat as the substratum of the
+contraries, such as is spoken of by those who describe the All as
+one nature-water or fire or what is intermediate between them. What
+is intermediate seems preferable; for fire, earth, air, and water
+are already involved with pairs of contraries. There is, therefore,
+much to be said for those who make the underlying substance different
+from these four; of the rest, the next best choice is air, as presenting
+sensible differences in a less degree than the others; and after air,
+water. All, however, agree in this, that they differentiate their
+One by means of the contraries, such as density and rarity and more
+and less, which may of course be generalized, as has already been
+said into excess and defect. Indeed this doctrine too (that the One
+and excess and defect are the principles of things) would appear to
+be of old standing, though in different forms; for the early thinkers
+made the two the active and the one the passive principle, whereas
+some of the more recent maintain the reverse. 
+
+To suppose then that the elements are three in number would seem,
+from these and similar considerations, a plausible view, as I said
+before. On the other hand, the view that they are more than three
+in number would seem to be untenable. 
+
+For the one substratum is sufficient to be acted on; but if we have
+four contraries, there will be two contrarieties, and we shall have
+to suppose an intermediate nature for each pair separately. If, on
+the other hand, the contrarieties, being two, can generate from each
+other, the second contrariety will be superfluous. Moreover, it is
+impossible that there should be more than one primary contrariety.
+For substance is a single genus of being, so that the principles can
+differ only as prior and posterior, not in genus; in a single genus
+there is always a single contrariety, all the other contrarieties
+in it being held to be reducible to one. 
+
+It is clear then that the number of elements is neither one nor more
+than two or three; but whether two or three is, as I said, a question
+of considerable difficulty. 
+
+Part 7
+
+We will now give our own account, approaching the question first with
+reference to becoming in its widest sense: for we shall be following
+the natural order of inquiry if we speak first of common characteristics,
+and then investigate the characteristics of special cases.
+
+We say that one thing comes to be from another thing, and one sort
+of thing from another sort of thing, both in the case of simple and
+of complex things. I mean the following. We can say (1) 'man becomes
+musical', (2) what is 'not-musical becomes musical', or (3), the 'not-musical
+man becomes a musical man'. Now what becomes in (1) and (2)-'man'
+and 'not musical'-I call simple, and what each becomes-'musical'-simple
+also. But when (3) we say the 'not-musical man becomes a musical man',
+both what becomes and what it becomes are complex. 
+
+As regards one of these simple 'things that become' we say not only
+'this becomes so-and-so', but also 'from being this, comes to be so-and-so',
+as 'from being not-musical comes to be musical'; as regards the other
+we do not say this in all cases, as we do not say (1) 'from being
+a man he came to be musical' but only 'the man became musical'.
+
+When a 'simple' thing is said to become something, in one case (1)
+it survives through the process, in the other (2) it does not. For
+man remains a man and is such even when he becomes musical, whereas
+what is not musical or is unmusical does not continue to exist, either
+simply or combined with the subject. 
+
+These distinctions drawn, one can gather from surveying the various
+cases of becoming in the way we are describing that, as we say, there
+must always be an underlying something, namely that which becomes,
+and that this, though always one numerically, in form at least is
+not one. (By that I mean that it can be described in different ways.)
+For 'to be man' is not the same as 'to be unmusical'. One part survives,
+the other does not: what is not an opposite survives (for 'man' survives),
+but 'not-musical' or 'unmusical' does not survive, nor does the compound
+of the two, namely 'unmusical man'. 
+
+We speak of 'becoming that from this' instead of 'this becoming that'
+more in the case of what does not survive the change-'becoming musical
+from unmusical', not 'from man'-but there are exceptions, as we sometimes
+use the latter form of expression even of what survives; we speak
+of 'a statue coming to be from bronze', not of the 'bronze becoming
+a statue'. The change, however, from an opposite which does not survive
+is described indifferently in both ways, 'becoming that from this'
+or 'this becoming that'. We say both that 'the unmusical becomes musical',
+and that 'from unmusical he becomes musical'. And so both forms are
+used of the complex, 'becoming a musical man from an unmusical man',
+and unmusical man becoming a musical man'. 
+
+But there are different senses of 'coming to be'. In some cases we
+do not use the expression 'come to be', but 'come to be so-and-so'.
+Only substances are said to 'come to be' in the unqualified sense.
+
+Now in all cases other than substance it is plain that there must
+be some subject, namely, that which becomes. For we know that when
+a thing comes to be of such a quantity or quality or in such a relation,
+time, or place, a subject is always presupposed, since substance alone
+is not predicated of another subject, but everything else of substance.
+
+But that substances too, and anything else that can be said 'to be'
+without qualification, come to be from some substratum, will appear
+on examination. For we find in every case something that underlies
+from which proceeds that which comes to be; for instance, animals
+and plants from seed. 
+
+Generally things which come to be, come to be in different ways: (1)
+by change of shape, as a statue; (2) by addition, as things which
+grow; (3) by taking away, as the Hermes from the stone; (4) by putting
+together, as a house; (5) by alteration, as things which 'turn' in
+respect of their material substance. 
+
+It is plain that these are all cases of coming to be from a substratum.
+
+Thus, clearly, from what has been said, whatever comes to be is always
+complex. There is, on the one hand, (a) something which comes into
+existence, and again (b) something which becomes that-the latter (b)
+in two senses, either the subject or the opposite. By the 'opposite'
+I mean the 'unmusical', by the 'subject' 'man', and similarly I call
+the absence of shape or form or order the 'opposite', and the bronze
+or stone or gold the 'subject'. 
+
+Plainly then, if there are conditions and principles which constitute
+natural objects and from which they primarily are or have come to
+be-have come to be, I mean, what each is said to be in its essential
+nature, not what each is in respect of a concomitant attribute-plainly,
+I say, everything comes to be from both subject and form. For 'musical
+man' is composed (in a way) of 'man' and 'musical': you can analyse
+it into the definitions of its elements. It is clear then that what
+comes to be will come to be from these elements. 
+
+Now the subject is one numerically, though it is two in form. (For
+it is the man, the gold-the 'matter' generally-that is counted, for
+it is more of the nature of a 'this', and what comes to be does not
+come from it in virtue of a concomitant attribute; the privation,
+on the other hand, and the contrary are incidental in the process.)
+And the positive form is one-the order, the acquired art of music,
+or any similar predicate. 
+
+There is a sense, therefore, in which we must declare the principles
+to be two, and a sense in which they are three; a sense in which the
+contraries are the principles-say for example the musical and the
+unmusical, the hot and the cold, the tuned and the untuned-and a sense
+in which they are not, since it is impossible for the contraries to
+be acted on by each other. But this difficulty also is solved by the
+fact that the substratum is different from the contraries, for it
+is itself not a contrary. The principles therefore are, in a way,
+not more in number than the contraries, but as it were two, nor yet
+precisely two, since there is a difference of essential nature, but
+three. For 'to be man' is different from 'to be unmusical', and 'to
+be unformed' from 'to be bronze'. 
+
+We have now stated the number of the principles of natural objects
+which are subject to generation, and how the number is reached: and
+it is clear that there must be a substratum for the contraries, and
+that the contraries must be two. (Yet in another way of putting it
+this is not necessary, as one of the contraries will serve to effect
+the change by its successive absence and presence.) 
+
+The underlying nature is an object of scientific knowledge, by an
+analogy. For as the bronze is to the statue, the wood to the bed,
+or the matter and the formless before receiving form to any thing
+which has form, so is the underlying nature to substance, i.e. the
+'this' or existent. 
+
+This then is one principle (though not one or existent in the same
+sense as the 'this'), and the definition was one as we agreed; then
+further there is its contrary, the privation. In what sense these
+are two, and in what sense more, has been stated above. Briefly, we
+explained first that only the contraries were principles, and later
+that a substratum was indispensable, and that the principles were
+three; our last statement has elucidated the difference between the
+contraries, the mutual relation of the principles, and the nature
+of the substratum. Whether the form or the substratum is the essential
+nature of a physical object is not yet clear. But that the principles
+are three, and in what sense, and the way in which each is a principle,
+is clear. 
+
+So much then for the question of the number and the nature of the
+principles. 
+
+Part 8
+
+We will now proceed to show that the difficulty of the early thinkers,
+as well as our own, is solved in this way alone. 
+
+The first of those who studied science were misled in their search
+for truth and the nature of things by their inexperience, which as
+it were thrust them into another path. So they say that none of the
+things that are either comes to be or passes out of existence, because
+what comes to be must do so either from what is or from what is not,
+both of which are impossible. For what is cannot come to be (because
+it is already), and from what is not nothing could have come to be
+(because something must be present as a substratum). So too they exaggerated
+the consequence of this, and went so far as to deny even the existence
+of a plurality of things, maintaining that only Being itself is. Such
+then was their opinion, and such the reason for its adoption.
+
+Our explanation on the other hand is that the phrases 'something comes
+to be from what is or from what is not', 'what is not or what is does
+something or has something done to it or becomes some particular thing',
+are to be taken (in the first way of putting our explanation) in the
+same sense as 'a doctor does something or has something done to him',
+'is or becomes something from being a doctor.' These expressions may
+be taken in two senses, and so too, clearly, may 'from being', and
+'being acts or is acted on'. A doctor builds a house, not qua doctor,
+but qua housebuilder, and turns gray, not qua doctor, but qua dark-haired.
+On the other hand he doctors or fails to doctor qua doctor. But we
+are using words most appropriately when we say that a doctor does
+something or undergoes something, or becomes something from being
+a doctor, if he does, undergoes, or becomes qua doctor. Clearly then
+also 'to come to be so-and-so from not-being' means 'qua not-being'.
+
+It was through failure to make this distinction that those thinkers
+gave the matter up, and through this error that they went so much
+farther astray as to suppose that nothing else comes to be or exists
+apart from Being itself, thus doing away with all becoming.
+
+We ourselves are in agreement with them in holding that nothing can
+be said without qualification to come from what is not. But nevertheless
+we maintain that a thing may 'come to be from what is not'-that is,
+in a qualified sense. For a thing comes to be from the privation,
+which in its own nature is not-being,-this not surviving as a constituent
+of the result. Yet this causes surprise, and it is thought impossible
+that something should come to be in the way described from what is
+not. 
+
+In the same way we maintain that nothing comes to be from being, and
+that being does not come to be except in a qualified sense. In that
+way, however, it does, just as animal might come to be from animal,
+and an animal of a certain kind from an animal of a certain kind.
+Thus, suppose a dog to come to be from a horse. The dog would then,
+it is true, come to be from animal (as well as from an animal of a
+certain kind) but not as animal, for that is already there. But if
+anything is to become an animal, not in a qualified sense, it will
+not be from animal: and if being, not from being-nor from not-being
+either, for it has been explained that by 'from not being' we mean
+from not-being qua not-being. 
+
+Note further that we do not subvert the principle that everything
+either is or is not. 
+
+This then is one way of solving the difficulty. Another consists in
+pointing out that the same things can be explained in terms of potentiality
+and actuality. But this has been done with greater precision elsewhere.
+So, as we said, the difficulties which constrain people to deny the
+existence of some of the things we mentioned are now solved. For it
+was this reason which also caused some of the earlier thinkers to
+turn so far aside from the road which leads to coming to be and passing
+away and change generally. If they had come in sight of this nature,
+all their ignorance would have been dispelled. 
+
+Part 9
+
+Others, indeed, have apprehended the nature in question, but not adequately.
+
+In the first place they allow that a thing may come to be without
+qualification from not being, accepting on this point the statement
+of Parmenides. Secondly, they think that if the substratum is one
+numerically, it must have also only a single potentiality-which is
+a very different thing. 
+
+Now we distinguish matter and privation, and hold that one of these,
+namely the matter, is not-being only in virtue of an attribute which
+it has, while the privation in its own nature is not-being; and that
+the matter is nearly, in a sense is, substance, while the privation
+in no sense is. They, on the other hand, identify their Great and
+Small alike with not being, and that whether they are taken together
+as one or separately. Their triad is therefore of quite a different
+kind from ours. For they got so far as to see that there must be some
+underlying nature, but they make it one-for even if one philosopher
+makes a dyad of it, which he calls Great and Small, the effect is
+the same, for he overlooked the other nature. For the one which persists
+is a joint cause, with the form, of what comes to be-a mother, as
+it were. But the negative part of the contrariety may often seem,
+if you concentrate your attention on it as an evil agent, not to exist
+at all. 
+
+For admitting with them that there is something divine, good, and
+desirable, we hold that there are two other principles, the one contrary
+to it, the other such as of its own nature to desire and yearn for
+it. But the consequence of their view is that the contrary desires
+its wtextinction. Yet the form cannot desire itself, for it is not
+defective; nor can the contrary desire it, for contraries are mutually
+destructive. The truth is that what desires the form is matter, as
+the female desires the male and the ugly the beautiful-only the ugly
+or the female not per se but per accidens. 
+
+The matter comes to be and ceases to be in one sense, while in another
+it does not. As that which contains the privation, it ceases to be
+in its own nature, for what ceases to be-the privation-is contained
+within it. But as potentiality it does not cease to be in its own
+nature, but is necessarily outside the sphere of becoming and ceasing
+to be. For if it came to be, something must have existed as a primary
+substratum from which it should come and which should persist in it;
+but this is its own special nature, so that it will be before coming
+to be. (For my definition of matter is just this-the primary substratum
+of each thing, from which it comes to be without qualification, and
+which persists in the result.) And if it ceases to be it will pass
+into that at the last, so it will have ceased to be before ceasing
+to be. 
+
+The accurate determination of the first principle in respect of form,
+whether it is one or many and what it is or what they are, is the
+province of the primary type of science; so these questions may stand
+over till then. But of the natural, i.e. perishable, forms we shall
+speak in the expositions which follow. 
+
+The above, then, may be taken as sufficient to establish that there
+are principles and what they are and how many there are. Now let us
+make a fresh start and proceed. 
+
+----------------------------------------------------------------------
+
+BOOK II
+
+Part 1 
+
+Of things that exist, some exist by nature, some from other causes.
+
+'By nature' the animals and their parts exist, and the plants and
+the simple bodies (earth, fire, air, water)-for we say that these
+and the like exist 'by nature'. 
+
+All the things mentioned present a feature in which they differ from
+things which are not constituted by nature. Each of them has within
+itself a principle of motion and of stationariness (in respect of
+place, or of growth and decrease, or by way of alteration). On the
+other hand, a bed and a coat and anything else of that sort, qua receiving
+these designations i.e. in so far as they are products of art-have
+no innate impulse to change. But in so far as they happen to be composed
+of stone or of earth or of a mixture of the two, they do have such
+an impulse, and just to that extent which seems to indicate that nature
+is a source or cause of being moved and of being at rest in that to
+which it belongs primarily, in virtue of itself and not in virtue
+of a concomitant attribute. 
+
+I say 'not in virtue of a concomitant attribute', because (for instance)
+a man who is a doctor might cure himself. Nevertheless it is not in
+so far as he is a patient that he possesses the art of medicine: it
+merely has happened that the same man is doctor and patient-and that
+is why these attributes are not always found together. So it is with
+all other artificial products. None of them has in itself the source
+of its own production. But while in some cases (for instance houses
+and the other products of manual labour) that principle is in something
+else external to the thing, in others those which may cause a change
+in themselves in virtue of a concomitant attribute-it lies in the
+things themselves (but not in virtue of what they are). 
+
+'Nature' then is what has been stated. Things 'have a nature'which
+have a principle of this kind. Each of them is a substance; for it
+is a subject, and nature always implies a subject in which it inheres.
+
+The term 'according to nature' is applied to all these things and
+also to the attributes which belong to them in virtue of what they
+are, for instance the property of fire to be carried upwards-which
+is not a 'nature' nor 'has a nature' but is 'by nature' or 'according
+to nature'. 
+
+What nature is, then, and the meaning of the terms 'by nature' and
+'according to nature', has been stated. That nature exists, it would
+be absurd to try to prove; for it is obvious that there are many things
+of this kind, and to prove what is obvious by what is not is the mark
+of a man who is unable to distinguish what is self-evident from what
+is not. (This state of mind is clearly possible. A man blind from
+birth might reason about colours. Presumably therefore such persons
+must be talking about words without any thought to correspond.)
+
+Some identify the nature or substance of a natural object with that
+immediate constituent of it which taken by itself is without arrangement,
+e.g. the wood is the 'nature' of the bed, and the bronze the 'nature'
+of the statue. 
+
+As an indication of this Antiphon points out that if you planted a
+bed and the rotting wood acquired the power of sending up a shoot,
+it would not be a bed that would come up, but wood-which shows that
+the arrangement in accordance with the rules of the art is merely
+an incidental attribute, whereas the real nature is the other, which,
+further, persists continuously through the process of making.
+
+But if the material of each of these objects has itself the same relation
+to something else, say bronze (or gold) to water, bones (or wood)
+to earth and so on, that (they say) would be their nature and essence.
+Consequently some assert earth, others fire or air or water or some
+or all of these, to be the nature of the things that are. For whatever
+any one of them supposed to have this character-whether one thing
+or more than one thing-this or these he declared to be the whole of
+substance, all else being its affections, states, or dispositions.
+Every such thing they held to be eternal (for it could not pass into
+anything else), but other things to come into being and cease to be
+times without number. 
+
+This then is one account of 'nature', namely that it is the immediate
+material substratum of things which have in themselves a principle
+of motion or change. 
+
+Another account is that 'nature' is the shape or form which is specified
+in the definition of the thing. 
+
+For the word 'nature' is applied to what is according to nature and
+the natural in the same way as 'art' is applied to what is artistic
+or a work of art. We should not say in the latter case that there
+is anything artistic about a thing, if it is a bed only potentially,
+not yet having the form of a bed; nor should we call it a work of
+art. The same is true of natural compounds. What is potentially flesh
+or bone has not yet its own 'nature', and does not exist until it
+receives the form specified in the definition, which we name in defining
+what flesh or bone is. Thus in the second sense of 'nature' it would
+be the shape or form (not separable except in statement) of things
+which have in themselves a source of motion. (The combination of the
+two, e.g. man, is not 'nature' but 'by nature' or 'natural'.)
+
+The form indeed is 'nature' rather than the matter; for a thing is
+more properly said to be what it is when it has attained to fulfilment
+than when it exists potentially. Again man is born from man, but not
+bed from bed. That is why people say that the figure is not the nature
+of a bed, but the wood is-if the bed sprouted not a bed but wood would
+come up. But even if the figure is art, then on the same principle
+the shape of man is his nature. For man is born from man.
+
+We also speak of a thing's nature as being exhibited in the process
+of growth by which its nature is attained. The 'nature' in this sense
+is not like 'doctoring', which leads not to the art of doctoring but
+to health. Doctoring must start from the art, not lead to it. But
+it is not in this way that nature (in the one sense) is related to
+nature (in the other). What grows qua growing grows from something
+into something. Into what then does it grow? Not into that from which
+it arose but into that to which it tends. The shape then is nature.
+
+'Shape' and 'nature', it should be added, are in two senses. For the
+privation too is in a way form. But whether in unqualified coming
+to be there is privation, i.e. a contrary to what comes to be, we
+must consider later. 
+
+Part 2
+
+We have distinguished, then, the different ways in which the term
+'nature' is used. 
+
+The next point to consider is how the mathematician differs from the
+physicist. Obviously physical bodies contain surfaces and volumes,
+lines and points, and these are the subject-matter of mathematics.
+
+Further, is astronomy different from physics or a department of it?
+It seems absurd that the physicist should be supposed to know the
+nature of sun or moon, but not to know any of their essential attributes,
+particularly as the writers on physics obviously do discuss their
+shape also and whether the earth and the world are spherical or not.
+
+Now the mathematician, though he too treats of these things, nevertheless
+does not treat of them as the limits of a physical body; nor does
+he consider the attributes indicated as the attributes of such bodies.
+That is why he separates them; for in thought they are separable from
+motion, and it makes no difference, nor does any falsity result, if
+they are separated. The holders of the theory of Forms do the same,
+though they are not aware of it; for they separate the objects of
+physics, which are less separable than those of mathematics. This
+becomes plain if one tries to state in each of the two cases the definitions
+of the things and of their attributes. 'Odd' and 'even', 'straight'
+and 'curved', and likewise 'number', 'line', and 'figure', do not
+involve motion; not so 'flesh' and 'bone' and 'man'-these are defined
+like 'snub nose', not like 'curved'. 
+
+Similar evidence is supplied by the more physical of the branches
+of mathematics, such as optics, harmonics, and astronomy. These are
+in a way the converse of geometry. While geometry investigates physical
+lines but not qua physical, optics investigates mathematical lines,
+but qua physical, not qua mathematical. 
+
+Since 'nature' has two senses, the form and the matter, we must investigate
+its objects as we would the essence of snubness. That is, such things
+are neither independent of matter nor can be defined in terms of matter
+only. Here too indeed one might raise a difficulty. Since there are
+two natures, with which is the physicist concerned? Or should he investigate
+the combination of the two? But if the combination of the two, then
+also each severally. Does it belong then to the same or to different
+sciences to know each severally? 
+
+If we look at the ancients, physics would to be concerned with the
+matter. (It was only very slightly that Empedocles and Democritus
+touched on the forms and the essence.) 
+
+But if on the other hand art imitates nature, and it is the part of
+the same discipline to know the form and the matter up to a point
+(e.g. the doctor has a knowledge of health and also of bile and phlegm,
+in which health is realized, and the builder both of the form of the
+house and of the matter, namely that it is bricks and beams, and so
+forth): if this is so, it would be the part of physics also to know
+nature in both its senses. 
+
+Again, 'that for the sake of which', or the end, belongs to the same
+department of knowledge as the means. But the nature is the end or
+'that for the sake of which'. For if a thing undergoes a continuous
+change and there is a stage which is last, this stage is the end or
+'that for the sake of which'. (That is why the poet was carried away
+into making an absurd statement when he said 'he has the end for the
+sake of which he was born'. For not every stage that is last claims
+to be an end, but only that which is best.) 
+
+For the arts make their material (some simply 'make' it, others make
+it serviceable), and we use everything as if it was there for our
+sake. (We also are in a sense an end. 'That for the sake of which'
+has two senses: the distinction is made in our work On Philosophy.)
+The arts, therefore, which govern the matter and have knowledge are
+two, namely the art which uses the product and the art which directs
+the production of it. That is why the using art also is in a sense
+directive; but it differs in that it knows the form, whereas the art
+which is directive as being concerned with production knows the matter.
+For the helmsman knows and prescribes what sort of form a helm should
+have, the other from what wood it should be made and by means of what
+operations. In the products of art, however, we make the material
+with a view to the function, whereas in the products of nature the
+matter is there all along. 
+
+Again, matter is a relative term: to each form there corresponds a
+special matter. How far then must the physicist know the form or essence?
+Up to a point, perhaps, as the doctor must know sinew or the smith
+bronze (i.e. until he understands the purpose of each): and the physicist
+is concerned only with things whose forms are separable indeed, but
+do not exist apart from matter. Man is begotten by man and by the
+sun as well. The mode of existence and essence of the separable it
+is the business of the primary type of philosophy to define.
+
+Part 3
+
+Now that we have established these distinctions, we must proceed to
+consider causes, their character and number. Knowledge is the object
+of our inquiry, and men do not think they know a thing till they have
+grasped the 'why' of (which is to grasp its primary cause). So clearly
+we too must do this as regards both coming to be and passing away
+and every kind of physical change, in order that, knowing their principles,
+we may try to refer to these principles each of our problems.
+
+In one sense, then, (1) that out of which a thing comes to be and
+which persists, is called 'cause', e.g. the bronze of the statue,
+the silver of the bowl, and the genera of which the bronze and the
+silver are species. 
+
+In another sense (2) the form or the archetype, i.e. the statement
+of the essence, and its genera, are called 'causes' (e.g. of the octave
+the relation of 2:1, and generally number), and the parts in the definition.
+
+Again (3) the primary source of the change or coming to rest; e.g.
+the man who gave advice is a cause, the father is cause of the child,
+and generally what makes of what is made and what causes change of
+what is changed. 
+
+Again (4) in the sense of end or 'that for the sake of which' a thing
+is done, e.g. health is the cause of walking about. ('Why is he walking
+about?' we say. 'To be healthy', and, having said that, we think we
+have assigned the cause.) The same is true also of all the intermediate
+steps which are brought about through the action of something else
+as means towards the end, e.g. reduction of flesh, purging, drugs,
+or surgical instruments are means towards health. All these things
+are 'for the sake of' the end, though they differ from one another
+in that some are activities, others instruments. 
+
+This then perhaps exhausts the number of ways in which the term 'cause'
+is used. 
+
+As the word has several senses, it follows that there are several
+causes of the same thing not merely in virtue of a concomitant attribute),
+e.g. both the art of the sculptor and the bronze are causes of the
+statue. These are causes of the statue qua statue, not in virtue of
+anything else that it may be-only not in the same way, the one being
+the material cause, the other the cause whence the motion comes. Some
+things cause each other reciprocally, e.g. hard work causes fitness
+and vice versa, but again not in the same way, but the one as end,
+the other as the origin of change. Further the same thing is the cause
+of contrary results. For that which by its presence brings about one
+result is sometimes blamed for bringing about the contrary by its
+absence. Thus we ascribe the wreck of a ship to the absence of the
+pilot whose presence was the cause of its safety. 
+
+All the causes now mentioned fall into four familiar divisions. The
+letters are the causes of syllables, the material of artificial products,
+fire, &c., of bodies, the parts of the whole, and the premisses of
+the conclusion, in the sense of 'that from which'. Of these pairs
+the one set are causes in the sense of substratum, e.g. the parts,
+the other set in the sense of essence-the whole and the combination
+and the form. But the seed and the doctor and the adviser, and generally
+the maker, are all sources whence the change or stationariness originates,
+while the others are causes in the sense of the end or the good of
+the rest; for 'that for the sake of which' means what is best and
+the end of the things that lead up to it. (Whether we say the 'good
+itself or the 'apparent good' makes no difference.) 
+
+Such then is the number and nature of the kinds of cause.
+
+Now the modes of causation are many, though when brought under heads
+they too can be reduced in number. For 'cause' is used in many senses
+and even within the same kind one may be prior to another (e.g. the
+doctor and the expert are causes of health, the relation 2:1 and number
+of the octave), and always what is inclusive to what is particular.
+Another mode of causation is the incidental and its genera, e.g. in
+one way 'Polyclitus', in another 'sculptor' is the cause of a statue,
+because 'being Polyclitus' and 'sculptor' are incidentally conjoined.
+Also the classes in which the incidental attribute is included; thus
+'a man' could be said to be the cause of a statue or, generally, 'a
+living creature'. An incidental attribute too may be more or less
+remote, e.g. suppose that 'a pale man' or 'a musical man' were said
+to be the cause of the statue. 
+
+All causes, both proper and incidental, may be spoken of either as
+potential or as actual; e.g. the cause of a house being built is either
+'house-builder' or 'house-builder building'. 
+
+Similar distinctions can be made in the things of which the causes
+are causes, e.g. of 'this statue' or of 'statue' or of 'image' generally,
+of 'this bronze' or of 'bronze' or of 'material' generally. So too
+with the incidental attributes. Again we may use a complex expression
+for either and say, e.g. neither 'Polyclitus' nor 'sculptor' but 'Polyclitus,
+sculptor'. 
+
+All these various uses, however, come to six in number, under each
+of which again the usage is twofold. Cause means either what is particular
+or a genus, or an incidental attribute or a genus of that, and these
+either as a complex or each by itself; and all six either as actual
+or as potential. The difference is this much, that causes which are
+actually at work and particular exist and cease to exist simultaneously
+with their effect, e.g. this healing person with this being-healed
+person and that house-building man with that being-built house; but
+this is not always true of potential causes--the house and the housebuilder
+do not pass away simultaneously. 
+
+In investigating the cause of each thing it is always necessary to
+seek what is most precise (as also in other things): thus man builds
+because he is a builder, and a builder builds in virtue of his art
+of building. This last cause then is prior: and so generally.
+
+Further, generic effects should be assigned to generic causes, particular
+effects to particular causes, e.g. statue to sculptor, this statue
+to this sculptor; and powers are relative to possible effects, actually
+operating causes to things which are actually being effected.
+
+This must suffice for our account of the number of causes and the
+modes of causation. 
+
+Part 4
+
+But chance also and spontaneity are reckoned among causes: many things
+are said both to be and to come to be as a result of chance and spontaneity.
+We must inquire therefore in what manner chance and spontaneity are
+present among the causes enumerated, and whether they are the same
+or different, and generally what chance and spontaneity are.
+
+Some people even question whether they are real or not. They say that
+nothing happens by chance, but that everything which we ascribe to
+chance or spontaneity has some definite cause, e.g. coming 'by chance'
+into the market and finding there a man whom one wanted but did not
+expect to meet is due to one's wish to go and buy in the market. Similarly
+in other cases of chance it is always possible, they maintain, to
+find something which is the cause; but not chance, for if chance were
+real, it would seem strange indeed, and the question might be raised,
+why on earth none of the wise men of old in speaking of the causes
+of generation and decay took account of chance; whence it would seem
+that they too did not believe that anything is by chance. But there
+is a further circumstance that is surprising. Many things both come
+to be and are by chance and spontaneity, and although know that each
+of them can be ascribed to some cause (as the old argument said which
+denied chance), nevertheless they speak of some of these things as
+happening by chance and others not. For this reason also they ought
+to have at least referred to the matter in some way or other.
+
+Certainly the early physicists found no place for chance among the
+causes which they recognized-love, strife, mind, fire, or the like.
+This is strange, whether they supposed that there is no such thing
+as chance or whether they thought there is but omitted to mention
+it-and that too when they sometimes used it, as Empedocles does when
+he says that the air is not always separated into the highest region,
+but 'as it may chance'. At any rate he says in his cosmogony that
+'it happened to run that way at that time, but it often ran otherwise.'
+He tells us also that most of the parts of animals came to be by chance.
+
+There are some too who ascribe this heavenly sphere and all the worlds
+to spontaneity. They say that the vortex arose spontaneously, i.e.
+the motion that separated and arranged in its present order all that
+exists. This statement might well cause surprise. For they are asserting
+that chance is not responsible for the existence or generation of
+animals and plants, nature or mind or something of the kind being
+the cause of them (for it is not any chance thing that comes from
+a given seed but an olive from one kind and a man from another); and
+yet at the same time they assert that the heavenly sphere and the
+divinest of visible things arose spontaneously, having no such cause
+as is assigned to animals and plants. Yet if this is so, it is a fact
+which deserves to be dwelt upon, and something might well have been
+said about it. For besides the other absurdities of the statement,
+it is the more absurd that people should make it when they see nothing
+coming to be spontaneously in the heavens, but much happening by chance
+among the things which as they say are not due to chance; whereas
+we should have expected exactly the opposite. 
+
+Others there are who, indeed, believe that chance is a cause, but
+that it is inscrutable to human intelligence, as being a divine thing
+and full of mystery. 
+
+Thus we must inquire what chance and spontaneity are, whether they
+are the same or different, and how they fit into our division of causes.
+
+Part 5
+
+First then we observe that some things always come to pass in the
+same way, and others for the most part. It is clearly of neither of
+these that chance is said to be the cause, nor can the 'effect of
+chance' be identified with any of the things that come to pass by
+necessity and always, or for the most part. But as there is a third
+class of events besides these two-events which all say are 'by chance'-it
+is plain that there is such a thing as chance and spontaneity; for
+we know that things of this kind are due to chance and that things
+due to chance are of this kind. 
+
+But, secondly, some events are for the sake of something, others not.
+Again, some of the former class are in accordance with deliberate
+intention, others not, but both are in the class of things which are
+for the sake of something. Hence it is clear that even among the things
+which are outside the necessary and the normal, there are some in
+connexion withwhich the phrase 'for the sake of something' is applicable.
+(Events that are for the sake of something include whatever may be
+done as a result of thought or of nature.) Things of this kind, then,
+when they come to pass incidental are said to be 'by chance'. For
+just as a thing is something either in virtue of itself or incidentally,
+so may it be a cause. For instance, the housebuilding faculty is in
+virtue of itself the cause of a house, whereas the pale or the musical
+is the incidental cause. That which is per se cause of the effect
+is determinate, but the incidental cause is indeterminable, for the
+possible attributes of an individual are innumerable. To resume then;
+when a thing of this kind comes to pass among events which are for
+the sake of something, it is said to be spontaneous or by chance.
+(The distinction between the two must be made later-for the present
+it is sufficient if it is plain that both are in the sphere of things
+done for the sake of something.) 
+
+Example: A man is engaged in collecting subscriptions for a feast.
+He would have gone to such and such a place for the purpose of getting
+the money, if he had known. He actually went there for another purpose
+and it was only incidentally that he got his money by going there;
+and this was not due to the fact that he went there as a rule or necessarily,
+nor is the end effected (getting the money) a cause present in himself-it
+belongs to the class of things that are intentional and the result
+of intelligent deliberation. It is when these conditions are satisfied
+that the man is said to have gone 'by chance'. If he had gone of deliberate
+purpose and for the sake of this-if he always or normally went there
+when he was collecting payments-he would not be said to have gone
+'by chance'. 
+
+It is clear then that chance is an incidental cause in the sphere
+of those actions for the sake of something which involve purpose.
+Intelligent reflection, then, and chance are in the same sphere, for
+purpose implies intelligent reflection. 
+
+It is necessary, no doubt, that the causes of what comes to pass by
+chance be indefinite; and that is why chance is supposed to belong
+to the class of the indefinite and to be inscrutable to man, and why
+it might be thought that, in a way, nothing occurs by chance. For
+all these statements are correct, because they are well grounded.
+Things do, in a way, occur by chance, for they occur incidentally
+and chance is an incidental cause. But strictly it is not the cause-without
+qualification-of anything; for instance, a housebuilder is the cause
+of a house; incidentally, a fluteplayer may be so. 
+
+And the causes of the man's coming and getting the money (when he
+did not come for the sake of that) are innumerable. He may have wished
+to see somebody or been following somebody or avoiding somebody, or
+may have gone to see a spectacle. Thus to say that chance is a thing
+contrary to rule is correct. For 'rule' applies to what is always
+true or true for the most part, whereas chance belongs to a third
+type of event. Hence, to conclude, since causes of this kind are indefinite,
+chance too is indefinite. (Yet in some cases one might raise the question
+whether any incidental fact might be the cause of the chance occurrence,
+e.g. of health the fresh air or the sun's heat may be the cause, but
+having had one's hair cut cannot; for some incidental causes are more
+relevant to the effect than others.) 
+
+Chance or fortune is called 'good' when the result is good, 'evil'
+when it is evil. The terms 'good fortune' and 'ill fortune' are used
+when either result is of considerable magnitude. Thus one who comes
+within an ace of some great evil or great good is said to be fortunate
+or unfortunate. The mind affirms the essence of the attribute, ignoring
+the hair's breadth of difference. Further, it is with reason that
+good fortune is regarded as unstable; for chance is unstable, as none
+of the things which result from it can be invariable or normal.
+
+Both are then, as I have said, incidental causes-both chance and spontaneity-in
+the sphere of things which are capable of coming to pass not necessarily,
+nor normally, and with reference to such of these as might come to
+pass for the sake of something. 
+
+Part 6
+
+They differ in that 'spontaneity' is the wider term. Every result
+of chance is from what is spontaneous, but not everything that is
+from what is spontaneous is from chance. 
+
+Chance and what results from chance are appropriate to agents that
+are capable of good fortune and of moral action generally. Therefore
+necessarily chance is in the sphere of moral actions. This is indicated
+by the fact that good fortune is thought to be the same, or nearly
+the same, as happiness, and happiness to be a kind of moral action,
+since it is well-doing. Hence what is not capable of moral action
+cannot do anything by chance. Thus an inanimate thing or a lower animal
+or a child cannot do anything by chance, because it is incapable of
+deliberate intention; nor can 'good fortune' or 'ill fortune' be ascribed
+to them, except metaphorically, as Protarchus, for example, said that
+the stones of which altars are made are fortunate because they are
+held in honour, while their fellows are trodden under foot. Even these
+things, however, can in a way be affected by chance, when one who
+is dealing with them does something to them by chance, but not otherwise.
+
+The spontaneous on the other hand is found both in the lower animals
+and in many inanimate objects. We say, for example, that the horse
+came 'spontaneously', because, though his coming saved him, he did
+not come for the sake of safety. Again, the tripod fell 'of itself',
+because, though when it fell it stood on its feet so as to serve for
+a seat, it did not fall for the sake of that. 
+
+Hence it is clear that events which (1) belong to the general class
+of things that may come to pass for the sake of something, (2) do
+not come to pass for the sake of what actually results, and (3) have
+an external cause, may be described by the phrase 'from spontaneity'.
+These 'spontaneous' events are said to be 'from chance' if they have
+the further characteristics of being the objects of deliberate intention
+and due to agents capable of that mode of action. This is indicated
+by the phrase 'in vain', which is used when A which is for the sake
+of B, does not result in B. For instance, taking a walk is for the
+sake of evacuation of the bowels; if this does not follow after walking,
+we say that we have walked 'in vain' and that the walking was 'vain'.
+This implies that what is naturally the means to an end is 'in vain',
+when it does not effect the end towards which it was the natural means-for
+it would be absurd for a man to say that he had bathed in vain because
+the sun was not eclipsed, since the one was not done with a view to
+the other. Thus the spontaneous is even according to its derivation
+the case in which the thing itself happens in vain. The stone that
+struck the man did not fall for the purpose of striking him; therefore
+it fell spontaneously, because it might have fallen by the action
+of an agent and for the purpose of striking. The difference between
+spontaneity and what results by chance is greatest in things that
+come to be by nature; for when anything comes to be contrary to nature,
+we do not say that it came to be by chance, but by spontaneity. Yet
+strictly this too is different from the spontaneous proper; for the
+cause of the latter is external, that of the former internal.
+
+We have now explained what chance is and what spontaneity is, and
+in what they differ from each other. Both belong to the mode of causation
+'source of change', for either some natural or some intelligent agent
+is always the cause; but in this sort of causation the number of possible
+causes is infinite. 
+
+Spontaneity and chance are causes of effects which though they might
+result from intelligence or nature, have in fact been caused by something
+incidentally. Now since nothing which is incidental is prior to what
+is per se, it is clear that no incidental cause can be prior to a
+cause per se. Spontaneity and chance, therefore, are posterior to
+intelligence and nature. Hence, however true it may be that the heavens
+are due to spontaneity, it will still be true that intelligence and
+nature will be prior causes of this All and of many things in it besides.
+
+Part 7
+
+It is clear then that there are causes, and that the number of them
+is what we have stated. The number is the same as that of the things
+comprehended under the question 'why'. The 'why' is referred ultimately
+either (1), in things which do not involve motion, e.g. in mathematics,
+to the 'what' (to the definition of 'straight line' or 'commensurable',
+&c.), or (2) to what initiated a motion, e.g. 'why did they go to
+war?-because there had been a raid'; or (3) we are inquiring 'for
+the sake of what?'-'that they may rule'; or (4), in the case of things
+that come into being, we are looking for the matter. The causes, therefore,
+are these and so many in number. 
+
+Now, the causes being four, it is the business of the physicist to
+know about them all, and if he refers his problems back to all of
+them, he will assign the 'why' in the way proper to his science-the
+matter, the form, the mover, 'that for the sake of which'. The last
+three often coincide; for the 'what' and 'that for the sake of which'
+are one, while the primary source of motion is the same in species
+as these (for man generates man), and so too, in general, are all
+things which cause movement by being themselves moved; and such as
+are not of this kind are no longer inside the province of physics,
+for they cause motion not by possessing motion or a source of motion
+in themselves, but being themselves incapable of motion. Hence there
+are three branches of study, one of things which are incapable of
+motion, the second of things in motion, but indestructible, the third
+of destructible things. 
+
+The question 'why', then, is answered by reference to the matter,
+to the form, and to the primary moving cause. For in respect of coming
+to be it is mostly in this last way that causes are investigated-'what
+comes to be after what? what was the primary agent or patient?' and
+so at each step of the series. 
+
+Now the principles which cause motion in a physical way are two, of
+which one is not physical, as it has no principle of motion in itself.
+Of this kind is whatever causes movement, not being itself moved,
+such as (1) that which is completely unchangeable, the primary reality,
+and (2) the essence of that which is coming to be, i.e. the form;
+for this is the end or 'that for the sake of which'. Hence since nature
+is for the sake of something, we must know this cause also. We must
+explain the 'why' in all the senses of the term, namely, (1) that
+from this that will necessarily result ('from this' either without
+qualification or in most cases); (2) that 'this must be so if that
+is to be so' (as the conclusion presupposes the premisses); (3) that
+this was the essence of the thing; and (4) because it is better thus
+(not without qualification, but with reference to the essential nature
+in each case). 
+
+Part 8
+
+We must explain then (1) that Nature belongs to the class of causes
+which act for the sake of something; (2) about the necessary and its
+place in physical problems, for all writers ascribe things to this
+cause, arguing that since the hot and the cold, &c., are of such and
+such a kind, therefore certain things necessarily are and come to
+be-and if they mention any other cause (one his 'friendship and strife',
+another his 'mind'), it is only to touch on it, and then good-bye
+to it. 
+
+A difficulty presents itself: why should not nature work, not for
+the sake of something, nor because it is better so, but just as the
+sky rains, not in order to make the corn grow, but of necessity? What
+is drawn up must cool, and what has been cooled must become water
+and descend, the result of this being that the corn grows. Similarly
+if a man's crop is spoiled on the threshing-floor, the rain did not
+fall for the sake of this-in order that the crop might be spoiled-but
+that result just followed. Why then should it not be the same with
+the parts in nature, e.g. that our teeth should come up of necessity-the
+front teeth sharp, fitted for tearing, the molars broad and useful
+for grinding down the food-since they did not arise for this end,
+but it was merely a coincident result; and so with all other parts
+in which we suppose that there is purpose? Wherever then all the parts
+came about just what they would have been if they had come be for
+an end, such things survived, being organized spontaneously in a fitting
+way; whereas those which grew otherwise perished and continue to perish,
+as Empedocles says his 'man-faced ox-progeny' did. 
+
+Such are the arguments (and others of the kind) which may cause difficulty
+on this point. Yet it is impossible that this should be the true view.
+For teeth and all other natural things either invariably or normally
+come about in a given way; but of not one of the results of chance
+or spontaneity is this true. We do not ascribe to chance or mere coincidence
+the frequency of rain in winter, but frequent rain in summer we do;
+nor heat in the dog-days, but only if we have it in winter. If then,
+it is agreed that things are either the result of coincidence or for
+an end, and these cannot be the result of coincidence or spontaneity,
+it follows that they must be for an end; and that such things are
+all due to nature even the champions of the theory which is before
+us would agree. Therefore action for an end is present in things which
+come to be and are by nature. 
+
+Further, where a series has a completion, all the preceding steps
+are for the sake of that. Now surely as in intelligent action, so
+in nature; and as in nature, so it is in each action, if nothing interferes.
+Now intelligent action is for the sake of an end; therefore the nature
+of things also is so. Thus if a house, e.g. had been a thing made
+by nature, it would have been made in the same way as it is now by
+art; and if things made by nature were made also by art, they would
+come to be in the same way as by nature. Each step then in the series
+is for the sake of the next; and generally art partly completes what
+nature cannot bring to a finish, and partly imitates her. If, therefore,
+artificial products are for the sake of an end, so clearly also are
+natural products. The relation of the later to the earlier terms of
+the series is the same in both. This is most obvious in the animals
+other than man: they make things neither by art nor after inquiry
+or deliberation. Wherefore people discuss whether it is by intelligence
+or by some other faculty that these creatures work,spiders, ants,
+and the like. By gradual advance in this direction we come to see
+clearly that in plants too that is produced which is conducive to
+the end-leaves, e.g. grow to provide shade for the fruit. If then
+it is both by nature and for an end that the swallow makes its nest
+and the spider its web, and plants grow leaves for the sake of the
+fruit and send their roots down (not up) for the sake of nourishment,
+it is plain that this kind of cause is operative in things which come
+to be and are by nature. And since 'nature' means two things, the
+matter and the form, of which the latter is the end, and since all
+the rest is for the sake of the end, the form must be the cause in
+the sense of 'that for the sake of which'. 
+
+Now mistakes come to pass even in the operations of art: the grammarian
+makes a mistake in writing and the doctor pours out the wrong dose.
+Hence clearly mistakes are possible in the operations of nature also.
+If then in art there are cases in which what is rightly produced serves
+a purpose, and if where mistakes occur there was a purpose in what
+was attempted, only it was not attained, so must it be also in natural
+products, and monstrosities will be failures in the purposive effort.
+Thus in the original combinations the 'ox-progeny' if they failed
+to reach a determinate end must have arisen through the corruption
+of some principle corresponding to what is now the seed.
+
+Further, seed must have come into being first, and not straightway
+the animals: the words 'whole-natured first...' must have meant seed.
+
+Again, in plants too we find the relation of means to end, though
+the degree of organization is less. Were there then in plants also
+'olive-headed vine-progeny', like the 'man-headed ox-progeny', or
+not? An absurd suggestion; yet there must have been, if there were
+such things among animals. 
+
+Moreover, among the seeds anything must have come to be at random.
+But the person who asserts this entirely does away with 'nature' and
+what exists 'by nature'. For those things are natural which, by a
+continuous movement originated from an internal principle, arrive
+at some completion: the same completion is not reached from every
+principle; nor any chance completion, but always the tendency in each
+is towards the same end, if there is no impediment. 
+
+The end and the means towards it may come about by chance. We say,
+for instance, that a stranger has come by chance, paid the ransom,
+and gone away, when he does so as if he had come for that purpose,
+though it was not for that that he came. This is incidental, for chance
+is an incidental cause, as I remarked before. But when an event takes
+place always or for the most part, it is not incidental or by chance.
+In natural products the sequence is invariable, if there is no impediment.
+
+It is absurd to suppose that purpose is not present because we do
+not observe the agent deliberating. Art does not deliberate. If the
+ship-building art were in the wood, it would produce the same results
+by nature. If, therefore, purpose is present in art, it is present
+also in nature. The best illustration is a doctor doctoring himself:
+nature is like that. 
+
+It is plain then that nature is a cause, a cause that operates for
+a purpose. 
+
+Part 9
+
+As regards what is 'of necessity', we must ask whether the necessity
+is 'hypothetical', or 'simple' as well. The current view places what
+is of necessity in the process of production, just as if one were
+to suppose that the wall of a house necessarily comes to be because
+what is heavy is naturally carried downwards and what is light to
+the top, wherefore the stones and foundations take the lowest place,
+with earth above because it is lighter, and wood at the top of all
+as being the lightest. Whereas, though the wall does not come to be
+without these, it is not due to these, except as its material cause:
+it comes to be for the sake of sheltering and guarding certain things.
+Similarly in all other things which involve production for an end;
+the product cannot come to be without things which have a necessary
+nature, but it is not due to these (except as its material); it comes
+to be for an end. For instance, why is a saw such as it is? To effect
+so-and-so and for the sake of so-and-so. This end, however, cannot
+be realized unless the saw is made of iron. It is, therefore, necessary
+for it to be of iron, it we are to have a saw and perform the operation
+of sawing. What is necessary then, is necessary on a hypothesis; it
+is not a result necessarily determined by antecedents. Necessity is
+in the matter, while 'that for the sake of which' is in the definition.
+
+Necessity in mathematics is in a way similar to necessity in things
+which come to be through the operation of nature. Since a straight
+line is what it is, it is necessary that the angles of a triangle
+should equal two right angles. But not conversely; though if the angles
+are not equal to two right angles, then the straight line is not what
+it is either. But in things which come to be for an end, the reverse
+is true. If the end is to exist or does exist, that also which precedes
+it will exist or does exist; otherwise just as there, if-the conclusion
+is not true, the premiss will not be true, so here the end or 'that
+for the sake of which' will not exist. For this too is itself a starting-point,
+but of the reasoning, not of the action; while in mathematics the
+starting-point is the starting-point of the reasoning only, as there
+is no action. If then there is to be a house, such-and-such things
+must be made or be there already or exist, or generally the matter
+relative to the end, bricks and stones if it is a house. But the end
+is not due to these except as the matter, nor will it come to exist
+because of them. Yet if they do not exist at all, neither will the
+house, or the saw-the former in the absence of stones, the latter
+in the absence of iron-just as in the other case the premisses will
+not be true, if the angles of the triangle are not equal to two right
+angles. 
+
+The necessary in nature, then, is plainly what we call by the name
+of matter, and the changes in it. Both causes must be stated by the
+physicist, but especially the end; for that is the cause of the matter,
+not vice versa; and the end is 'that for the sake of which', and the
+beginning starts from the definition or essence; as in artificial
+products, since a house is of such-and-such a kind, certain things
+must necessarily come to be or be there already, or since health is
+this, these things must necessarily come to be or be there already.
+Similarly if man is this, then these; if these, then those. Perhaps
+the necessary is present also in the definition. For if one defines
+the operation of sawing as being a certain kind of dividing, then
+this cannot come about unless the saw has teeth of a certain kind;
+and these cannot be unless it is of iron. For in the definition too
+there are some parts that are, as it were, its matter. 
+
+----------------------------------------------------------------------
+
+BOOK III
+
+Part 1 
+
+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.
+
+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. 
+
+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
+quantum nor quale 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. 
+
+Now each of these belongs to all its subjects in either of two ways:
+namely (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.
+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. 
+
+Def. The fulfilment of what exists potentially, in so far as it exists
+potentially, is motion-namely, of what is alterable qua 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 he and passing away: of
+what can be carried along, locomotion. 
+
+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 e.g. 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. 
+
+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, i.e. in definition,
+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. 
+
+Part 2
+
+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. 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 qua 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, e.g. the full-formed man begets man from what is potentially
+man. 
+
+Part 3
+
+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. 
+
+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.
+(If we ought to call the latter also 'agency', the word would be used
+in two senses.) 
+
+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 (e.g. 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. 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 qua 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, &c. A similar
+definition will apply to each of the other kinds of motion.
+
+Part 4
+
+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, e.g. a quality
+or a point-it is not necessary perhaps that such things should be
+put under either head. Hence it is incumbent on 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, 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. 
+
+(1) 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 (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. 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. 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. (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. 
+
+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.
+
+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.
+
+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? 
+
+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. 
+
+Part 5
+
+Now it is impossible that the infinite should be a thing which is
+itself infinite, separable from sensible objects. If the infinite
+is neither a magnitude nor an aggregate, but is itself a substance
+and not an attribute, it will be indivisible; for the divisible must
+be either a magnitude or an aggregate. But if indivisible, then not
+infinite, except in the sense (1) in which the voice is 'invisible'.
+But this is not the sense in which it is used by those who say that
+the infinite exists, nor that in which we are investigating it, namely
+as (2) 'that which cannot be gone through'. But if the infinite exists
+as an attribute, it would not be, qua infinite an element in substances,
+any more than the invisible would be an element of speech, though
+the voice is invisible. 
+
+Further, how can the infinite be itself any thing, unless both number
+and magnitude, of which it is an essential attribute, exist in that
+way? If they are not substances, a fortiori the infinite is not.
+
+It is plain, too, that the infinite cannot be an actual thing and
+a substance and principle. For any part of it that is taken will be
+infinite, if it has parts: for 'to be infinite' and 'the infinite'
+are the same, if it is a substance and not predicated of a subject.
+Hence it will be either indivisible or divisible into infinites. But
+the same thing cannot be many infinites. (Yet just as part of air
+is air, so a part of the infinite would be infinite, if it is supposed
+to be a substance and principle.) Therefore the infinite must be without
+parts and indivisible. But this cannot be true of what is infinite
+in full completion: for it must be a definite quantity. 
+
+Suppose then that infinity belongs to substance as an attribute. But,
+if so, it cannot, as we have said, be described as a principle, but
+rather that of which it is an attribute-the air or the even number.
+
+Thus the view of those who speak after the manner of the Pythagoreans
+is absurd. With the same breath they treat the infinite as substance,
+and divide it into parts. 
+
+This discussion, however, involves the more general question whether
+the infinite can be present in mathematical objects and things which
+are intelligible and do not have extension, as well as among sensible
+objects. Our inquiry (as physicists) is limited to its special subject-matter,
+the objects of sense, and we have to ask whether there is or is not
+among them a body which is infinite in the direction of increase.
+
+We may begin with a dialectical argument and show as follows that
+there is no such thing. If 'bounded by a surface' is the definition
+of body there cannot be an infinite body either intelligible or sensible.
+Nor can number taken in abstraction be infinite, for number or that
+which has number is numerable. If then the numerable can be numbered,
+it would also be possible to go through the infinite. 
+
+If, on the other hand, we investigate the question more in accordance
+with principles appropriate to physics, we are led as follows to the
+same result. 
+
+The infinite body must be either (1) compound, or (2) simple; yet
+neither alternative is possible. 
+
+(1) Compound the infinite body will not be, if the elements are finite
+in number. For they must be more than one, and the contraries must
+always balance, and no one of them can be infinite. If one of the
+bodies falls in any degree short of the other in potency-suppose fire
+is finite in amount while air is infinite and a given quantity of
+fire exceeds in power the same amount of air in any ratio provided
+it is numerically definite-the infinite body will obviously prevail
+over and annihilate the finite body. On the other hand, it is impossible
+that each should be infinite. 'Body' is what has extension in all
+directions and the infinite is what is boundlessly extended, so that
+the infinite body would be extended in all directions ad infinitum.
+
+Nor (2) can the infinite body be one and simple, whether it is, as
+some hold, a thing over and above the elements (from which they generate
+the elements) or is not thus qualified. 
+
+(a) We must consider the former alternative; for there are some people
+who make this the infinite, and not air or water, in order that the
+other elements may not be annihilated by the element which is infinite.
+They have contrariety with each other-air is cold, water moist, fire
+hot; if one were infinite, the others by now would have ceased to
+be. As it is, they say, the infinite is different from them and is
+their source. 
+
+It is impossible, however, that there should be such a body; not because
+it is infinite on that point a general proof can be given which applies
+equally to all, air, water, or anything else-but simply because there
+is, as a matter of fact, no such sensible body, alongside the so-called
+elements. Everything can be resolved into the elements of which it
+is composed. Hence the body in question would have been present in
+our world here, alongside air and fire and earth and water: but nothing
+of the kind is observed. 
+
+(b) Nor can fire or any other of the elements be infinite. For generally,
+and apart from the question of how any of them could be infinite,
+the All, even if it were limited, cannot either be or become one of
+them, as Heraclitus says that at some time all things become fire.
+(The same argument applies also to the one which the physicists suppose
+to exist alongside the elements: for everything changes from contrary
+to contrary, e.g. from hot to cold). 
+
+The preceding consideration of the various cases serves to show us
+whether it is or is not possible that there should be an infinite
+sensible body. The following arguments give a general demonstration
+that it is not possible. 
+
+It is the nature of every kind of sensible body to be somewhere, and
+there is a place appropriate to each, the same for the part and for
+the whole, e.g. for the whole earth and for a single clod, and for
+fire and for a spark. 
+
+Suppose (a) that the infinite sensible body is homogeneous. Then each
+part will be either immovable or always being carried along. Yet neither
+is possible. For why downwards rather than upwards or in any other
+direction? I mean, e.g, if you take a clod, where will it be moved
+or where will it be at rest? For ex hypothesi the place of the body
+akin to it is infinite. Will it occupy the whole place, then? And
+how? What then will be the nature of its rest and of its movement,
+or where will they be? It will either be at home everywhere-then it
+will not be moved; or it will be moved everywhere-then it will not
+come to rest. 
+
+But if (b) the All has dissimilar parts, the proper places of the
+parts will be dissimilar also, and the body of the All will have no
+unity except that of contact. Then, further, the parts will be either
+finite or infinite in variety of kind. (i) Finite they cannot be,
+for if the All is to be infinite, some of them would have to be infinite,
+while the others were not, e.g. fire or water will be infinite. But,
+as we have seen before, such an element would destroy what is contrary
+to it. (This indeed is the reason why none of the physicists made
+fire or earth the one infinite body, but either water or air or what
+is intermediate between them, because the abode of each of the two
+was plainly determinate, while the others have an ambiguous place
+between up and down.) 
+
+But (ii) if the parts are infinite in number and simple, their proper
+places too will be infinite in number, and the same will be true of
+the elements themselves. If that is impossible, and the places are
+finite, the whole too must be finite; for the place and the body cannot
+but fit each other. Neither is the whole place larger than what can
+be filled by the body (and then the body would no longer be infinite),
+nor is the body larger than the place; for either there would be an
+empty space or a body whose nature it is to be nowhere. 
+
+Anaxagoras gives an absurd account of why the infinite is at rest.
+He says that the infinite itself is the cause of its being fixed.
+This because it is in itself, since nothing else contains it-on the
+assumption that wherever anything is, it is there by its own nature.
+But this is not true: a thing could be somewhere by compulsion, and
+not where it is its nature to be. 
+
+Even if it is true as true can be that the whole is not moved (for
+what is fixed by itself and is in itself must be immovable), yet we
+must explain why it is not its nature to be moved. It is not enough
+just to make this statement and then decamp. Anything else might be
+in a state of rest, but there is no reason why it should not be its
+nature to be moved. The earth is not carried along, and would not
+be carried along if it were infinite, provided it is held together
+by the centre. But it would not be because there was no other region
+in which it could be carried along that it would remain at the centre,
+but because this is its nature. Yet in this case also we may say that
+it fixes itself. If then in the case of the earth, supposed to be
+infinite, it is at rest, not because it is infinite, but because it
+has weight and what is heavy rests at the centre and the earth is
+at the centre, similarly the infinite also would rest in itself, not
+because it is infinite and fixes itself, but owing to some other cause.
+
+Another difficulty emerges at the same time. Any part of the infinite
+body ought to remain at rest. Just as the infinite remains at rest
+in itself because it fixes itself, so too any part of it you may take
+will remain in itself. The appropriate places of the whole and of
+the part are alike, e.g. of the whole earth and of a clod the appropriate
+place is the lower region; of fire as a whole and of a spark, the
+upper region. If, therefore, to be in itself is the place of the infinite,
+that also will be appropriate to the part. Therefore it will remain
+in itself. 
+
+In general, the view that there is an infinite body is plainly incompatible
+with the doctrine that there is necessarily a proper place for each
+kind of body, if every sensible body has either weight or lightness,
+and if a body has a natural locomotion towards the centre if it is
+heavy, and upwards if it is light. This would need to be true of the
+infinite also. But neither character can belong to it: it cannot be
+either as a whole, nor can it be half the one and half the other.
+For how should you divide it? or how can the infinite have the one
+part up and the other down, or an extremity and a centre?
+
+Further, every sensible body is in place, and the kinds or differences
+of place are up-down, before-behind, right-left; and these distinctions
+hold not only in relation to us and by arbitrary agreement, but also
+in the whole itself. But in the infinite body they cannot exist. In
+general, if it is impossible that there should be an infinite place,
+and if every body is in place, there cannot be an infinite body.
+
+Surely what is in a special place is in place, and what is in place
+is in a special place. Just, then, as the infinite cannot be quantity-that
+would imply that it has a particular quantity, e,g, two or three cubits;
+quantity just means these-so a thing's being in place means that it
+is somewhere, and that is either up or down or in some other of the
+six differences of position: but each of these is a limit.
+
+It is plain from these arguments that there is no body which is actually
+infinite. 
+
+Part 6
+
+But on the other hand to suppose that the infinite does not exist
+in any way leads obviously to many impossible consequences: there
+will be a beginning and an end of time, a magnitude will not be divisible
+into magnitudes, number will not be infinite. If, then, in view of
+the above considerations, neither alternative seems possible, an arbiter
+must be called in; and clearly there is a sense in which the infinite
+exists and another in which it does not. 
+
+We must keep in mind that the word 'is' means either what potentially
+is or what fully is. Further, a thing is infinite either by addition
+or by division. 
+
+Now, as we have seen, magnitude is not actually infinite. But by division
+it is infinite. (There is no difficulty in refuting the theory of
+indivisible lines.) The alternative then remains that the infinite
+has a potential existence. 
+
+But the phrase 'potential existence' is ambiguous. When we speak of
+the potential existence of a statue we mean that there will be an
+actual statue. It is not so with the infinite. There will not be an
+actual infinite. The word 'is' has many senses, and we say that the
+infinite 'is' in the sense in which we say 'it is day' or 'it is the
+games', because one thing after another is always coming into existence.
+For of these things too the distinction between potential and actual
+existence holds. We say that there are Olympic games, both in the
+sense that they may occur and that they are actually occurring.
+
+The infinite exhibits itself in different ways-in time, in the generations
+of man, and in the division of magnitudes. For generally the infinite
+has this mode of existence: one thing is always being taken after
+another, and each thing that is taken is always finite, but always
+different. Again, 'being' has more than one sense, so that we must
+not regard the infinite as a 'this', such as a man or a horse, but
+must suppose it to exist in the sense in which we speak of the day
+or the games as existing things whose being has not come to them like
+that of a substance, but consists in a process of coming to be or
+passing away; definite if you like at each stage, yet always different.
+
+But when this takes place in spatial magnitudes, what is taken perists,
+while in the succession of time and of men it takes place by the passing
+away of these in such a way that the source of supply never gives
+out. 
+
+In a way the infinite by addition is the same thing as the infinite
+by division. In a finite magnitude, the infinite by addition comes
+about in a way inverse to that of the other. For in proportion as
+we see division going on, in the same proportion we see addition being
+made to what is already marked off. For if we take a determinate part
+of a finite magnitude and add another part determined by the same
+ratio (not taking in the same amount of the original whole), and so
+on, we shall not traverse the given magnitude. But if we increase
+the ratio of the part, so as always to take in the same amount, we
+shall traverse the magnitude, for every finite magnitude is exhausted
+by means of any determinate quantity however small. 
+
+The infinite, then, exists in no other way, but in this way it does
+exist, potentially and by reduction. It exists fully in the sense
+in which we say 'it is day' or 'it is the games'; and potentially
+as matter exists, not independently as what is finite does.
+
+By addition then, also, there is potentially an infinite, namely,
+what we have described as being in a sense the same as the infinite
+in respect of division. For it will always be possible to take something
+ah extra. Yet the sum of the parts taken will not exceed every determinate
+magnitude, just as in the direction of division every determinate
+magnitude is surpassed in smallness and there will be a smaller part.
+
+But in respect of addition there cannot be an infinite which even
+potentially exceeds every assignable magnitude, unless it has the
+attribute of being actually infinite, as the physicists hold to be
+true of the body which is outside the world, whose essential nature
+is air or something of the kind. But if there cannot be in this way
+a sensible body which is infinite in the full sense, evidently there
+can no more be a body which is potentially infinite in respect of
+addition, except as the inverse of the infinite by division, as we
+have said. It is for this reason that Plato also made the infinites
+two in number, because it is supposed to be possible to exceed all
+limits and to proceed ad infinitum in the direction both of increase
+and of reduction. Yet though he makes the infinites two, he does not
+use them. For in the numbers the infinite in the direction of reduction
+is not present, as the monad is the smallest; nor is the infinite
+in the direction of increase, for the parts number only up to the
+decad. 
+
+The infinite turns out to be the contrary of what it is said to be.
+It is not what has nothing outside it that is infinite, but what always
+has something outside it. This is indicated by the fact that rings
+also that have no bezel are described as 'endless', because it is
+always possible to take a part which is outside a given part. The
+description depends on a certain similarity, but it is not true in
+the full sense of the word. This condition alone is not sufficient:
+it is necessary also that the next part which is taken should never
+be the same. In the circle, the latter condition is not satisfied:
+it is only the adjacent part from which the new part is different.
+
+Our definition then is as follows: 
+A quantity is infinite if it is such that we can always take a part
+outside what has been already taken. On the other hand, what has nothing
+outside it is complete and whole. For thus we define the whole-that
+from which nothing is wanting, as a whole man or a whole box. What
+is true of each particular is true of the whole as such-the whole
+is that of which nothing is outside. On the other hand that from which
+something is absent and outside, however small that may be, is not
+'all'. 'Whole' and 'complete' are either quite identical or closely
+akin. Nothing is complete (teleion) which has no end (telos); and
+the end is a limit. 
+
+Hence Parmenides must be thought to have spoken better than Melissus.
+The latter says that the whole is infinite, but the former describes
+it as limited, 'equally balanced from the middle'. For to connect
+the infinite with the all and the whole is not like joining two pieces
+of string; for it is from this they get the dignity they ascribe to
+the infinite-its containing all things and holding the all in itself-from
+its having a certain similarity to the whole. It is in fact the matter
+of the completeness which belongs to size, and what is potentially
+a whole, though not in the full sense. It is divisible both in the
+direction of reduction and of the inverse addition. It is a whole
+and limited; not, however, in virtue of its own nature, but in virtue
+of what is other than it. It does not contain, but, in so far as it
+is infinite, is contained. Consequently, also, it is unknowable, qua
+infinite; for the matter has no form. (Hence it is plain that the
+infinite stands in the relation of part rather than of whole. For
+the matter is part of the whole, as the bronze is of the bronze statue.)
+If it contains in the case of sensible things, in the case of intelligible
+things the great and the small ought to contain them. But it is absurd
+and impossible to suppose that the unknowable and indeterminate should
+contain and determine. 
+
+Part 7
+
+It is reasonable that there should not be held to be an infinite in
+respect of addition such as to surpass every magnitude, but that there
+should be thought to be such an infinite in the direction of division.
+For the matter and the infinite are contained inside what contains
+them, while it is the form which contains. It is natural too to suppose
+that in number there is a limit in the direction of the minimum, and
+that in the other direction every assigned number is surpassed. In
+magnitude, on the contrary, every assigned magnitude is surpassed
+in the direction of smallness, while in the other direction there
+is no infinite magnitude. The reason is that what is one is indivisible
+whatever it may be, e.g. a man is one man, not many. Number on the
+other hand is a plurality of 'ones' and a certain quantity of them.
+Hence number must stop at the indivisible: for 'two' and 'three' are
+merely derivative terms, and so with each of the other numbers. But
+in the direction of largeness it is always possible to think of a
+larger number: for the number of times a magnitude can be bisected
+is infinite. Hence this infinite is potential, never actual: the number
+of parts that can be taken always surpasses any assigned number. But
+this number is not separable from the process of bisection, and its
+infinity is not a permanent actuality but consists in a process of
+coming to be, like time and the number of time. 
+
+With magnitudes the contrary holds. What is continuous is divided
+ad infinitum, but there is no infinite in the direction of increase.
+For the size which it can potentially be, it can also actually be.
+Hence since no sensible magnitude is infinite, it is impossible to
+exceed every assigned magnitude; for if it were possible there would
+be something bigger than the heavens. 
+
+The infinite is not the same in magnitude and movement and time, in
+the sense of a single nature, but its secondary sense depends on its
+primary sense, i.e. movement is called infinite in virtue of the magnitude
+covered by the movement (or alteration or growth), and time because
+of the movement. (I use these terms for the moment. Later I shall
+explain what each of them means, and also why every magnitude is divisible
+into magnitudes.) 
+
+Our account does not rob the mathematicians of their science, by disproving
+the actual existence of the infinite in the direction of increase,
+in the sense of the untraversable. In point of fact they do not need
+the infinite and do not use it. They postulate only that the finite
+straight line may be produced as far as they wish. It is possible
+to have divided in the same ratio as the largest quantity another
+magnitude of any size you like. Hence, for the purposes of proof,
+it will make no difference to them to have such an infinite instead,
+while its existence will be in the sphere of real magnitudes.
+
+In the fourfold scheme of causes, it is plain that the infinite is
+a cause in the sense of matter, and that its essence is privation,
+the subject as such being what is continuous and sensible. All the
+other thinkers, too, evidently treat the infinite as matter-that is
+why it is inconsistent in them to make it what contains, and not what
+is contained. 
+
+Part 8
+
+It remains to dispose of the arguments which are supposed to support
+the view that the infinite exists not only potentially but as a separate
+thing. Some have no cogency; others can be met by fresh objections
+that are valid. 
+
+(1) In order that coming to be should not fail, it is not necessary
+that there should be a sensible body which is actually infinite. The
+passing away of one thing may be the coming to be of another, the
+All being limited. 
+
+(2) There is a difference between touching and being limited. The
+former is relative to something and is the touching of something (for
+everything that touches touches something), and further is an attribute
+of some one of the things which are limited. On the other hand, what
+is limited is not limited in relation to anything. Again, contact
+is not necessarily possible between any two things taken at random.
+
+(3) To rely on mere thinking is absurd, for then the excess or defect
+is not in the thing but in the thought. One might think that one of
+us is bigger than he is and magnify him ad infinitum. But it does
+not follow that he is bigger than the size we are, just because some
+one thinks he is, but only because he is the size he is. The thought
+is an accident. 
+
+(a) Time indeed and movement are infinite, and also thinking, in the
+sense that each part that is taken passes in succession out of existence.
+
+(b) Magnitude is not infinite either in the way of reduction or of
+magnification in thought. 
+
+This concludes my account of the way in which the infinite exists,
+and of the way in which it does not exist, and of what it is.
+
+----------------------------------------------------------------------
+
+BOOK IV
+
+Part 1 
+
+The physicist must have a knowledge of Place, too, as well as of
+the infinite-namely, whether there is such a thing or not, and the
+manner of its existence and what it is-both because all suppose that
+things which exist are somewhere (the non-existent is nowhere--where
+is the goat-stag or the sphinx?), and because 'motion' in its most
+general and primary sense is change of place, which we call 'locomotion'.
+
+The question, what is place? presents many difficulties. An examination
+of all the relevant facts seems to lead to divergent conclusions.
+Moreover, we have inherited nothing from previous thinkers, whether
+in the way of a statement of difficulties or of a solution.
+
+The existence of place is held to be obvious from the fact of mutual
+replacement. Where water now is, there in turn, when the water has
+gone out as from a vessel, air is present. When therefore another
+body occupies this same place, the place is thought to be different
+from all the bodies which come to be in it and replace one another.
+What now contains air formerly contained water, so that clearly the
+place or space into which and out of which they passed was something
+different from both. 
+
+Further, the typical locomotions of the elementary natural bodies-namely,
+fire, earth, and the like-show not only that place is something, but
+also that it exerts a certain influence. Each is carried to its own
+place, if it is not hindered, the one up, the other down. Now these
+are regions or kinds of place-up and down and the rest of the six
+directions. Nor do such distinctions (up and down and right and left,
+&c.) hold only in relation to us. To us they are not always the same
+but change with the direction in which we are turned: that is why
+the same thing may be both right and left, up and down, before and
+behind. But in nature each is distinct, taken apart by itself. It
+is not every chance direction which is 'up', but where fire and what
+is light are carried; similarly, too, 'down' is not any chance direction
+but where what has weight and what is made of earth are carried-the
+implication being that these places do not differ merely in relative
+position, but also as possessing distinct potencies. This is made
+plain also by the objects studied by mathematics. Though they have
+no real place, they nevertheless, in respect of their position relatively
+to us, have a right and left as attributes ascribed to them only in
+consequence of their relative position, not having by nature these
+various characteristics. Again, the theory that the void exists involves
+the existence of place: for one would define void as place bereft
+of body. 
+
+These considerations then would lead us to suppose that place is something
+distinct from bodies, and that every sensible body is in place. Hesiod
+too might be held to have given a correct account of it when he made
+chaos first. At least he says: 
+
+'First of all things came chaos to being, then broad-breasted earth,'
+implying that things need to have space first, because he thought,
+with most people, that everything is somewhere and in place. If this
+is its nature, the potency of place must be a marvellous thing, and
+take precedence of all other things. For that without which nothing
+else can exist, while it can exist without the others, must needs
+be first; for place does not pass out of existence when the things
+in it are annihilated. 
+
+True, but even if we suppose its existence settled, the question of
+its nature presents difficulty-whether it is some sort of 'bulk' of
+body or some entity other than that, for we must first determine its
+genus. 
+
+(1) Now it has three dimensions, length, breadth, depth, the dimensions
+by which all body also is bounded. But the place cannot be body; for
+if it were there would be two bodies in the same place. 
+
+(2) Further, if body has a place and space, clearly so too have surface
+and the other limits of body; for the same statement will apply to
+them: where the bounding planes of the water were, there in turn will
+be those of the air. But when we come to a point we cannot make a
+distinction between it and its place. Hence if the place of a point
+is not different from the point, no more will that of any of the others
+be different, and place will not be something different from each
+of them. 
+
+(3) What in the world then are we to suppose place to be? If it has
+the sort of nature described, it cannot be an element or composed
+of elements, whether these be corporeal or incorporeal: for while
+it has size, it has not body. But the elements of sensible bodies
+are bodies, while nothing that has size results from a combination
+of intelligible elements. 
+
+(4) Also we may ask: of what in things is space the cause? None of
+the four modes of causation can be ascribed to it. It is neither in
+the sense of the matter of existents (for nothing is composed of it),
+nor as the form and definition of things, nor as end, nor does it
+move existents. 
+
+(5) Further, too, if it is itself an existent, where will it be? Zeno's
+difficulty demands an explanation: for if everything that exists has
+a place, place too will have a place, and so on ad infinitum.
+
+(6) Again, just as every body is in place, so, too, every place has
+a body in it. What then shall we say about growing things? It follows
+from these premisses that their place must grow with them, if their
+place is neither less nor greater than they are. 
+
+By asking these questions, then, we must raise the whole problem about
+place-not only as to what it is, but even whether there is such a
+thing. 
+
+Part 2
+
+We may distinguish generally between predicating B of A because it
+(A) is itself, and because it is something else; and particularly
+between place which is common and in which all bodies are, and the
+special place occupied primarily by each. I mean, for instance, that
+you are now in the heavens because you are in the air and it is in
+the heavens; and you are in the air because you are on the earth;
+and similarly on the earth because you are in this place which contains
+no more than you. 
+
+Now if place is what primarily contains each body, it would be a limit,
+so that the place would be the form or shape of each body by which
+the magnitude or the matter of the magnitude is defined: for this
+is the limit of each body. 
+
+If, then, we look at the question in this way the place of a thing
+is its form. But, if we regard the place as the extension of the magnitude,
+it is the matter. For this is different from the magnitude: it is
+what is contained and defined by the form, as by a bounding plane.
+Matter or the indeterminate is of this nature; when the boundary and
+attributes of a sphere are taken away, nothing but the matter is left.
+
+This is why Plato in the Timaeus says that matter and space are the
+same; for the 'participant' and space are identical. (It is true,
+indeed, that the account he gives there of the 'participant' is different
+from what he says in his so-called 'unwritten teaching'. Nevertheless,
+he did identify place and space.) I mention Plato because, while all
+hold place to be something, he alone tried to say what it is.
+
+In view of these facts we should naturally expect to find difficulty
+in determining what place is, if indeed it is one of these two things,
+matter or form. They demand a very close scrutiny, especially as it
+is not easy to recognize them apart. 
+
+But it is at any rate not difficult to see that place cannot be either
+of them. The form and the matter are not separate from the thing,
+whereas the place can be separated. As we pointed out, where air was,
+water in turn comes to be, the one replacing the other; and similarly
+with other bodies. Hence the place of a thing is neither a part nor
+a state of it, but is separable from it. For place is supposed to
+be something like a vessel-the vessel being a transportable place.
+But the vessel is no part of the thing. 
+
+In so far then as it is separable from the thing, it is not the form:
+qua containing, it is different from the matter. 
+
+Also it is held that what is anywhere is both itself something and
+that there is a different thing outside it. (Plato of course, if we
+may digress, ought to tell us why the form and the numbers are not
+in place, if 'what participates' is place-whether what participates
+is the Great and the Small or the matter, as he called it in writing
+in the Timaeus.) 
+
+Further, how could a body be carried to its own place, if place was
+the matter or the form? It is impossible that what has no reference
+to motion or the distinction of up and down can be place. So place
+must be looked for among things which have these characteristics.
+
+If the place is in the thing (it must be if it is either shape or
+matter) place will have a place: for both the form and the indeterminate
+undergo change and motion along with the thing, and are not always
+in the same place, but are where the thing is. Hence the place will
+have a place. 
+
+Further, when water is produced from air, the place has been destroyed,
+for the resulting body is not in the same place. What sort of destruction
+then is that? 
+
+This concludes my statement of the reasons why space must be something,
+and again of the difficulties that may be raised about its essential
+nature. 
+
+Part 3
+
+The next step we must take is to see in how many senses one thing
+is said to be 'in' another. 
+
+(1) As the finger is 'in' the hand and generally the part 'in' the
+whole. 
+
+(2) As the whole is 'in' the parts: for there is no whole over and
+above the parts. 
+
+(3) As man is 'in' animal and generally species 'in' genus.
+
+(4) As the genus is 'in' the species and generally the part of the
+specific form 'in' the definition of the specific form. 
+
+(5) As health is 'in' the hot and the cold and generally the form
+'in' the matter. 
+
+(6) As the affairs of Greece centre 'in' the king, and generally events
+centre 'in' their primary motive agent. 
+
+(7) As the existence of a thing centres 'in its good and generally
+'in' its end, i.e. in 'that for the sake of which' it exists.
+
+(8) In the strictest sense of all, as a thing is 'in' a vessel, and
+generally 'in' place. 
+
+One might raise the question whether a thing can be in itself, or
+whether nothing can be in itself-everything being either nowhere or
+in something else. 
+
+The question is ambiguous; we may mean the thing qua itself or qua
+something else. 
+
+When there are parts of a whole-the one that in which a thing is,
+the other the thing which is in it-the whole will be described as
+being in itself. For a thing is described in terms of its parts, as
+well as in terms of the thing as a whole, e.g. a man is said to be
+white because the visible surface of him is white, or to be scientific
+because his thinking faculty has been trained. The jar then will not
+be in itself and the wine will not be in itself. But the jar of wine
+will: for the contents and the container are both parts of the same
+whole. 
+
+In this sense then, but not primarily, a thing can be in itself, namely,
+as 'white' is in body (for the visible surface is in body), and science
+is in the mind. 
+
+It is from these, which are 'parts' (in the sense at least of being
+'in' the man), that the man is called white, &c. But the jar and the
+wine in separation are not parts of a whole, though together they
+are. So when there are parts, a thing will be in itself, as 'white'
+is in man because it is in body, and in body because it resides in
+the visible surface. We cannot go further and say that it is in surface
+in virtue of something other than itself. (Yet it is not in itself:
+though these are in a way the same thing,) they differ in essence,
+each having a special nature and capacity, 'surface' and 'white'.
+
+Thus if we look at the matter inductively we do not find anything
+to be 'in' itself in any of the senses that have been distinguished;
+and it can be seen by argument that it is impossible. For each of
+two things will have to be both, e.g. the jar will have to be both
+vessel and wine, and the wine both wine and jar, if it is possible
+for a thing to be in itself; so that, however true it might be that
+they were in each other, the jar will receive the wine in virtue not
+of its being wine but of the wine's being wine, and the wine will
+be in the jar in virtue not of its being a jar but of the jar's being
+a jar. Now that they are different in respect of their essence is
+evident; for 'that in which something is' and 'that which is in it'
+would be differently defined. 
+
+Nor is it possible for a thing to be in itself even incidentally:
+for two things would at the same time in the same thing. The jar would
+be in itself-if a thing whose nature it is to receive can be in itself;
+and that which it receives, namely (if wine) wine, will be in it.
+
+Obviously then a thing cannot be in itself primarily. 
+Zeno's problem-that if Place is something it must be in something-is
+not difficult to solve. There is nothing to prevent the first place
+from being 'in' something else-not indeed in that as 'in' place, but
+as health is 'in' the hot as a positive determination of it or as
+the hot is 'in' body as an affection. So we escape the infinite regress.
+
+Another thing is plain: since the vessel is no part of what is in
+it (what contains in the strict sense is different from what is contained),
+place could not be either the matter or the form of the thing contained,
+but must different-for the latter, both the matter and the shape,
+are parts of what is contained. 
+
+This then may serve as a critical statement of the difficulties involved.
+
+Part 4
+
+What then after all is place? The answer to this question may be elucidated
+as follows. 
+
+Let us take for granted about it the various characteristics which
+are supposed correctly to belong to it essentially. We assume then-
+
+(1) Place is what contains that of which it is the place.
+
+(2) Place is no part of the thing. 
+(3) The immediate place of a thing is neither less nor greater than
+the thing. 
+
+(4) Place can be left behind by the thing and is separable. In addition:
+
+(5) All place admits of the distinction of up and down, and each of
+the bodies is naturally carried to its appropriate place and rests
+there, and this makes the place either up or down. 
+
+Having laid these foundations, we must complete the theory. We ought
+to try to make our investigation such as will render an account of
+place, and will not only solve the difficulties connected with it,
+but will also show that the attributes supposed to belong to it do
+really belong to it, and further will make clear the cause of the
+trouble and of the difficulties about it. Such is the most satisfactory
+kind of exposition. 
+
+First then we must understand that place would not have been thought
+of, if there had not been a special kind of motion, namely that with
+respect to place. It is chiefly for this reason that we suppose the
+heaven also to be in place, because it is in constant movement. Of
+this kind of change there are two species-locomotion on the one hand
+and, on the other, increase and diminution. For these too involve
+variation of place: what was then in this place has now in turn changed
+to what is larger or smaller. 
+
+Again, when we say a thing is 'moved', the predicate either (1) belongs
+to it actually, in virtue of its own nature, or (2) in virtue of something
+conjoined with it. In the latter case it may be either (a) something
+which by its own nature is capable of being moved, e.g. the parts
+of the body or the nail in the ship, or (b) something which is not
+in itself capable of being moved, but is always moved through its
+conjunction with something else, as 'whiteness' or 'science'. These
+have changed their place only because the subjects to which they belong
+do so. 
+
+We say that a thing is in the world, in the sense of in place, because
+it is in the air, and the air is in the world; and when we say it
+is in the air, we do not mean it is in every part of the air, but
+that it is in the air because of the outer surface of the air which
+surrounds it; for if all the air were its place, the place of a thing
+would not be equal to the thing-which it is supposed to be, and which
+the primary place in which a thing is actually is. 
+
+When what surrounds, then, is not separate from the thing, but is
+in continuity with it, the thing is said to be in what surrounds it,
+not in the sense of in place, but as a part in a whole. But when the
+thing is separate and in contact, it is immediately 'in' the inner
+surface of the surrounding body, and this surface is neither a part
+of what is in it nor yet greater than its extension, but equal to
+it; for the extremities of things which touch are coincident.
+
+Further, if one body is in continuity with another, it is not moved
+in that but with that. On the other hand it is moved in that if it
+is separate. It makes no difference whether what contains is moved
+or not. 
+
+Again, when it is not separate it is described as a part in a whole,
+as the pupil in the eye or the hand in the body: when it is separate,
+as the water in the cask or the wine in the jar. For the hand is moved
+with the body and the water in the cask. 
+
+It will now be plain from these considerations what place is. There
+are just four things of which place must be one-the shape, or the
+matter, or some sort of extension between the bounding surfaces of
+the containing body, or this boundary itself if it contains no extension
+over and above the bulk of the body which comes to be in it.
+
+Three of these it obviously cannot be: 
+(1) The shape is supposed to be place because it surrounds, for the
+extremities of what contains and of what is contained are coincident.
+Both the shape and the place, it is true, are boundaries. But not
+of the same thing: the form is the boundary of the thing, the place
+is the boundary of the body which contains it. 
+
+(2) The extension between the extremities is thought to be something,
+because what is contained and separate may often be changed while
+the container remains the same (as water may be poured from a vessel)-the
+assumption being that the extension is something over and above the
+body displaced. But there is no such extension. One of the bodies
+which change places and are naturally capable of being in contact
+with the container falls in whichever it may chance to be.
+
+If there were an extension which were such as to exist independently
+and be permanent, there would be an infinity of places in the same
+thing. For when the water and the air change places, all the portions
+of the two together will play the same part in the whole which was
+previously played by all the water in the vessel; at the same time
+the place too will be undergoing change; so that there will be another
+place which is the place of the place, and many places will be coincident.
+There is not a different place of the part, in which it is moved,
+when the whole vessel changes its place: it is always the same: for
+it is in the (proximate) place where they are that the air and the
+water (or the parts of the water) succeed each other, not in that
+place in which they come to be, which is part of the place which is
+the place of the whole world. 
+
+(3) The matter, too, might seem to be place, at least if we consider
+it in what is at rest and is thus separate but in continuity. For
+just as in change of quality there is something which was formerly
+black and is now white, or formerly soft and now hard-this is just
+why we say that the matter exists-so place, because it presents a
+similar phenomenon, is thought to exist-only in the one case we say
+so because what was air is now water, in the other because where air
+formerly was there a is now water. But the matter, as we said before,
+is neither separable from the thing nor contains it, whereas place
+has both characteristics. 
+
+Well, then, if place is none of the three-neither the form nor the
+matter nor an extension which is always there, different from, and
+over and above, the extension of the thing which is displaced-place
+necessarily is the one of the four which is left, namely, the boundary
+of the containing body at which it is in contact with the contained
+body. (By the contained body is meant what can be moved by way of
+locomotion.) 
+
+Place is thought to be something important and hard to grasp, both
+because the matter and the shape present themselves along with it,
+and because the displacement of the body that is moved takes place
+in a stationary container, for it seems possible that there should
+be an interval which is other than the bodies which are moved. The
+air, too, which is thought to be incorporeal, contributes something
+to the belief: it is not only the boundaries of the vessel which seem
+to be place, but also what is between them, regarded as empty. Just,
+in fact, as the vessel is transportable place, so place is a non-portable
+vessel. So when what is within a thing which is moved, is moved and
+changes its place, as a boat on a river, what contains plays the part
+of a vessel rather than that of place. Place on the other hand is
+rather what is motionless: so it is rather the whole river that is
+place, because as a whole it is motionless. 
+
+Hence we conclude that the innermost motionless boundary of what contains
+is place. 
+
+This explains why the middle of the heaven and the surface which faces
+us of the rotating system are held to be 'up' and 'down' in the strict
+and fullest sense for all men: for the one is always at rest, while
+the inner side of the rotating body remains always coincident with
+itself. Hence since the light is what is naturally carried up, and
+the heavy what is carried down, the boundary which contains in the
+direction of the middle of the universe, and the middle itself, are
+down, and that which contains in the direction of the outermost part
+of the universe, and the outermost part itself, are up. 
+
+For this reason, too, place is thought to be a kind of surface, and
+as it were a vessel, i.e. a container of the thing. 
+
+Further, place is coincident with the thing, for boundaries are coincident
+with the bounded. 
+
+Part 5
+
+If then a body has another body outside it and containing it, it is
+in place, and if not, not. That is why, even if there were to be water
+which had not a container, the parts of it, on the one hand, will
+be moved (for one part is contained in another), while, on the other
+hand, the whole will be moved in one sense, but not in another. For
+as a whole it does not simultaneously change its place, though it
+will be moved in a circle: for this place is the place of its parts.
+(Some things are moved, not up and down, but in a circle; others up
+and down, such things namely as admit of condensation and rarefaction.)
+
+As was explained, some things are potentially in place, others actually.
+So, when you have a homogeneous substance which is continuous, the
+parts are potentially in place: when the parts are separated, but
+in contact, like a heap, they are actually in place. 
+
+Again, (1) some things are per se in place, namely every body which
+is movable either by way of locomotion or by way of increase is per
+se somewhere, but the heaven, as has been said, is not anywhere as
+a whole, nor in any place, if at least, as we must suppose, no body
+contains it. On the line on which it is moved, its parts have place:
+for each is contiguous the next. 
+
+But (2) other things are in place indirectly, through something conjoined
+with them, as the soul and the heaven. The latter is, in a way, in
+place, for all its parts are: for on the orb one part contains another.
+That is why the upper part is moved in a circle, while the All is
+not anywhere. For what is somewhere is itself something, and there
+must be alongside it some other thing wherein it is and which contains
+it. But alongside the All or the Whole there is nothing outside the
+All, and for this reason all things are in the heaven; for the heaven,
+we may say, is the All. Yet their place is not the same as the heaven.
+It is part of it, the innermost part of it, which is in contact with
+the movable body; and for this reason the earth is in water, and this
+in the air, and the air in the aether, and the aether in heaven, but
+we cannot go on and say that the heaven is in anything else.
+
+It is clear, too, from these considerations that all the problems
+which were raised about place will be solved when it is explained
+in this way: 
+
+(1) There is no necessity that the place should grow with the body
+in it, 
+
+(2) Nor that a point should have a place, 
+(3) Nor that two bodies should be in the same place, 
+(4) Nor that place should be a corporeal interval: for what is between
+the boundaries of the place is any body which may chance to be there,
+not an interval in body. 
+
+Further, (5) place is also somewhere, not in the sense of being in
+a place, but as the limit is in the limited; for not everything that
+is is in place, but only movable body. 
+
+Also (6) it is reasonable that each kind of body should be carried
+to its own place. For a body which is next in the series and in contact
+(not by compulsion) is akin, and bodies which are united do not affect
+each other, while those which are in contact interact on each other.
+
+Nor (7) is it without reason that each should remain naturally in
+its proper place. For this part has the same relation to its place,
+as a separable part to its whole, as when one moves a part of water
+or air: so, too, air is related to water, for the one is like matter,
+the other form-water is the matter of air, air as it were the actuality
+of water, for water is potentially air, while air is potentially water,
+though in another way. 
+
+These distinctions will be drawn more carefully later. On the present
+occasion it was necessary to refer to them: what has now been stated
+obscurely will then be made more clear. If the matter and the fulfilment
+are the same thing (for water is both, the one potentially, the other
+completely), water will be related to air in a way as part to whole.
+That is why these have contact: it is organic union when both become
+actually one. 
+
+This concludes my account of place-both of its existence and of its
+nature. 
+
+Part 6
+
+The investigation of similar questions about the void, also, must
+be held to belong to the physicist-namely whether it exists or not,
+and how it exists or what it is-just as about place. The views taken
+of it involve arguments both for and against, in much the same sort
+of way. For those who hold that the void exists regard it as a sort
+of place or vessel which is supposed to be 'full' when it holds the
+bulk which it is capable of containing, 'void' when it is deprived
+of that-as if 'void' and 'full' and 'place' denoted the same thing,
+though the essence of the three is different. 
+
+We must begin the inquiry by putting down the account given by those
+who say that it exists, then the account of those who say that it
+does not exist, and third the current view on these questions.
+
+Those who try to show that the void does not exist do not disprove
+what people really mean by it, but only their erroneous way of speaking;
+this is true of Anaxagoras and of those who refute the existence of
+the void in this way. They merely give an ingenious demonstration
+that air is something--by straining wine-skins and showing the resistance
+of the air, and by cutting it off in clepsydras. But people really
+mean that there is an empty interval in which there is no sensible
+body. They hold that everything which is in body is body and say that
+what has nothing in it at all is void (so what is full of air is void).
+It is not then the existence of air that needs to be proved, but the
+non-existence of an interval, different from the bodies, either separable
+or actual-an interval which divides the whole body so as to break
+its continuity, as Democritus and Leucippus hold, and many other physicists-or
+even perhaps as something which is outside the whole body, which remains
+continuous. 
+
+These people, then, have not reached even the threshold of the problem,
+but rather those who say that the void exists. 
+
+(1) They argue, for one thing, that change in place (i.e. locomotion
+and increase) would not be. For it is maintained that motion would
+seem not to exist, if there were no void, since what is full cannot
+contain anything more. If it could, and there were two bodies in the
+same place, it would also be true that any number of bodies could
+be together; for it is impossible to draw a line of division beyond
+which the statement would become untrue. If this were possible, it
+would follow also that the smallest body would contain the greatest;
+for 'many a little makes a mickle': thus if many equal bodies can
+be together, so also can many unequal bodies. 
+
+Melissus, indeed, infers from these considerations that the All is
+immovable; for if it were moved there must, he says, be void, but
+void is not among the things that exist. 
+
+This argument, then, is one way in which they show that there is a
+void. 
+
+(2) They reason from the fact that some things are observed to contract
+and be compressed, as people say that a cask will hold the wine which
+formerly filled it, along with the skins into which the wine has been
+decanted, which implies that the compressed body contracts into the
+voids present in it. 
+
+Again (3) increase, too, is thought to take always by means of void,
+for nutriment is body, and it is impossible for two bodies to be together.
+A proof of this they find also in what happens to ashes, which absorb
+as much water as the empty vessel. 
+
+The Pythagoreans, too, (4) held that void exists and that it enters
+the heaven itself, which as it were inhales it, from the infinite
+air. Further it is the void which distinguishes the natures of things,
+as if it were like what separates and distinguishes the terms of a
+series. This holds primarily in the numbers, for the void distinguishes
+their nature. 
+
+These, then, and so many, are the main grounds on which people have
+argued for and against the existence of the void. 
+
+Part 7
+
+As a step towards settling which view is true, we must determine the
+meaning of the name. 
+
+The void is thought to be place with nothing in it. The reason for
+this is that people take what exists to be body, and hold that while
+every body is in place, void is place in which there is no body, so
+that where there is no body, there must be void. 
+
+Every body, again, they suppose to be tangible; and of this nature
+is whatever has weight or lightness. 
+
+Hence, by a syllogism, what has nothing heavy or light in it, is void.
+
+This result, then, as I have said, is reached by syllogism. It would
+be absurd to suppose that the point is void; for the void must be
+place which has in it an interval in tangible body. 
+
+But at all events we observe then that in one way the void is described
+as what is not full of body perceptible to touch; and what has heaviness
+and lightness is perceptible to touch. So we would raise the question:
+what would they say of an interval that has colour or sound-is it
+void or not? Clearly they would reply that if it could receive what
+is tangible it was void, and if not, not. 
+
+In another way void is that in which there is no 'this' or corporeal
+substance. So some say that the void is the matter of the body (they
+identify the place, too, with this), and in this they speak incorrectly;
+for the matter is not separable from the things, but they are inquiring
+about the void as about something separable. 
+
+Since we have determined the nature of place, and void must, if it
+exists, be place deprived of body, and we have stated both in what
+sense place exists and in what sense it does not, it is plain that
+on this showing void does not exist, either unseparated or separated;
+the void is meant to be, not body but rather an interval in body.
+This is why the void is thought to be something, viz. because place
+is, and for the same reasons. For the fact of motion in respect of
+place comes to the aid both of those who maintain that place is something
+over and above the bodies that come to occupy it, and of those who
+maintain that the void is something. They state that the void is the
+condition of movement in the sense of that in which movement takes
+place; and this would be the kind of thing that some say place is.
+
+But there is no necessity for there being a void if there is movement.
+It is not in the least needed as a condition of movement in general,
+for a reason which, incidentally, escaped Melissus; viz. that the
+full can suffer qualitative change. 
+
+But not even movement in respect of place involves a void; for bodies
+may simultaneously make room for one another, though there is no interval
+separate and apart from the bodies that are in movement. And this
+is plain even in the rotation of continuous things, as in that of
+liquids. 
+
+And things can also be compressed not into a void but because they
+squeeze out what is contained in them (as, for instance, when water
+is compressed the air within it is squeezed out); and things can increase
+in size not only by the entrance of something but also by qualitative
+change; e.g. if water were to be transformed into air. 
+
+In general, both the argument about increase of size and that about
+water poured on to the ashes get in their own way. For either not
+any and every part of the body is increased, or bodies may be increased
+otherwise than by the addition of body, or there may be two bodies
+in the same place (in which case they are claiming to solve a quite
+general difficulty, but are not proving the existence of void), or
+the whole body must be void, if it is increased in every part and
+is increased by means of void. The same argument applies to the ashes.
+
+It is evident, then, that it is easy to refute the arguments by which
+they prove the existence of the void. 
+
+Part 8
+
+Let us explain again that there is no void existing separately, as
+some maintain. If each of the simple bodies has a natural locomotion,
+e.g. fire upward and earth downward and towards the middle of the
+universe, it is clear that it cannot be the void that is the condition
+of locomotion. What, then, will the void be the condition of? It is
+thought to be the condition of movement in respect of place, and it
+is not the condition of this. 
+
+Again, if void is a sort of place deprived of body, when there is
+a void where will a body placed in it move to? It certainly cannot
+move into the whole of the void. The same argument applies as against
+those who think that place is something separate, into which things
+are carried; viz. how will what is placed in it move, or rest? Much
+the same argument will apply to the void as to the 'up' and 'down'
+in place, as is natural enough since those who maintain the existence
+of the void make it a place. 
+
+And in what way will things be present either in place-or in the void?
+For the expected result does not take place when a body is placed
+as a whole in a place conceived of as separate and permanent; for
+a part of it, unless it be placed apart, will not be in a place but
+in the whole. Further, if separate place does not exist, neither will
+void. 
+
+If people say that the void must exist, as being necessary if there
+is to be movement, what rather turns out to be the case, if one the
+matter, is the opposite, that not a single thing can be moved if there
+is a void; for as with those who for a like reason say the earth is
+at rest, so, too, in the void things must be at rest; for there is
+no place to which things can move more or less than to another; since
+the void in so far as it is void admits no difference. 
+
+The second reason is this: all movement is either compulsory or according
+to nature, and if there is compulsory movement there must also be
+natural (for compulsory movement is contrary to nature, and movement
+contrary to nature is posterior to that according to nature, so that
+if each of the natural bodies has not a natural movement, none of
+the other movements can exist); but how can there be natural movement
+if there is no difference throughout the void or the infinite? For
+in so far as it is infinite, there will be no up or down or middle,
+and in so far as it is a void, up differs no whit from down; for as
+there is no difference in what is nothing, there is none in the void
+(for the void seems to be a non-existent and a privation of being),
+but natural locomotion seems to be differentiated, so that the things
+that exist by nature must be differentiated. Either, then, nothing
+has a natural locomotion, or else there is no void. 
+
+Further, in point of fact things that are thrown move though that
+which gave them their impulse is not touching them, either by reason
+of mutual replacement, as some maintain, or because the air that has
+been pushed pushes them with a movement quicker than the natural locomotion
+of the projectile wherewith it moves to its proper place. But in a
+void none of these things can take place, nor can anything be moved
+save as that which is carried is moved. 
+
+Further, no one could say why a thing once set in motion should stop
+anywhere; for why should it stop here rather than here? So that a
+thing will either be at rest or must be moved ad infinitum, unless
+something more powerful get in its way. 
+
+Further, things are now thought to move into the void because it yields;
+but in a void this quality is present equally everywhere, so that
+things should move in all directions. 
+
+Further, the truth of what we assert is plain from the following considerations.
+We see the same weight or body moving faster than another for two
+reasons, either because there is a difference in what it moves through,
+as between water, air, and earth, or because, other things being equal,
+the moving body differs from the other owing to excess of weight or
+of lightness. 
+
+Now the medium causes a difference because it impedes the moving thing,
+most of all if it is moving in the opposite direction, but in a secondary
+degree even if it is at rest; and especially a medium that is not
+easily divided, i.e. a medium that is somewhat dense. A, then, will
+move through B in time G, and through D, which is thinner, in time
+E (if the length of B is egual to D), in proportion to the density
+of the hindering body. For let B be water and D air; then by so much
+as air is thinner and more incorporeal than water, A will move through
+D faster than through B. Let the speed have the same ratio to the
+speed, then, that air has to water. Then if air is twice as thin,
+the body will traverse B in twice the time that it does D, and the
+time G will be twice the time E. And always, by so much as the medium
+is more incorporeal and less resistant and more easily divided, the
+faster will be the movement. 
+
+Now there is no ratio in which the void is exceeded by body, as there
+is no ratio of 0 to a number. For if 4 exceeds 3 by 1, and 2 by more
+than 1, and 1 by still more than it exceeds 2, still there is no ratio
+by which it exceeds 0; for that which exceeds must be divisible into
+the excess + that which is exceeded, so that will be what it exceeds
+0 by + 0. For this reason, too, a line does not exceed a point unless
+it is composed of points! Similarly the void can bear no ratio to
+the full, and therefore neither can movement through the one to movement
+through the other, but if a thing moves through the thickest medium
+such and such a distance in such and such a time, it moves through
+the void with a speed beyond any ratio. For let Z be void, equal in
+magnitude to B and to D. Then if A is to traverse and move through
+it in a certain time, H, a time less than E, however, the void will
+bear this ratio to the full. But in a time equal to H, A will traverse
+the part O of A. And it will surely also traverse in that time any
+substance Z which exceeds air in thickness in the ratio which the
+time E bears to the time H. For if the body Z be as much thinner than
+D as E exceeds H, A, if it moves through Z, will traverse it in a
+time inverse to the speed of the movement, i.e. in a time equal to
+H. If, then, there is no body in Z, A will traverse Z still more quickly.
+But we supposed that its traverse of Z when Z was void occupied the
+time H. So that it will traverse Z in an equal time whether Z be full
+or void. But this is impossible. It is plain, then, that if there
+is a time in which it will move through any part of the void, this
+impossible result will follow: it will be found to traverse a certain
+distance, whether this be full or void, in an equal time; for there
+will be some body which is in the same ratio to the other body as
+the time is to the time. 
+
+To sum the matter up, the cause of this result is obvious, viz. that
+between any two movements there is a ratio (for they occupy time,
+and there is a ratio between any two times, so long as both are finite),
+but there is no ratio of void to full. 
+
+These are the consequences that result from a difference in the media;
+the following depend upon an excess of one moving body over another.
+We see that bodies which have a greater impulse either of weight or
+of lightness, if they are alike in other respects, move faster over
+an equal space, and in the ratio which their magnitudes bear to each
+other. Therefore they will also move through the void with this ratio
+of speed. But that is impossible; for why should one move faster?
+(In moving through plena it must be so; for the greater divides them
+faster by its force. For a moving thing cleaves the medium either
+by its shape, or by the impulse which the body that is carried along
+or is projected possesses.) Therefore all will possess equal velocity.
+But this is impossible. 
+
+It is evident from what has been said, then, that, if there is a void,
+a result follows which is the very opposite of the reason for which
+those who believe in a void set it up. They think that if movement
+in respect of place is to exist, the void cannot exist, separated
+all by itself; but this is the same as to say that place is a separate
+cavity; and this has already been stated to be impossible.
+
+But even if we consider it on its own merits the so-called vacuum
+will be found to be really vacuous. For as, if one puts a cube in
+water, an amount of water equal to the cube will be displaced; so
+too in air; but the effect is imperceptible to sense. And indeed always
+in the case of any body that can be displaced, must, if it is not
+compressed, be displaced in the direction in which it is its nature
+to be displaced-always either down, if its locomotion is downwards
+as in the case of earth, or up, if it is fire, or in both directions-whatever
+be the nature of the inserted body. Now in the void this is impossible;
+for it is not body; the void must have penetrated the cube to a distance
+equal to that which this portion of void formerly occupied in the
+void, just as if the water or air had not been displaced by the wooden
+cube, but had penetrated right through it. 
+
+But the cube also has a magnitude equal to that occupied by the void;
+a magnitude which, if it is also hot or cold, or heavy or light, is
+none the less different in essence from all its attributes, even if
+it is not separable from them; I mean the volume of the wooden cube.
+So that even if it were separated from everything else and were neither
+heavy nor light, it will occupy an equal amount of void, and fill
+the same place, as the part of place or of the void equal to itself.
+How then will the body of the cube differ from the void or place that
+is equal to it? And if there can be two such things, why cannot there
+be any number coinciding? 
+
+This, then, is one absurd and impossible implication of the theory.
+It is also evident that the cube will have this same volume even if
+it is displaced, which is an attribute possessed by all other bodies
+also. Therefore if this differs in no respect from its place, why
+need we assume a place for bodies over and above the volume of each,
+if their volume be conceived of as free from attributes? It contributes
+nothing to the situation if there is an equal interval attached to
+it as well. [Further it ought to be clear by the study of moving things
+what sort of thing void is. But in fact it is found nowhere in the
+world. For air is something, though it does not seem to be so-nor,
+for that matter, would water, if fishes were made of iron; for the
+discrimination of the tangible is by touch.] 
+
+It is clear, then, from these considerations that there is no separate
+void. 
+
+Part 9
+
+There are some who think that the existence of rarity and density
+shows that there is a void. If rarity and density do not exist, they
+say, neither can things contract and be compressed. But if this were
+not to take place, either there would be no movement at all, or the
+universe would bulge, as Xuthus said, or air and water must always
+change into equal amounts (e.g. if air has been made out of a cupful
+of water, at the same time out of an equal amount of air a cupful
+of water must have been made), or void must necessarily exist; for
+compression and expansion cannot take place otherwise. 
+
+Now, if they mean by the rare that which has many voids existing separately,
+it is plain that if void cannot exist separate any more than a place
+can exist with an extension all to itself, neither can the rare exist
+in this sense. But if they mean that there is void, not separately
+existent, but still present in the rare, this is less impossible,
+yet, first, the void turns out not to be a condition of all movement,
+but only of movement upwards (for the rare is light, which is the
+reason why they say fire is rare); second, the void turns out to be
+a condition of movement not as that in which it takes place, but in
+that the void carries things up as skins by being carried up themselves
+carry up what is continuous with them. Yet how can void have a local
+movement or a place? For thus that into which void moves is till then
+void of a void. 
+
+Again, how will they explain, in the case of what is heavy, its movement
+downwards? And it is plain that if the rarer and more void a thing
+is the quicker it will move upwards, if it were completely void it
+would move with a maximum speed! But perhaps even this is impossible,
+that it should move at all; the same reason which showed that in the
+void all things are incapable of moving shows that the void cannot
+move, viz. the fact that the speeds are incomparable. 
+
+Since we deny that a void exists, but for the rest the problem has
+been truly stated, that either there will be no movement, if there
+is not to be condensation and rarefaction, or the universe will bulge,
+or a transformation of water into air will always be balanced by an
+equal transformation of air into water (for it is clear that the air
+produced from water is bulkier than the water): it is necessary therefore,
+if compression does not exist, either that the next portion will be
+pushed outwards and make the outermost part bulge, or that somewhere
+else there must be an equal amount of water produced out of air, so
+that the entire bulk of the whole may be equal, or that nothing moves.
+For when anything is displaced this will always happen, unless it
+comes round in a circle; but locomotion is not always circular, but
+sometimes in a straight line. 
+
+These then are the reasons for which they might say that there is
+a void; our statement is based on the assumption that there is a single
+matter for contraries, hot and cold and the other natural contrarieties,
+and that what exists actually is produced from a potential existent,
+and that matter is not separable from the contraries but its being
+is different, and that a single matter may serve for colour and heat
+and cold. 
+
+The same matter also serves for both a large and a small body. This
+is evident; for when air is produced from water, the same matter has
+become something different, not by acquiring an addition to it, but
+has become actually what it was potentially, and, again, water is
+produced from air in the same way, the change being sometimes from
+smallness to greatness, and sometimes from greatness to smallness.
+Similarly, therefore, if air which is large in extent comes to have
+a smaller volume, or becomes greater from being smaller, it is the
+matter which is potentially both that comes to be each of the two.
+
+For as the same matter becomes hot from being cold, and cold from
+being hot, because it was potentially both, so too from hot it can
+become more hot, though nothing in the matter has become hot that
+was not hot when the thing was less hot; just as, if the arc or curve
+of a greater circle becomes that of a smaller, whether it remains
+the same or becomes a different curve, convexity has not come to exist
+in anything that was not convex but straight (for differences of degree
+do not depend on an intermission of the quality); nor can we get any
+portion of a flame, in which both heat and whiteness are not present.
+So too, then, is the earlier heat related to the later. So that the
+greatness and smallness, also, of the sensible volume are extended,
+not by the matter's acquiring anything new, but because the matter
+is potentially matter for both states; so that the same thing is dense
+and rare, and the two qualities have one matter. 
+
+The dense is heavy, and the rare is light. [Again, as the arc of a
+circle when contracted into a smaller space does not acquire a new
+part which is convex, but what was there has been contracted; and
+as any part of fire that one takes will be hot; so, too, it is all
+a question of contraction and expansion of the same matter.] There
+are two types in each case, both in the dense and in the rare; for
+both the heavy and the hard are thought to be dense, and contrariwise
+both the light and the soft are rare; and weight and hardness fail
+to coincide in the case of lead and iron. 
+
+From what has been said it is evident, then, that void does not exist
+either separate (either absolutely separate or as a separate element
+in the rare) or potentially, unless one is willing to call the condition
+of movement void, whatever it may be. At that rate the matter of the
+heavy and the light, qua matter of them, would be the void; for the
+dense and the rare are productive of locomotion in virtue of this
+contrariety, and in virtue of their hardness and softness productive
+of passivity and impassivity, i.e. not of locomotion but rather of
+qualitative change. 
+
+So much, then, for the discussion of the void, and of the sense in
+which it exists and the sense in which it does not exist.
+
+Part 10
+
+Next for discussion after the subjects mentioned is Time. The best
+plan will be to begin by working out the difficulties connected with
+it, making use of the current arguments. First, does it belong to
+the class of things that exist or to that of things that do not exist?
+Then secondly, what is its nature? To start, then: the following considerations
+would make one suspect that it either does not exist at all or barely,
+and in an obscure way. One part of it has been and is not, while the
+other is going to be and is not yet. Yet time-both infinite time and
+any time you like to take-is made up of these. One would naturally
+suppose that what is made up of things which do not exist could have
+no share in reality. 
+
+Further, if a divisible thing is to exist, it is necessary that, when
+it exists, all or some of its parts must exist. But of time some parts
+have been, while others have to be, and no part of it is though it
+is divisible. For what is 'now' is not a part: a part is a measure
+of the whole, which must be made up of parts. Time, on the other hand,
+is not held to be made up of 'nows'. 
+
+Again, the 'now' which seems to bound the past and the future-does
+it always remain one and the same or is it always other and other?
+It is hard to say. 
+
+(1) If it is always different and different, and if none of the parts
+in time which are other and other are simultaneous (unless the one
+contains and the other is contained, as the shorter time is by the
+longer), and if the 'now' which is not, but formerly was, must have
+ceased-to-be at some time, the 'nows' too cannot be simultaneous with
+one another, but the prior 'now' must always have ceased-to-be. But
+the prior 'now' cannot have ceased-to-be in itself (since it then
+existed); yet it cannot have ceased-to-be in another 'now'. For we
+may lay it down that one 'now' cannot be next to another, any more
+than point to point. If then it did not cease-to-be in the next 'now'
+but in another, it would exist simultaneously with the innumerable
+'nows' between the two-which is impossible. 
+
+Yes, but (2) neither is it possible for the 'now' to remain always
+the same. No determinate divisible thing has a single termination,
+whether it is continuously extended in one or in more than one dimension:
+but the 'now' is a termination, and it is possible to cut off a determinate
+time. Further, if coincidence in time (i.e. being neither prior nor
+posterior) means to be 'in one and the same "now"', then, if both
+what is before and what is after are in this same 'now', things which
+happened ten thousand years ago would be simultaneous with what has
+happened to-day, and nothing would be before or after anything else.
+
+This may serve as a statement of the difficulties about the attributes
+of time. 
+
+As to what time is or what is its nature, the traditional accounts
+give us as little light as the preliminary problems which we have
+worked through. 
+
+Some assert that it is (1) the movement of the whole, others that
+it is (2) the sphere itself. 
+
+(1) Yet part, too, of the revolution is a time, but it certainly is
+not a revolution: for what is taken is part of a revolution, not a
+revolution. Besides, if there were more heavens than one, the movement
+of any of them equally would be time, so that there would be many
+times at the same time. 
+
+(2) Those who said that time is the sphere of the whole thought so,
+no doubt, on the ground that all things are in time and all things
+are in the sphere of the whole. The view is too naive for it to be
+worth while to consider the impossibilities implied in it.
+
+But as time is most usually supposed to be (3) motion and a kind of
+change, we must consider this view. 
+
+Now (a) the change or movement of each thing is only in the thing
+which changes or where the thing itself which moves or changes may
+chance to be. But time is present equally everywhere and with all
+things. 
+
+Again, (b) change is always faster or slower, whereas time is not:
+for 'fast' and 'slow' are defined by time-'fast' is what moves much
+in a short time, 'slow' what moves little in a long time; but time
+is not defined by time, by being either a certain amount or a certain
+kind of it. 
+
+Clearly then it is not movement. (We need not distinguish at present
+between 'movement' and 'change'.) 
+
+Part 11
+
+But neither does time exist without change; for when the state of
+our own minds does not change at all, or we have not noticed its changing,
+we do not realize that time has elapsed, any more than those who are
+fabled to sleep among the heroes in Sardinia do when they are awakened;
+for they connect the earlier 'now' with the later and make them one,
+cutting out the interval because of their failure to notice it. So,
+just as, if the 'now' were not different but one and the same, there
+would not have been time, so too when its difference escapes our notice
+the interval does not seem to be time. If, then, the non-realization
+of the existence of time happens to us when we do not distinguish
+any change, but the soul seems to stay in one indivisible state, and
+when we perceive and distinguish we say time has elapsed, evidently
+time is not independent of movement and change. It is evident, then,
+that time is neither movement nor independent of movement.
+
+We must take this as our starting-point and try to discover-since
+we wish to know what time is-what exactly it has to do with movement.
+
+Now we perceive movement and time together: for even when it is dark
+and we are not being affected through the body, if any movement takes
+place in the mind we at once suppose that some time also has elapsed;
+and not only that but also, when some time is thought to have passed,
+some movement also along with it seems to have taken place. Hence
+time is either movement or something that belongs to movement. Since
+then it is not movement, it must be the other. 
+
+But what is moved is moved from something to something, and all magnitude
+is continuous. Therefore the movement goes with the magnitude. Because
+the magnitude is continuous, the movement too must be continuous,
+and if the movement, then the time; for the time that has passed is
+always thought to be in proportion to the movement. 
+
+The distinction of 'before' and 'after' holds primarily, then, in
+place; and there in virtue of relative position. Since then 'before'
+and 'after' hold in magnitude, they must hold also in movement, these
+corresponding to those. But also in time the distinction of 'before'
+and 'after' must hold, for time and movement always correspond with
+each other. The 'before' and 'after' in motion is identical in substratum
+with motion yet differs from it in definition, and is not identical
+with motion. 
+
+But we apprehend time only when we have marked motion, marking it
+by 'before' and 'after'; and it is only when we have perceived 'before'
+and 'after' in motion that we say that time has elapsed. Now we mark
+them by judging that A and B are different, and that some third thing
+is intermediate to them. When we think of the extremes as different
+from the middle and the mind pronounces that the 'nows' are two, one
+before and one after, it is then that we say that there is time, and
+this that we say is time. For what is bounded by the 'now' is thought
+to be time-we may assume this. 
+
+When, therefore, we perceive the 'now' one, and neither as before
+and after in a motion nor as an identity but in relation to a 'before'
+and an 'after', no time is thought to have elapsed, because there
+has been no motion either. On the other hand, when we do perceive
+a 'before' and an 'after', then we say that there is time. For time
+is just this-number of motion in respect of 'before' and 'after'.
+
+Hence time is not movement, but only movement in so far as it admits
+of enumeration. A proof of this: we discriminate the more or the less
+by number, but more or less movement by time. Time then is a kind
+of number. (Number, we must note, is used in two senses-both of what
+is counted or the countable and also of that with which we count.
+Time obviously is what is counted, not that with which we count: there
+are different kinds of thing.) Just as motion is a perpetual succession,
+so also is time. But every simultaneous time is self-identical; for
+the 'now' as a subject is an identity, but it accepts different attributes.
+The 'now' measures time, in so far as time involves the 'before and
+after'. 
+
+The 'now' in one sense is the same, in another it is not the same.
+In so far as it is in succession, it is different (which is just what
+its being was supposed to mean), but its substratum is an identity:
+for motion, as was said, goes with magnitude, and time, as we maintain,
+with motion. Similarly, then, there corresponds to the point the body
+which is carried along, and by which we are aware of the motion and
+of the 'before and after' involved in it. This is an identical substratum
+(whether a point or a stone or something else of the kind), but it
+has different attributes as the sophists assume that Coriscus' being
+in the Lyceum is a different thing from Coriscus' being in the market-place.
+And the body which is carried along is different, in so far as it
+is at one time here and at another there. But the 'now' corresponds
+to the body that is carried along, as time corresponds to the motion.
+For it is by means of the body that is carried along that we become
+aware of the 'before and after' the motion, and if we regard these
+as countable we get the 'now'. Hence in these also the 'now' as substratum
+remains the same (for it is what is before and after in movement),
+but what is predicated of it is different; for it is in so far as
+the 'before and after' is numerable that we get the 'now'. This is
+what is most knowable: for, similarly, motion is known because of
+that which is moved, locomotion because of that which is carried.
+what is carried is a real thing, the movement is not. Thus what is
+called 'now' in one sense is always the same; in another it is not
+the same: for this is true also of what is carried. 
+
+Clearly, too, if there were no time, there would be no 'now', and
+vice versa. just as the moving body and its locomotion involve each
+other mutually, so too do the number of the moving body and the number
+of its locomotion. For the number of the locomotion is time, while
+the 'now' corresponds to the moving body, and is like the unit of
+number. 
+
+Time, then, also is both made continuous by the 'now' and divided
+at it. For here too there is a correspondence with the locomotion
+and the moving body. For the motion or locomotion is made one by the
+thing which is moved, because it is one-not because it is one in its
+own nature (for there might be pauses in the movement of such a thing)-but
+because it is one in definition: for this determines the movement
+as 'before' and 'after'. Here, too there is a correspondence with
+the point; for the point also both connects and terminates the length-it
+is the beginning of one and the end of another. But when you take
+it in this way, using the one point as two, a pause is necessary,
+if the same point is to be the beginning and the end. The 'now' on
+the other hand, since the body carried is moving, is always different.
+
+Hence time is not number in the sense in which there is 'number' of
+the same point because it is beginning and end, but rather as the
+extremities of a line form a number, and not as the parts of the line
+do so, both for the reason given (for we can use the middle point
+as two, so that on that analogy time might stand still), and further
+because obviously the 'now' is no part of time nor the section any
+part of the movement, any more than the points are parts of the line-for
+it is two lines that are parts of one line. 
+
+In so far then as the 'now' is a boundary, it is not time, but an
+attribute of it; in so far as it numbers, it is number; for boundaries
+belong only to that which they bound, but number (e.g. ten) is the
+number of these horses, and belongs also elsewhere. 
+
+It is clear, then, that time is 'number of movement in respect of
+the before and after', and is continuous since it is an attribute
+of what is continuous. 
+
+Part 12
+
+The smallest number, in the strict sense of the word 'number', is
+two. But of number as concrete, sometimes there is a minimum, sometimes
+not: e.g. of a 'line', the smallest in respect of multiplicity is
+two (or, if you like, one), but in respect of size there is no minimum;
+for every line is divided ad infinitum. Hence it is so with time.
+In respect of number the minimum is one (or two); in point of extent
+there is no minimum. 
+
+It is clear, too, that time is not described as fast or slow, but
+as many or few and as long or short. For as continuous it is long
+or short and as a number many or few, but it is not fast or slow-any
+more than any number with which we number is fast or slow.
+
+Further, there is the same time everywhere at once, but not the same
+time before and after, for while the present change is one, the change
+which has happened and that which will happen are different. Time
+is not number with which we count, but the number of things which
+are counted, and this according as it occurs before or after is always
+different, for the 'nows' are different. And the number of a hundred
+horses and a hundred men is the same, but the things numbered are
+different-the horses from the men. Further, as a movement can be one
+and the same again and again, so too can time, e.g. a year or a spring
+or an autumn. 
+
+Not only do we measure the movement by the time, but also the time
+by the movement, because they define each other. The time marks the
+movement, since it is its number, and the movement the time. We describe
+the time as much or little, measuring it by the movement, just as
+we know the number by what is numbered, e.g. the number of the horses
+by one horse as the unit. For we know how many horses there are by
+the use of the number; and again by using the one horse as unit we
+know the number of the horses itself. So it is with the time and the
+movement; for we measure the movement by the time and vice versa.
+It is natural that this should happen; for the movement goes with
+the distance and the time with the movement, because they are quanta
+and continuous and divisible. The movement has these attributes because
+the distance is of this nature, and the time has them because of the
+movement. And we measure both the distance by the movement and the
+movement by the distance; for we say that the road is long, if the
+journey is long, and that this is long, if the road is long-the time,
+too, if the movement, and the movement, if the time. 
+
+Time is a measure of motion and of being moved, and it measures the
+motion by determining a motion which will measure exactly the whole
+motion, as the cubit does the length by determining an amount which
+will measure out the whole. Further 'to be in time' means for movement,
+that both it and its essence are measured by time (for simultaneously
+it measures both the movement and its essence, and this is what being
+in time means for it, that its essence should be measured).
+
+Clearly then 'to be in time' has the same meaning for other things
+also, namely, that their being should be measured by time. 'To be
+in time' is one of two things: (1) to exist when time exists, (2)
+as we say of some things that they are 'in number'. The latter means
+either what is a part or mode of number-in general, something which
+belongs to number-or that things have a number. 
+
+Now, since time is number, the 'now' and the 'before' and the like
+are in time, just as 'unit' and 'odd' and 'even' are in number, i.e.
+in the sense that the one set belongs to number, the other to time.
+But things are in time as they are in number. If this is so, they
+are contained by time as things in place are contained by place.
+
+Plainly, too, to be in time does not mean to co-exist with time, any
+more than to be in motion or in place means to co-exist with motion
+or place. For if 'to be in something' is to mean this, then all things
+will be in anything, and the heaven will be in a grain; for when the
+grain is, then also is the heaven. But this is a merely incidental
+conjunction, whereas the other is necessarily involved: that which
+is in time necessarily involves that there is time when it is, and
+that which is in motion that there is motion when it is.
+
+Since what is 'in time' is so in the same sense as what is in number
+is so, a time greater than everything in time can be found. So it
+is necessary that all the things in time should be contained by time,
+just like other things also which are 'in anything', e.g. the things
+'in place' by place. 
+
+A thing, then, will be affected by time, just as we are accustomed
+to say that time wastes things away, and that all things grow old
+through time, and that there is oblivion owing to the lapse of time,
+but we do not say the same of getting to know or of becoming young
+or fair. For time is by its nature the cause rather of decay, since
+it is the number of change, and change removes what is. 
+
+Hence, plainly, things which are always are not, as such, in time,
+for they are not contained time, nor is their being measured by time.
+A proof of this is that none of them is affected by time, which indicates
+that they are not in time. 
+
+Since time is the measure of motion, it will be the measure of rest
+too-indirectly. For all rest is in time. For it does not follow that
+what is in time is moved, though what is in motion is necessarily
+moved. For time is not motion, but 'number of motion': and what is
+at rest, also, can be in the number of motion. Not everything that
+is not in motion can be said to be 'at rest'-but only that which can
+be moved, though it actually is not moved, as was said above.
+
+'To be in number' means that there is a number of the thing, and that
+its being is measured by the number in which it is. Hence if a thing
+is 'in time' it will be measured by time. But time will measure what
+is moved and what is at rest, the one qua moved, the other qua at
+rest; for it will measure their motion and rest respectively.
+
+Hence what is moved will not be measurable by the time simply in so
+far as it has quantity, but in so far as its motion has quantity.
+Thus none of the things which are neither moved nor at rest are in
+time: for 'to be in time' is 'to be measured by time', while time
+is the measure of motion and rest. 
+
+Plainly, then, neither will everything that does not exist be in time,
+i.e. those non-existent things that cannot exist, as the diagonal
+cannot be commensurate with the side. 
+
+Generally, if time is directly the measure of motion and indirectly
+of other things, it is clear that a thing whose existence is measured
+by it will have its existence in rest or motion. Those things therefore
+which are subject to perishing and becoming-generally, those which
+at one time exist, at another do not-are necessarily in time: for
+there is a greater time which will extend both beyond their existence
+and beyond the time which measures their existence. Of things which
+do not exist but are contained by time some were, e.g. Homer once
+was, some will be, e.g. a future event; this depends on the direction
+in which time contains them; if on both, they have both modes of existence.
+As to such things as it does not contain in any way, they neither
+were nor are nor will be. These are those nonexistents whose opposites
+always are, as the incommensurability of the diagonal always is-and
+this will not be in time. Nor will the commensurability, therefore;
+hence this eternally is not, because it is contrary to what eternally
+is. A thing whose contrary is not eternal can be and not be, and it
+is of such things that there is coming to be and passing away.
+
+Part 13
+
+The 'now' is the link of time, as has been said (for it connects past
+and future time), and it is a limit of time (for it is the beginning
+of the one and the end of the other). But this is not obvious as it
+is with the point, which is fixed. It divides potentially, and in
+so far as it is dividing the 'now' is always different, but in so
+far as it connects it is always the same, as it is with mathematical
+lines. For the intellect it is not always one and the same point,
+since it is other and other when one divides the line; but in so far
+as it is one, it is the same in every respect. 
+
+So the 'now' also is in one way a potential dividing of time, in another
+the termination of both parts, and their unity. And the dividing and
+the uniting are the same thing and in the same reference, but in essence
+they are not the same. 
+
+So one kind of 'now' is described in this way: another is when the
+time is near this kind of 'now'. 'He will come now' because he will
+come to-day; 'he has come now' because he came to-day. But the things
+in the Iliad have not happened 'now', nor is the flood 'now'-not that
+the time from now to them is not continuous, but because they are
+not near. 
+
+'At some time' means a time determined in relation to the first of
+the two types of 'now', e.g. 'at some time' Troy was taken, and 'at
+some time' there will be a flood; for it must be determined with reference
+to the 'now'. There will thus be a determinate time from this 'now'
+to that, and there was such in reference to the past event. But if
+there be no time which is not 'sometime', every time will be determined.
+
+Will time then fail? Surely not, if motion always exists. Is time
+then always different or does the same time recur? Clearly time is,
+in the same way as motion is. For if one and the same motion sometimes
+recurs, it will be one and the same time, and if not, not.
+
+Since the 'now' is an end and a beginning of time, not of the same
+time however, but the end of that which is past and the beginning
+of that which is to come, it follows that, as the circle has its convexity
+and its concavity, in a sense, in the same thing, so time is always
+at a beginning and at an end. And for this reason it seems to be always
+different; for the 'now' is not the beginning and the end of the same
+thing; if it were, it would be at the same time and in the same respect
+two opposites. And time will not fail; for it is always at a beginning.
+
+'Presently' or 'just' refers to the part of future time which is near
+the indivisible present 'now' ('When do you walk? 'Presently', because
+the time in which he is going to do so is near), and to the part of
+past time which is not far from the 'now' ('When do you walk?' 'I
+have just been walking'). But to say that Troy has just been taken-we
+do not say that, because it is too far from the 'now'. 'Lately', too,
+refers to the part of past time which is near the present 'now'. 'When
+did you go?' 'Lately', if the time is near the existing now. 'Long
+ago' refers to the distant past. 
+
+'Suddenly' refers to what has departed from its former condition in
+a time imperceptible because of its smallness; but it is the nature
+of all change to alter things from their former condition. In time
+all things come into being and pass away; for which reason some called
+it the wisest of all things, but the Pythagorean Paron called it the
+most stupid, because in it we also forget; and his was the truer view.
+It is clear then that it must be in itself, as we said before, the
+condition of destruction rather than of coming into being (for change,
+in itself, makes things depart from their former condition), and only
+incidentally of coming into being, and of being. A sufficient evidence
+of this is that nothing comes into being without itself moving somehow
+and acting, but a thing can be destroyed even if it does not move
+at all. And this is what, as a rule, we chiefly mean by a thing's
+being destroyed by time. Still, time does not work even this change;
+even this sort of change takes place incidentally in time.
+
+We have stated, then, that time exists and what it is, and in how
+many senses we speak of the 'now', and what 'at some time', 'lately',
+'presently' or 'just', 'long ago', and 'suddenly' mean. 
+
+Part 14
+
+These distinctions having been drawn, it is evident that every change
+and everything that moves is in time; for the distinction of faster
+and slower exists in reference to all change, since it is found in
+every instance. In the phrase 'moving faster' I refer to that which
+changes before another into the condition in question, when it moves
+over the same interval and with a regular movement; e.g. in the case
+of locomotion, if both things move along the circumference of a circle,
+or both along a straight line; and similarly in all other cases. But
+what is before is in time; for we say 'before' and 'after' with reference
+to the distance from the 'now', and the 'now' is the boundary of the
+past and the future; so that since 'nows' are in time, the before
+and the after will be in time too; for in that in which the 'now'
+is, the distance from the 'now' will also be. But 'before' is used
+contrariwise with reference to past and to future time; for in the
+past we call 'before' what is farther from the 'now', and 'after'
+what is nearer, but in the future we call the nearer 'before' and
+the farther 'after'. So that since the 'before' is in time, and every
+movement involves a 'before', evidently every change and every movement
+is in time. 
+
+It is also worth considering how time can be related to the soul;
+and why time is thought to be in everything, both in earth and in
+sea and in heaven. Is because it is an attribute, or state, or movement
+(since it is the number of movement) and all these things are movable
+(for they are all in place), and time and movement are together, both
+in respect of potentiality and in respect of actuality? 
+
+Whether if soul did not exist time would exist or not, is a question
+that may fairly be asked; for if there cannot be some one to count
+there cannot be anything that can be counted, so that evidently there
+cannot be number; for number is either what has been, or what can
+be, counted. But if nothing but soul, or in soul reason, is qualified
+to count, there would not be time unless there were soul, but only
+that of which time is an attribute, i.e. if movement can exist without
+soul, and the before and after are attributes of movement, and time
+is these qua numerable. 
+
+One might also raise the question what sort of movement time is the
+number of. Must we not say 'of any kind'? For things both come into
+being in time and pass away, and grow, and are altered in time, and
+are moved locally; thus it is of each movement qua movement that time
+is the number. And so it is simply the number of continuous movement,
+not of any particular kind of it. 
+
+But other things as well may have been moved now, and there would
+be a number of each of the two movements. Is there another time, then,
+and will there be two equal times at once? Surely not. For a time
+that is both equal and simultaneous is one and the same time, and
+even those that are not simultaneous are one in kind; for if there
+were dogs, and horses, and seven of each, it would be the same number.
+So, too, movements that have simultaneous limits have the same time,
+yet the one may in fact be fast and the other not, and one may be
+locomotion and the other alteration; still the time of the two changes
+is the same if their number also is equal and simultaneous; and for
+this reason, while the movements are different and separate, the time
+is everywhere the same, because the number of equal and simultaneous
+movements is everywhere one and the same. 
+
+Now there is such a thing as locomotion, and in locomotion there is
+included circular movement, and everything is measured by some one
+thing homogeneous with it, units by a unit, horses by a horse, and
+similarly times by some definite time, and, as we said, time is measured
+by motion as well as motion by time (this being so because by a motion
+definite in time the quantity both of the motion and of the time is
+measured): if, then, what is first is the measure of everything homogeneous
+with it, regular circular motion is above all else the measure, because
+the number of this is the best known. Now neither alteration nor increase
+nor coming into being can be regular, but locomotion can be. This
+also is why time is thought to be the movement of the sphere, viz.
+because the other movements are measured by this, and time by this
+movement. 
+
+This also explains the common saying that human affairs form a circle,
+and that there is a circle in all other things that have a natural
+movement and coming into being and passing away. This is because all
+other things are discriminated by time, and end and begin as though
+conforming to a cycle; for even time itself is thought to be a circle.
+And this opinion again is held because time is the measure of this
+kind of locomotion and is itself measured by such. So that to say
+that the things that come into being form a circle is to say that
+there is a circle of time; and this is to say that it is measured
+by the circular movement; for apart from the measure nothing else
+to be measured is observed; the whole is just a plurality of measures.
+
+It is said rightly, too, that the number of the sheep and of the dogs
+is the same number if the two numbers are equal, but not the same
+decad or the same ten; just as the equilateral and the scalene are
+not the same triangle, yet they are the same figure, because they
+are both triangles. For things are called the same so-and-so if they
+do not differ by a differentia of that thing, but not if they do;
+e.g. triangle differs from triangle by a differentia of triangle,
+therefore they are different triangles; but they do not differ by
+a differentia of figure, but are in one and the same division of it.
+For a figure of the one kind is a circle and a figure of another kind
+of triangle, and a triangle of one kind is equilateral and a triangle
+of another kind scalene. They are the same figure, then, that, triangle,
+but not the same triangle. Therefore the number of two groups also-is
+the same number (for their number does not differ by a differentia
+of number), but it is not the same decad; for the things of which
+it is asserted differ; one group are dogs, and the other horses.
+
+We have now discussed time-both time itself and the matters appropriate
+to the consideration of it. 
+
+----------------------------------------------------------------------
+
+BOOK V
+
+Part 1 
+
+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. Again (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. And above all there is (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. 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.
+We have, then, the following factors: (a) on the one hand that which
+directly causes motion, and (b) on the other hand that which is in
+motion: further, we have (c) that in which motion takes place, namely
+time, and (distinct from these three, d) that from which and (e)
+that to which it proceeds: 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. 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. 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. 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. 
+
+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: from subject to subject,
+from subject to nonsubject, from non-subject to subject, or from non-subject
+to non-subject, where by 'subject' I mean what is affirmatively expressed.
+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. 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'. 
+
+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-qualitative, quantitative,
+and local. 
+
+Part 2
+
+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. 
+
+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. Or (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.) 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. 
+
+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. 
+
+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. 
+
+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. 
+
+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. 
+
+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. 
+
+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. 
+
+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. 
+
+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. 
+
+Part 3
+
+Let us now proceed to define the terms 'together' and 'apart', 'in
+contact', 'between', 'in succession', 'contiguous', and 'continuous',
+and to show in what circumstances each of these terms is naturally
+applicable. 
+
+Things are said to be together in place when they are in one place
+(in the strictest sense of the word 'place') and to be apart when
+they are in different places. 
+
+Things are said to be in contact when their extremities are together.
+
+That which a changing thing, if it changes continuously in a natural
+manner, naturally reaches before it reaches that to which it changes
+last, is between. Thus 'between' implies the presence of at least
+three things: for in a process of change it is the contrary that is
+'last': and a thing is moved continuously if it leaves no gap or only
+the smallest possible gap in the material-not in the time (for a gap
+in the time does not prevent things having a 'between', while, on
+the other hand, there is nothing to prevent the highest note sounding
+immediately after the lowest) but in the material in which the motion
+takes place. This is manifestly true not only in local changes but
+in every other kind as well. (Now every change implies a pair of opposites,
+and opposites may be either contraries or contradictories; since then
+contradiction admits of no mean term, it is obvious that 'between'
+must imply a pair of contraries) That is locally contrary which is
+most distant in a straight line: for the shortest line is definitely
+limited, and that which is definitely limited constitutes a measure.
+
+A thing is 'in succession' when it is after the beginning in position
+or in form or in some other respect in which it is definitely so regarded,
+and when further there is nothing of the same kind as itself between
+it and that to which it is in succession, e.g. a line or lines if
+it is a line, a unit or units if it is a unit, a house if it is a
+house (there is nothing to prevent something of a different kind being
+between). For that which is in succession is in succession to a particular
+thing, and is something posterior: for one is not 'in succession'
+to two, nor is the first day of the month to be second: in each case
+the latter is 'in succession' to the former. 
+
+A thing that is in succession and touches is 'contiguous'. The 'continuous'
+is a subdivision of the contiguous: things are called continuous when
+the touching limits of each become one and the same and are, as the
+word implies, contained in each other: continuity is impossible if
+these extremities are two. This definition makes it plain that continuity
+belongs to things that naturally in virtue of their mutual contact
+form a unity. And in whatever way that which holds them together is
+one, so too will the whole be one, e.g. by a rivet or glue or contact
+or organic union. 
+
+It is obvious that of these terms 'in succession' is first in order
+of analysis: for that which touches is necessarily in succession,
+but not everything that is in succession touches: and so succession
+is a property of things prior in definition, e.g. numbers, while contact
+is not. And if there is continuity there is necessarily contact, but
+if there is contact, that alone does not imply continuity: for the
+extremities of things may be 'together' without necessarily being
+one: but they cannot be one without being necessarily together. So
+natural junction is last in coming to be: for the extremities must
+necessarily come into contact if they are to be naturally joined:
+but things that are in contact are not all naturally joined, while
+there is no contact clearly there is no natural junction either. Hence,
+if as some say 'point' and 'unit' have an independent existence of
+their own, it is impossible for the two to be identical: for points
+can touch while units can only be in succession. Moreover, there can
+always be something between points (for all lines are intermediate
+between points), whereas it is not necessary that there should possibly
+be anything between units: for there can be nothing between the numbers
+one and two. 
+
+We have now defined what is meant by 'together' and 'apart', 'contact',
+'between' and 'in succession', 'contiguous' and 'continuous': and
+we have shown in what circumstances each of these terms is applicable.
+
+Part 4
+
+There are many senses in which motion is said to be 'one': for we
+use the term 'one' in many senses. 
+
+Motion is one generically according to the different categories to
+which it may be assigned: thus any locomotion is one generically with
+any other locomotion, whereas alteration is different generically
+from locomotion. 
+
+Motion is one specifically when besides being one generically it also
+takes place in a species incapable of subdivision: e.g. colour has
+specific differences: therefore blackening and whitening differ specifically;
+but at all events every whitening will be specifically the same with
+every other whitening and every blackening with every other blackening.
+But white is not further subdivided by specific differences: hence
+any whitening is specifically one with any other whitening. Where
+it happens that the genus is at the same time a species, it is clear
+that the motion will then in a sense be one specifically though not
+in an unqualified sense: learning is an example of this, knowledge
+being on the one hand a species of apprehension and on the other hand
+a genus including the various knowledges. A difficulty, however, may
+be raised as to whether a motion is specifically one when the same
+thing changes from the same to the same, e.g. when one point changes
+again and again from a particular place to a particular place: if
+this motion is specifically one, circular motion will be the same
+as rectilinear motion, and rolling the same as walking. But is not
+this difficulty removed by the principle already laid down that if
+that in which the motion takes place is specifically different (as
+in the present instance the circular path is specifically different
+from the straight) the motion itself is also different? We have explained,
+then, what is meant by saying that motion is one generically or one
+specifically. 
+
+Motion is one in an unqualified sense when it is one essentially or
+numerically: and the following distinctions will make clear what this
+kind of motion is. There are three classes of things in connexion
+with which we speak of motion, the 'that which', the 'that in which',
+and the 'that during which'. I mean that there must he something that
+is in motion, e.g. a man or gold, and it must be in motion in something,
+e.g. a place or an affection, and during something, for all motion
+takes place during a time. Of these three it is the thing in which
+the motion takes place that makes it one generically or specifically,
+it is the thing moved that makes the motion one in subject, and it
+is the time that makes it consecutive: but it is the three together
+that make it one without qualification: to effect this, that in which
+the motion takes place (the species) must be one and incapable of
+subdivision, that during which it takes place (the time) must be one
+and unintermittent, and that which is in motion must be one-not in
+an accidental sense (i.e. it must be one as the white that blackens
+is one or Coriscus who walks is one, not in the accidental sense in
+which Coriscus and white may be one), nor merely in virtue of community
+of nature (for there might be a case of two men being restored to
+health at the same time in the same way, e.g. from inflammation of
+the eye, yet this motion is not really one, but only specifically
+one). 
+
+Suppose, however, that Socrates undergoes an alteration specifically
+the same but at one time and again at another: in this case if it
+is possible for that which ceased to be again to come into being and
+remain numerically the same, then this motion too will be one: otherwise
+it will be the same but not one. And akin to this difficulty there
+is another; viz. is health one? and generally are the states and affections
+in bodies severally one in essence although (as is clear) the things
+that contain them are obviously in motion and in flux? Thus if a person's
+health at daybreak and at the present moment is one and the same,
+why should not this health be numerically one with that which he recovers
+after an interval? The same argument applies in each case. There is,
+however, we may answer, this difference: that if the states are two
+then it follows simply from this fact that the activities must also
+in point of number be two (for only that which is numerically one
+can give rise to an activity that is numerically one), but if the
+state is one, this is not in itself enough to make us regard the activity
+also as one: for when a man ceases walking, the walking no longer
+is, but it will again be if he begins to walk again. But, be this
+as it may, if in the above instance the health is one and the same,
+then it must be possible for that which is one and the same to come
+to be and to cease to be many times. However, these difficulties lie
+outside our present inquiry. 
+
+Since every motion is continuous, a motion that is one in an unqualified
+sense must (since every motion is divisible) be continuous, and a
+continuous motion must be one. There will not be continuity between
+any motion and any other indiscriminately any more than there is between
+any two things chosen at random in any other sphere: there can be
+continuity only when the extremities of the two things are one. Now
+some things have no extremities at all: and the extremities of others
+differ specifically although we give them the same name of 'end':
+how should e.g. the 'end' of a line and the 'end' of walking touch
+or come to be one? Motions that are not the same either specifically
+or generically may, it is true, be consecutive (e.g. a man may run
+and then at once fall ill of a fever), and again, in the torch-race
+we have consecutive but not continuous locomotion: for according to
+our definition there can be continuity only when the ends of the two
+things are one. Hence motions may be consecutive or successive in
+virtue of the time being continuous, but there can be continuity only
+in virtue of the motions themselves being continuous, that is when
+the end of each is one with the end of the other. Motion, therefore,
+that is in an unqualified sense continuous and one must be specifically
+the same, of one thing, and in one time. Unity is required in respect
+of time in order that there may be no interval of immobility, for
+where there is intermission of motion there must be rest, and a motion
+that includes intervals of rest will be not one but many, so that
+a motion that is interrupted by stationariness is not one or continuous,
+and it is so interrupted if there is an interval of time. And though
+of a motion that is not specifically one (even if the time is unintermittent)
+the time is one, the motion is specifically different, and so cannot
+really be one, for motion that is one must be specifically one, though
+motion that is specifically one is not necessarily one in an unqualified
+sense. We have now explained what we mean when we call a motion one
+without qualification. 
+
+Further, a motion is also said to be one generically, specifically,
+or essentially when it is complete, just as in other cases completeness
+and wholeness are characteristics of what is one: and sometimes a
+motion even if incomplete is said to be one, provided only that it
+is continuous. 
+
+And besides the cases already mentioned there is another in which
+a motion is said to be one, viz. when it is regular: for in a sense
+a motion that is irregular is not regarded as one, that title belonging
+rather to that which is regular, as a straight line is regular, the
+irregular being as such divisible. But the difference would seem to
+be one of degree. In every kind of motion we may have regularity or
+irregularity: thus there may be regular alteration, and locomotion
+in a regular path, e.g. in a circle or on a straight line, and it
+is the same with regard to increase and decrease. The difference that
+makes a motion irregular is sometimes to be found in its path: thus
+a motion cannot be regular if its path is an irregular magnitude,
+e.g. a broken line, a spiral, or any other magnitude that is not such
+that any part of it taken at random fits on to any other that may
+be chosen. Sometimes it is found neither in the place nor in the time
+nor in the goal but in the manner of the motion: for in some cases
+the motion is differentiated by quickness and slowness: thus if its
+velocity is uniform a motion is regular, if not it is irregular. So
+quickness and slowness are not species of motion nor do they constitute
+specific differences of motion, because this distinction occurs in
+connexion with all the distinct species of motion. The same is true
+of heaviness and lightness when they refer to the same thing: e.g.
+they do not specifically distinguish earth from itself or fire from
+itself. Irregular motion, therefore, while in virtue of being continuous
+it is one, is so in a lesser degree, as is the case with locomotion
+in a broken line: and a lesser degree of something always means an
+admixture of its contrary. And since every motion that is one can
+be both regular and irregular, motions that are consecutive but not
+specifically the same cannot be one and continuous: for how should
+a motion composed of alteration and locomotion be regular? If a motion
+is to be regular its parts ought to fit one another. 
+
+Part 5
+
+We have further to determine what motions are contrary to each other,
+and to determine similarly how it is with rest. And we have first
+to decide whether contrary motions are motions respectively from and
+to the same thing, e.g. a motion from health and a motion to health
+(where the opposition, it would seem, is of the same kind as that
+between coming to be and ceasing to be); or motions respectively from
+contraries, e.g. a motion from health and a motion from disease; or
+motions respectively to contraries, e.g. a motion to health and a
+motion to disease; or motions respectively from a contrary and to
+the opposite contrary, e.g. a motion from health and a motion to disease;
+or motions respectively from a contrary to the opposite contrary and
+from the latter to the former, e.g. a motion from health to disease
+and a motion from disease to health: for motions must be contrary
+to one another in one or more of these ways, as there is no other
+way in which they can be opposed. 
+
+Now motions respectively from a contrary and to the opposite contrary,
+e.g. a motion from health and a motion to disease, are not contrary
+motions: for they are one and the same. (Yet their essence is not
+the same, just as changing from health is different from changing
+to disease.) Nor are motion respectively from a contrary and from
+the opposite contrary contrary motions, for a motion from a contrary
+is at the same time a motion to a contrary or to an intermediate (of
+this, however, we shall speak later), but changing to a contrary rather
+than changing from a contrary would seem to be the cause of the contrariety
+of motions, the latter being the loss, the former the gain, of contrariness.
+Moreover, each several motion takes its name rather from the goal
+than from the starting-point of change, e.g. motion to health we call
+convalescence, motion to disease sickening. Thus we are left with
+motions respectively to contraries, and motions respectively to contraries
+from the opposite contraries. Now it would seem that motions to contraries
+are at the same time motions from contraries (though their essence
+may not be the same; 'to health' is distinct, I mean, from 'from disease',
+and 'from health' from 'to disease'). 
+
+Since then change differs from motion (motion being change from a
+particular subject to a particular subject), it follows that contrary
+motions are motions respectively from a contrary to the opposite contrary
+and from the latter to the former, e.g. a motion from health to disease
+and a motion from disease to health. Moreover, the consideration of
+particular examples will also show what kinds of processes are generally
+recognized as contrary: thus falling ill is regarded as contrary to
+recovering one's health, these processes having contrary goals, and
+being taught as contrary to being led into error by another, it being
+possible to acquire error, like knowledge, either by one's own agency
+or by that of another. Similarly we have upward locomotion and downward
+locomotion, which are contrary lengthwise, locomotion to the right
+and locomotion to the left, which are contrary breadthwise, and forward
+locomotion and backward locomotion, which too are contraries. On the
+other hand, a process simply to a contrary, e.g. that denoted by the
+expression 'becoming white', where no starting-point is specified,
+is a change but not a motion. And in all cases of a thing that has
+no contrary we have as contraries change from and change to the same
+thing. Thus coming to be is contrary to ceasing to be, and losing
+to gaining. But these are changes and not motions. And wherever a
+pair of contraries admit of an intermediate, motions to that intermediate
+must be held to be in a sense motions to one or other of the contraries:
+for the intermediate serves as a contrary for the purposes of the
+motion, in whichever direction the change may be, e.g. grey in a motion
+from grey to white takes the place of black as starting-point, in
+a motion from white to grey it takes the place of black as goal, and
+in a motion from black to grey it takes the place of white as goal:
+for the middle is opposed in a sense to either of the extremes, as
+has been said above. Thus we see that two motions are contrary to
+each other only when one is a motion from a contrary to the opposite
+contrary and the other is a motion from the latter to the former.
+
+Part 6
+
+But since a motion appears to have contrary to it not only another
+motion but also a state of rest, we must determine how this is so.
+A motion has for its contrary in the strict sense of the term another
+motion, but it also has for an opposite a state of rest (for rest
+is the privation of motion and the privation of anything may be called
+its contrary), and motion of one kind has for its opposite rest of
+that kind, e.g. local motion has local rest. This statement, however,
+needs further qualification: there remains the question, is the opposite
+of remaining at a particular place motion from or motion to that place?
+It is surely clear that since there are two subjects between which
+motion takes place, motion from one of these (A) to its contrary (B)
+has for its opposite remaining in A while the reverse motion has for
+its opposite remaining in B. At the same time these two are also contrary
+to each other: for it would be absurd to suppose that there are contrary
+motions and not opposite states of rest. States of rest in contraries
+are opposed. To take an example, a state of rest in health is (1)
+contrary to a state of rest in disease, and (2) the motion to which
+it is contrary is that from health to disease. For (2) it would be
+absurd that its contrary motion should be that from disease to health,
+since motion to that in which a thing is at rest is rather a coming
+to rest, the coming to rest being found to come into being simultaneously
+with the motion; and one of these two motions it must be. And (1)
+rest in whiteness is of course not contrary to rest in health.
+
+Of all things that have no contraries there are opposite changes (viz.
+change from the thing and change to the thing, e.g. change from being
+and change to being), but no motion. So, too, of such things there
+is no remaining though there is absence of change. Should there be
+a particular subject, absence of change in its being will be contrary
+to absence of change in its not-being. And here a difficulty may be
+raised: if not-being is not a particular something, what is it, it
+may be asked, that is contrary to absence of change in a thing's being?
+and is this absence of change a state of rest? If it is, then either
+it is not true that every state of rest is contrary to a motion or
+else coming to be and ceasing to be are motion. It is clear then that,
+since we exclude these from among motions, we must not say that this
+absence of change is a state of rest: we must say that it is similar
+to a state of rest and call it absence of change. And it will have
+for its contrary either nothing or absence of change in the thing's
+not-being, or the ceasing to be of the thing: for such ceasing to
+be is change from it and the thing's coming to be is change to it.
+
+Again, a further difficulty may be raised. How is it, it may be asked,
+that whereas in local change both remaining and moving may be natural
+or unnatural, in the other changes this is not so? e.g. alteration
+is not now natural and now unnatural, for convalescence is no more
+natural or unnatural than falling ill, whitening no more natural or
+unnatural than blackening; so, too, with increase and decrease: these
+are not contrary to each other in the sense that either of them is
+natural while the other is unnatural, nor is one increase contrary
+to another in this sense; and the same account may be given of becoming
+and perishing: it is not true that becoming is natural and perishing
+unnatural (for growing old is natural), nor do we observe one becoming
+to be natural and another unnatural. We answer that if what happens
+under violence is unnatural, then violent perishing is unnatural and
+as such contrary to natural perishing. Are there then also some becomings
+that are violent and not the result of natural necessity, and are
+therefore contrary to natural becomings, and violent increases and
+decreases, e.g. the rapid growth to maturity of profligates and the
+rapid ripening of seeds even when not packed close in the earth? And
+how is it with alterations? Surely just the same: we may say that
+some alterations are violent while others are natural, e.g. patients
+alter naturally or unnaturally according as they throw off fevers
+on the critical days or not. But, it may be objected, then we shall
+have perishings contrary to one another, not to becoming. Certainly:
+and why should not this in a sense be so? Thus it is so if one perishing
+is pleasant and another painful: and so one perishing will be contrary
+to another not in an unqualified sense, but in so far as one has this
+quality and the other that. 
+
+Now motions and states of rest universally exhibit contrariety in
+the manner described above, e.g. upward motion and rest above are
+respectively contrary to downward motion and rest below, these being
+instances of local contrariety; and upward locomotion belongs naturally
+to fire and downward to earth, i.e. the locomotions of the two are
+contrary to each other. And again, fire moves up naturally and down
+unnaturally: and its natural motion is certainly contrary to its unnatural
+motion. Similarly with remaining: remaining above is contrary to motion
+from above downwards, and to earth this remaining comes unnaturally,
+this motion naturally. So the unnatural remaining of a thing is contrary
+to its natural motion, just as we find a similar contrariety in the
+motion of the same thing: one of its motions, the upward or the downward,
+will be natural, the other unnatural. 
+
+Here, however, the question arises, has every state of rest that is
+not permanent a becoming, and is this becoming a coming to a standstill?
+If so, there must be a becoming of that which is at rest unnaturally,
+e.g. of earth at rest above: and therefore this earth during the time
+that it was being carried violently upward was coming to a standstill.
+But whereas the velocity of that which comes to a standstill seems
+always to increase, the velocity of that which is carried violently
+seems always to decrease: so it will he in a state of rest without
+having become so. Moreover 'coming to a standstill' is generally recognized
+to be identical or at least concomitant with the locomotion of a thing
+to its proper place. 
+
+There is also another difficulty involved in the view that remaining
+in a particular place is contrary to motion from that place. For when
+a thing is moving from or discarding something, it still appears to
+have that which is being discarded, so that if a state of rest is
+itself contrary to the motion from the state of rest to its contrary,
+the contraries rest and motion will be simultaneously predicable of
+the same thing. May we not say, however, that in so far as the thing
+is still stationary it is in a state of rest in a qualified sense?
+For, in fact, whenever a thing is in motion, part of it is at the
+starting-point while part is at the goal to which it is changing:
+and consequently a motion finds its true contrary rather in another
+motion than in a state of rest. 
+
+With regard to motion and rest, then, we have now explained in what
+sense each of them is one and under what conditions they exhibit contrariety.
+
+[With regard to coming to a standstill the question may be raised
+whether there is an opposite state of rest to unnatural as well as
+to natural motions. It would be absurd if this were not the case:
+for a thing may remain still merely under violence: thus we shall
+have a thing being in a non-permanent state of rest without having
+become so. But it is clear that it must be the case: for just as there
+is unnatural motion, so, too, a thing may be in an unnatural state
+of rest. Further, some things have a natural and an unnatural motion,
+e.g. fire has a natural upward motion and an unnatural downward motion:
+is it, then, this unnatural downward motion or is it the natural downward
+motion of earth that is contrary to the natural upward motion? Surely
+it is clear that both are contrary to it though not in the same sense:
+the natural motion of earth is contrary inasmuch as the motion of
+fire is also natural, whereas the upward motion of fire as being natural
+is contrary to the downward motion of fire as being unnatural. The
+same is true of the corresponding cases of remaining. But there would
+seem to be a sense in which a state of rest and a motion are opposites.]
+
+----------------------------------------------------------------------
+
+BOOK VI
+
+Part 1 
+
+Now if the terms 'continuous', 'in contact', and 'in succession'
+are understood as defined above things being 'continuous' if their
+extremities are one, 'in contact' if their extremities are together,
+and 'in succession' if there is nothing of their own kind intermediate
+between them-nothing that is continuous can be composed 'of indivisibles':
+e.g. a line cannot be composed of points, the line being continuous
+and the point indivisible. For the extremities of two points can neither
+be one (since of an indivisible there can be no extremity as distinct
+from some other part) nor together (since that which has no parts
+can have no extremity, the extremity and the thing of which it is
+the extremity being distinct). 
+
+Moreover, if that which is continuous is composed of points, these
+points must be either continuous or in contact with one another: and
+the same reasoning applies in the case of all indivisibles. Now for
+the reason given above they cannot be continuous: and one thing can
+be in contact with another only if whole is in contact with whole
+or part with part or part with whole. But since indivisibles have
+no parts, they must be in contact with one another as whole with whole.
+And if they are in contact with one another as whole with whole, they
+will not be continuous: for that which is continuous has distinct
+parts: and these parts into which it is divisible are different in
+this way, i.e. spatially separate. 
+
+Nor, again, can a point be in succession to a point or a moment to
+a moment in such a way that length can be composed of points or time
+of moments: for things are in succession if there is nothing of their
+own kind intermediate between them, whereas that which is intermediate
+between points is always a line and that which is intermediate between
+moments is always a period of time. 
+
+Again, if length and time could thus be composed of indivisibles,
+they could be divided into indivisibles, since each is divisible into
+the parts of which it is composed. But, as we saw, no continuous thing
+is divisible into things without parts. Nor can there be anything
+of any other kind intermediate between the parts or between the moments:
+for if there could be any such thing it is clear that it must be either
+indivisible or divisible, and if it is divisible, it must be divisible
+either into indivisibles or into divisibles that are infinitely divisible,
+in which case it is continuous. 
+
+Moreover, it is plain that everything continuous is divisible into
+divisibles that are infinitely divisible: for if it were divisible
+into indivisibles, we should have an indivisible in contact with an
+indivisible, since the extremities of things that are continuous with
+one another are one and are in contact. 
+
+The same reasoning applies equally to magnitude, to time, and to motion:
+either all of these are composed of indivisibles and are divisible
+into indivisibles, or none. This may be made clear as follows. If
+a magnitude is composed of indivisibles, the motion over that magnitude
+must be composed of corresponding indivisible motions: e.g. if the
+magnitude ABG is composed of the indivisibles A, B, G, each corresponding
+part of the motion DEZ of O over ABG is indivisible. Therefore, since
+where there is motion there must be something that is in motion, and
+where there is something in motion there must be motion, therefore
+the being-moved will also be composed of indivisibles. So O traversed
+A when its motion was D, B when its motion was E, and G similarly
+when its motion was Z. Now a thing that is in motion from one place
+to another cannot at the moment when it was in motion both be in motion
+and at the same time have completed its motion at the place to which
+it was in motion: e.g. if a man is walking to Thebes, he cannot be
+walking to Thebes and at the same time have completed his walk to
+Thebes: and, as we saw, O traverses a the partless section A in virtue
+of the presence of the motion D. Consequently, if O actually passed
+through A after being in process of passing through, the motion must
+be divisible: for at the time when O was passing through, it neither
+was at rest nor had completed its passage but was in an intermediate
+state: while if it is passing through and has completed its passage
+at the same moment, then that which is walking will at the moment
+when it is walking have completed its walk and will be in the place
+to which it is walking; that is to say, it will have completed its
+motion at the place to which it is in motion. And if a thing is in
+motion over the whole Kbg and its motion is the three D, E, and Z,
+and if it is not in motion at all over the partless section A but
+has completed its motion over it, then the motion will consist not
+of motions but of starts, and will take place by a thing's having
+completed a motion without being in motion: for on this assumption
+it has completed its passage through A without passing through it.
+So it will be possible for a thing to have completed a walk without
+ever walking: for on this assumption it has completed a walk over
+a particular distance without walking over that distance. Since, then,
+everything must be either at rest or in motion, and O is therefore
+at rest in each of the sections A, B, and G, it follows that a thing
+can be continuously at rest and at the same time in motion: for, as
+we saw, O is in motion over the whole ABG and at rest in any part
+(and consequently in the whole) of it. Moreover, if the indivisibles
+composing DEZ are motions, it would be possible for a thing in spite
+of the presence in it of motion to be not in motion but at rest, while
+if they are not motions, it would be possible for motion to be composed
+of something other than motions. 
+
+And if length and motion are thus indivisible, it is neither more
+nor less necessary that time also be similarly indivisible, that is
+to say be composed of indivisible moments: for if the whole distance
+is divisible and an equal velocity will cause a thing to pass through
+less of it in less time, the time must also be divisible, and conversely,
+if the time in which a thing is carried over the section A is divisible,
+this section A must also be divisible. 
+
+Part 2
+
+And since every magnitude is divisible into magnitudes-for we have
+shown that it is impossible for anything continuous to be composed
+of indivisible parts, and every magnitude is continuous-it necessarily
+follows that the quicker of two things traverses a greater magnitude
+in an equal time, an equal magnitude in less time, and a greater magnitude
+in less time, in conformity with the definition sometimes given of
+'the quicker'. Suppose that A is quicker than B. Now since of two
+things that which changes sooner is quicker, in the time ZH, in which
+A has changed from G to D, B will not yet have arrived at D but will
+be short of it: so that in an equal time the quicker will pass over
+a greater magnitude. More than this, it will pass over a greater magnitude
+in less time: for in the time in which A has arrived at D, B being
+the slower has arrived, let us say, at E. Then since A has occupied
+the whole time ZH in arriving at D, will have arrived at O in less
+time than this, say ZK. Now the magnitude GO that A has passed over
+is greater than the magnitude GE, and the time ZK is less than the
+whole time ZH: so that the quicker will pass over a greater magnitude
+in less time. And from this it is also clear that the quicker will
+pass over an equal magnitude in less time than the slower. For since
+it passes over the greater magnitude in less time than the slower,
+and (regarded by itself) passes over LM the greater in more time than
+LX the lesser, the time PRh in which it passes over LM will be more
+than the time PS, which it passes over LX: so that, the time PRh being
+less than the time PCh in which the slower passes over LX, the time
+PS will also be less than the time PX: for it is less than the time
+PRh, and that which is less than something else that is less than
+a thing is also itself less than that thing. Hence it follows that
+the quicker will traverse an equal magnitude in less time than the
+slower. Again, since the motion of anything must always occupy either
+an equal time or less or more time in comparison with that of another
+thing, and since, whereas a thing is slower if its motion occupies
+more time and of equal velocity if its motion occupies an equal time,
+the quicker is neither of equal velocity nor slower, it follows that
+the motion of the quicker can occupy neither an equal time nor more
+time. It can only be, then, that it occupies less time, and thus we
+get the necessary consequence that the quicker will pass over an equal
+magnitude (as well as a greater) in less time than the slower.
+
+And since every motion is in time and a motion may occupy any time,
+and the motion of everything that is in motion may be either quicker
+or slower, both quicker motion and slower motion may occupy any time:
+and this being so, it necessarily follows that time also is continuous.
+By continuous I mean that which is divisible into divisibles that
+are infinitely divisible: and if we take this as the definition of
+continuous, it follows necessarily that time is continuous. For since
+it has been shown that the quicker will pass over an equal magnitude
+in less time than the slower, suppose that A is quicker and B slower,
+and that the slower has traversed the magnitude GD in the time ZH.
+Now it is clear that the quicker will traverse the same magnitude
+in less time than this: let us say in the time ZO. Again, since the
+quicker has passed over the whole D in the time ZO, the slower will
+in the same time pass over GK, say, which is less than GD. And since
+B, the slower, has passed over GK in the time ZO, the quicker will
+pass over it in less time: so that the time ZO will again be divided.
+And if this is divided the magnitude GK will also be divided just
+as GD was: and again, if the magnitude is divided, the time will also
+be divided. And we can carry on this process for ever, taking the
+slower after the quicker and the quicker after the slower alternately,
+and using what has been demonstrated at each stage as a new point
+of departure: for the quicker will divide the time and the slower
+will divide the length. If, then, this alternation always holds good,
+and at every turn involves a division, it is evident that all time
+must be continuous. And at the same time it is clear that all magnitude
+is also continuous; for the divisions of which time and magnitude
+respectively are susceptible are the same and equal. 
+
+Moreover, the current popular arguments make it plain that, if time
+is continuous, magnitude is continuous also, inasmuch as a thing asses
+over half a given magnitude in half the time taken to cover the whole:
+in fact without qualification it passes over a less magnitude in less
+time; for the divisions of time and of magnitude will be the same.
+And if either is infinite, so is the other, and the one is so in the
+same way as the other; i.e. if time is infinite in respect of its
+extremities, length is also infinite in respect of its extremities:
+if time is infinite in respect of divisibility, length is also infinite
+in respect of divisibility: and if time is infinite in both respects,
+magnitude is also infinite in both respects. 
+
+Hence Zeno's argument makes a false assumption in asserting that it
+is impossible for a thing to pass over or severally to come in contact
+with infinite things in a finite time. For there are two senses in
+which length and time and generally anything continuous are called
+'infinite': they are called so either in respect of divisibility or
+in respect of their extremities. So while a thing in a finite time
+cannot come in contact with things quantitatively infinite, it can
+come in contact with things infinite in respect of divisibility: for
+in this sense the time itself is also infinite: and so we find that
+the time occupied by the passage over the infinite is not a finite
+but an infinite time, and the contact with the infinites is made by
+means of moments not finite but infinite in number. 
+
+The passage over the infinite, then, cannot occupy a finite time,
+and the passage over the finite cannot occupy an infinite time: if
+the time is infinite the magnitude must be infinite also, and if the
+magnitude is infinite, so also is the time. This may be shown as follows.
+Let AB be a finite magnitude, and let us suppose that it is traversed
+in infinite time G, and let a finite period GD of the time be taken.
+Now in this period the thing in motion will pass over a certain segment
+of the magnitude: let BE be the segment that it has thus passed over.
+(This will be either an exact measure of AB or less or greater than
+an exact measure: it makes no difference which it is.) Then, since
+a magnitude equal to BE will always be passed over in an equal time,
+and BE measures the whole magnitude, the whole time occupied in passing
+over AB will be finite: for it will be divisible into periods equal
+in number to the segments into which the magnitude is divisible. Moreover,
+if it is the case that infinite time is not occupied in passing over
+every magnitude, but it is possible to ass over some magnitude, say
+BE, in a finite time, and if this BE measures the whole of which it
+is a part, and if an equal magnitude is passed over in an equal time,
+then it follows that the time like the magnitude is finite. That infinite
+time will not be occupied in passing over BE is evident if the time
+be taken as limited in one direction: for as the part will be passed
+over in less time than the whole, the time occupied in traversing
+this part must be finite, the limit in one direction being given.
+The same reasoning will also show the falsity of the assumption that
+infinite length can be traversed in a finite time. It is evident,
+then, from what has been said that neither a line nor a surface nor
+in fact anything continuous can be indivisible. 
+
+This conclusion follows not only from the present argument but from
+the consideration that the opposite assumption implies the divisibility
+of the indivisible. For since the distinction of quicker and slower
+may apply to motions occupying any period of time and in an equal
+time the quicker passes over a greater length, it may happen that
+it will pass over a length twice, or one and a half times, as great
+as that passed over by the slower: for their respective velocities
+may stand to one another in this proportion. Suppose, then, that the
+quicker has in the same time been carried over a length one and a
+half times as great as that traversed by the slower, and that the
+respective magnitudes are divided, that of the quicker, the magnitude
+ABGD, into three indivisibles, and that of the slower into the two
+indivisibles EZ, ZH. Then the time may also be divided into three
+indivisibles, for an equal magnitude will be passed over in an equal
+time. Suppose then that it is thus divided into KL, Lm, MN. Again,
+since in the same time the slower has been carried over Ez, ZH, the
+time may also be similarly divided into two. Thus the indivisible
+will be divisible, and that which has no parts will be passed over
+not in an indivisible but in a greater time. It is evident, therefore,
+that nothing continuous is without parts. 
+
+Part 3
+
+The present also is necessarily indivisible-the present, that is,
+not in the sense in which the word is applied to one thing in virtue
+of another, but in its proper and primary sense; in which sense it
+is inherent in all time. For the present is something that is an extremity
+of the past (no part of the future being on this side of it) and also
+of the future (no part of the past being on the other side of it):
+it is, as we have said, a limit of both. And if it is once shown that
+it is essentially of this character and one and the same, it will
+at once be evident also that it is indivisible. 
+
+Now the present that is the extremity of both times must be one and
+the same: for if each extremity were different, the one could not
+be in succession to the other, because nothing continuous can be composed
+of things having no parts: and if the one is apart from the other,
+there will be time intermediate between them, because everything continuous
+is such that there is something intermediate between its limits and
+described by the same name as itself. But if the intermediate thing
+is time, it will be divisible: for all time has been shown to be divisible.
+Thus on this assumption the present is divisible. But if the present
+is divisible, there will be part of the past in the future and part
+of the future in the past: for past time will be marked off from future
+time at the actual point of division. Also the present will be a present
+not in the proper sense but in virtue of something else: for the division
+which yields it will not be a division proper. Furthermore, there
+will be a part of the present that is past and a part that is future,
+and it will not always be the same part that is past or future: in
+fact one and the same present will not be simultaneous: for the time
+may be divided at many points. If, therefore, the present cannot possibly
+have these characteristics, it follows that it must be the same present
+that belongs to each of the two times. But if this is so it is evident
+that the present is also indivisible: for if it is divisible it will
+be involved in the same implications as before. It is clear, then,
+from what has been said that time contains something indivisible,
+and this is what we call a present. 
+
+We will now show that nothing can be in motion in a present. For if
+this is possible, there can be both quicker and slower motion in the
+present. Suppose then that in the present N the quicker has traversed
+the distance AB. That being so, the slower will in the same present
+traverse a distance less than AB, say AG. But since the slower will
+have occupied the whole present in traversing AG, the quicker will
+occupy less than this in traversing it. Thus we shall have a division
+of the present, whereas we found it to be indivisible. It is impossible,
+therefore, for anything to be in motion in a present. 
+
+Nor can anything be at rest in a present: for, as we were saying,
+only can be at rest which is naturally designed to be in motion but
+is not in motion when, where, or as it would naturally be so: since,
+therefore, nothing is naturally designed to be in motion in a present,
+it is clear that nothing can be at rest in a present either.
+
+Moreover, inasmuch as it is the same present that belongs to both
+the times, and it is possible for a thing to be in motion throughout
+one time and to be at rest throughout the other, and that which is
+in motion or at rest for the whole of a time will be in motion or
+at rest as the case may be in any part of it in which it is naturally
+designed to be in motion or at rest: this being so, the assumption
+that there can be motion or rest in a present will carry with it the
+implication that the same thing can at the same time be at rest and
+in motion: for both the times have the same extremity, viz. the present.
+
+Again, when we say that a thing is at rest, we imply that its condition
+in whole and in part is at the time of speaking uniform with what
+it was previously: but the present contains no 'previously': consequently,
+there can be no rest in it. 
+
+It follows then that the motion of that which is in motion and the
+rest of that which is at rest must occupy time. 
+
+Part 4
+
+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. 
+
+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: e.g. 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. 
+
+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: for, as
+we saw, a motion that is one cannot be the motion of more things than
+one): since this is so, the whole motion will be the motion of the
+magnitude ABG. 
+
+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.
+
+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. 
+
+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. 
+
+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. 
+
+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? 
+
+Part 5
+
+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.
+
+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. 
+
+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. 
+
+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. 
+
+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.
+
+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. 
+
+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-that
+which changes, that in which it changes, and 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. 
+
+Part 6
+
+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. 
+
+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. 
+
+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. 
+
+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. 
+
+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. 
+
+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.
+
+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. 
+
+Consequently no absolutely first stage of change can be represented
+by any particular part of space or time which the changing thing may
+occupy. 
+
+Part 7
+
+Now since the motion of everything that is in motion occupies a period
+of time, and a greater magnitude is traversed in a longer time, it
+is impossible that a thing should undergo a finite motion in an infinite
+time, if this is understood to mean not that the same motion or a
+part of it is continually repeated, but that the whole infinite time
+is occupied by the whole finite motion. In all cases where a thing
+is in motion with uniform velocity it is clear that the finite magnitude
+is traversed in a finite time. For if we take a part of the motion
+which shall be a measure of the whole, the whole motion is completed
+in as many equal periods of the time as there are parts of the motion.
+Consequently, since these parts are finite, both in size individually
+and in number collectively, the whole time must also be finite: for
+it will be a multiple of the portion, equal to the time occupied in
+completing the aforesaid part multiplied by the number of the parts.
+
+But it makes no difference even if the velocity is not uniform. For
+let us suppose that the line AB represents a finite stretch over which
+a thing has been moved in the given time, and let GD be the infinite
+time. Now if one part of the stretch must have been traversed before
+another part (this is clear, that in the earlier and in the later
+part of the time a different part of the stretch has been traversed:
+for as the time lengthens a different part of the motion will always
+be completed in it, whether the thing in motion changes with uniform
+velocity or not: and whether the rate of motion increases or diminishes
+or remains stationary this is none the less so), let us then take
+AE a part of the whole stretch of motion AB which shall be a measure
+of AB. Now this part of the motion occupies a certain period of the
+infinite time: it cannot itself occupy an infinite time, for we are
+assuming that that is occupied by the whole AB. And if again I take
+another part equal to AE, that also must occupy a finite time in consequence
+of the same assumption. And if I go on taking parts in this way, on
+the one hand there is no part which will be a measure of the infinite
+time (for the infinite cannot be composed of finite parts whether
+equal or unequal, because there must be some unity which will be a
+measure of things finite in multitude or in magnitude, which, whether
+they are equal or unequal, are none the less limited in magnitude);
+while on the other hand the finite stretch of motion AB is a certain
+multiple of AE: consequently the motion AB must be accomplished in
+a finite time. Moreover it is the same with coming to rest as with
+motion. And so it is impossible for one and the same thing to be infinitely
+in process of becoming or of perishing. The reasoning he will prove
+that in a finite time there cannot be an infinite extent of motion
+or of coming to rest, whether the motion is regular or irregular.
+For if we take a part which shall be a measure of the whole time,
+in this part a certain fraction, not the whole, of the magnitude will
+be traversed, because we assume that the traversing of the whole occupies
+all the time. Again, in another equal part of the time another part
+of the magnitude will be traversed: and similarly in each part of
+the time that we take, whether equal or unequal to the part originally
+taken. It makes no difference whether the parts are equal or not,
+if only each is finite: for it is clear that while the time is exhausted
+by the subtraction of its parts, the infinite magnitude will not be
+thus exhausted, since the process of subtraction is finite both in
+respect of the quantity subtracted and of the number of times a subtraction
+is made. Consequently the infinite magnitude will not be traversed
+in finite time: and it makes no difference whether the magnitude is
+infinite in only one direction or in both: for the same reasoning
+will hold good. 
+
+This having been proved, it is evident that neither can a finite magnitude
+traverse an infinite magnitude in a finite time, the reason being
+the same as that given above: in part of the time it will traverse
+a finite magnitude and in each several part likewise, so that in the
+whole time it will traverse a finite magnitude. 
+
+And since a finite magnitude will not traverse an infinite in a finite
+time, it is clear that neither will an infinite traverse a finite
+in a finite time. For if the infinite could traverse the finite, the
+finite could traverse the infinite; for it makes no difference which
+of the two is the thing in motion; either case involves the traversing
+of the infinite by the finite. For when the infinite magnitude A is
+in motion a part of it, say GD, will occupy the finite and then another,
+and then another, and so on to infinity. Thus the two results will
+coincide: the infinite will have completed a motion over the finite
+and the finite will have traversed the infinite: for it would seem
+to be impossible for the motion of the infinite over the finite to
+occur in any way other than by the finite traversing the infinite
+either by locomotion over it or by measuring it. Therefore, since
+this is impossible, the infinite cannot traverse the finite.
+
+Nor again will the infinite traverse the infinite in a finite time.
+Otherwise it would also traverse the finite, for the infinite includes
+the finite. We can further prove this in the same way by taking the
+time as our starting-point. 
+
+Since, then, it is established that in a finite time neither will
+the finite traverse the infinite, nor the infinite the finite, nor
+the infinite the infinite, it is evident also that in a finite time
+there cannot be infinite motion: for what difference does it make
+whether we take the motion or the magnitude to be infinite? If either
+of the two is infinite, the other must be so likewise: for all locomotion
+is in space. 
+
+Part 8
+
+Since everything to which motion or rest is natural is in motion or
+at rest in the natural time, place, and manner, that which is coming
+to a stand, when it is coming to a stand, must be in motion: for if
+it is not in motion it must be at rest: but that which is at rest
+cannot be coming to rest. From this it evidently follows that coming
+to a stand must occupy a period of time: for the motion of that which
+is in motion occupies a period of time, and that which is coming to
+a stand has been shown to be in motion: consequently coming to a stand
+must occupy a period of time. 
+
+Again, since the terms 'quicker' and 'slower' are used only of that
+which occupies a period of time, and the process of coming to a stand
+may be quicker or slower, the same conclusion follows. 
+
+And that which is coming to a stand must be coming to a stand in any
+part of the primary time in which it is coming to a stand. For if
+it is coming to a stand in neither of two parts into which the time
+may be divided, it cannot be coming to a stand in the whole time,
+with the result that that that which is coming to a stand will not
+be coming to a stand. If on the other hand it is coming to a stand
+in only one of the two parts of the time, the whole cannot be the
+primary time in which it is coming to a stand: for it is coming to
+a stand in the whole time not primarily but in virtue of something
+distinct from itself, the argument being the same as that which we
+used above about things in motion. 
+
+And just as there is no primary time in which that which is in motion
+is in motion, so too there is no primary time in which that which
+is coming to a stand is coming to a stand, there being no primary
+stage either of being in motion or of coming to a stand. For let AB
+be the primary time in which a thing is coming to a stand. Now AB
+cannot be without parts: for there cannot be motion in that which
+is without parts, because the moving thing would necessarily have
+been already moved for part of the time of its movement: and that
+which is coming to a stand has been shown to be in motion. But since
+Ab is therefore divisible, the thing is coming to a stand in every
+one of the parts of AB: for we have shown above that it is coming
+to a stand in every one of the parts in which it is primarily coming
+to a stand. Since then, that in which primarily a thing is coming
+to a stand must be a period of time and not something indivisible,
+and since all time is infinitely divisible, there cannot be anything
+in which primarily it is coming to a stand. 
+
+Nor again can there be a primary time at which the being at rest of
+that which is at rest occurred: for it cannot have occurred in that
+which has no parts, because there cannot be motion in that which is
+indivisible, and that in which rest takes place is the same as that
+in which motion takes place: for we defined a state of rest to be
+the state of a thing to which motion is natural but which is not in
+motion when (that is to say in that in which) motion would be natural
+to it. Again, our use of the phrase 'being at rest' also implies that
+the previous state of a thing is still unaltered, not one point only
+but two at least being thus needed to determine its presence: consequently
+that in which a thing is at rest cannot be without parts. Since, then
+it is divisible, it must be a period of time, and the thing must be
+at rest in every one of its parts, as may be shown by the same method
+as that used above in similar demonstrations. 
+
+So there can be no primary part of the time: and the reason is that
+rest and motion are always in a period of time, and a period of time
+has no primary part any more than a magnitude or in fact anything
+continuous: for everything continuous is divisible into an infinite
+number of parts. 
+
+And since everything that is in motion is in motion in a period of
+time and changes from something to something, when its motion is comprised
+within a particular period of time essentially-that is to say when
+it fills the whole and not merely a part of the time in question-it
+is impossible that in that time that which is in motion should be
+over against some particular thing primarily. For if a thing-itself
+and each of its parts-occupies the same space for a definite period
+of time, it is at rest: for it is in just these circumstances that
+we use the term 'being at rest'-when at one moment after another it
+can be said with truth that a thing, itself and its parts, occupies
+the same space. So if this is being at rest it is impossible for that
+which is changing to be as a whole, at the time when it is primarily
+changing, over against any particular thing (for the whole period
+of time is divisible), so that in one part of it after another it
+will be true to say that the thing, itself and its parts, occupies
+the same space. If this is not so and the aforesaid proposition is
+true only at a single moment, then the thing will be over against
+a particular thing not for any period of time but only at a moment
+that limits the time. It is true that at any moment it is always over
+against something stationary: but it is not at rest: for at a moment
+it is not possible for anything to be either in motion or at rest.
+So while it is true to say that that which is in motion is at a moment
+not in motion and is opposite some particular thing, it cannot in
+a period of time be over against that which is at rest: for that would
+involve the conclusion that that which is in locomotion is at rest.
+
+Part 9
+
+Zeno's reasoning, however, is fallacious, when he says that if everything
+when it occupies an equal space is at rest, and if that which is in
+locomotion is always occupying such a space at any moment, the flying
+arrow is therefore motionless. This is false, for time is not composed
+of indivisible moments any more than any other magnitude is composed
+of indivisibles. 
+
+Zeno's arguments about motion, which cause so much disquietude to
+those who try to solve the problems that they present, are four in
+number. The first asserts the non-existence of motion on the ground
+that that which is in locomotion must arrive at the half-way stage
+before it arrives at the goal. This we have discussed above.
+
+The second is the so-called 'Achilles', and it amounts to this, that
+in a race the quickest runner can never overtake the slowest, since
+the pursuer must first reach the point whence the pursued started,
+so that the slower must always hold a lead. This argument is the same
+in principle as that which depends on bisection, though it differs
+from it in that the spaces with which we successively have to deal
+are not divided into halves. The result of the argument is that the
+slower is not overtaken: but it proceeds along the same lines as the
+bisection-argument (for in both a division of the space in a certain
+way leads to the result that the goal is not reached, though the 'Achilles'
+goes further in that it affirms that even the quickest runner in legendary
+tradition must fail in his pursuit of the slowest), so that the solution
+must be the same. And the axiom that that which holds a lead is never
+overtaken is false: it is not overtaken, it is true, while it holds
+a lead: but it is overtaken nevertheless if it is granted that it
+traverses the finite distance prescribed. These then are two of his
+arguments. 
+
+The third is that already given above, to the effect that the flying
+arrow is at rest, which result follows from the assumption that time
+is composed of moments: if this assumption is not granted, the conclusion
+will not follow. 
+
+The fourth argument is that concerning the two rows of bodies, each
+row being composed of an equal number of bodies of equal size, passing
+each other on a race-course as they proceed with equal velocity in
+opposite directions, the one row originally occupying the space between
+the goal and the middle point of the course and the other that between
+the middle point and the starting-post. This, he thinks, involves
+the conclusion that half a given time is equal to double that time.
+The fallacy of the reasoning lies in the assumption that a body occupies
+an equal time in passing with equal velocity a body that is in motion
+and a body of equal size that is at rest; which is false. For instance
+(so runs the argument), let A, A...be the stationary bodies of equal
+size, B, B...the bodies, equal in number and in size to A, A...,originally
+occupying the half of the course from the starting-post to the middle
+of the A's, and G, G...those originally occupying the other half from
+the goal to the middle of the A's, equal in number, size, and velocity
+to B, B....Then three consequences follow: 
+
+First, as the B's and the G's pass one another, the first B reaches
+the last G at the same moment as the first G reaches the last B. Secondly
+at this moment the first G has passed all the A's, whereas the first
+B has passed only half the A's, and has consequently occupied only
+half the time occupied by the first G, since each of the two occupies
+an equal time in passing each A. Thirdly, at the same moment all the
+B's have passed all the G's: for the first G and the first B will
+simultaneously reach the opposite ends of the course, since (so says
+Zeno) the time occupied by the first G in passing each of the B's
+is equal to that occupied by it in passing each of the A's, because
+an equal time is occupied by both the first B and the first G in passing
+all the A's. This is the argument, but it presupposed the aforesaid
+fallacious assumption. 
+
+Nor in reference to contradictory change shall we find anything unanswerable
+in the argument that if a thing is changing from not-white, say, to
+white, and is in neither condition, then it will be neither white
+nor not-white: for the fact that it is not wholly in either condition
+will not preclude us from calling it white or not-white. We call a
+thing white or not-white not necessarily because it is be one or the
+other, but cause most of its parts or the most essential parts of
+it are so: not being in a certain condition is different from not
+being wholly in that condition. So, too, in the case of being and
+not-being and all other conditions which stand in a contradictory
+relation: while the changing thing must of necessity be in one of
+the two opposites, it is never wholly in either. 
+
+Again, in the case of circles and spheres and everything whose motion
+is confined within the space that it occupies, it is not true to say
+the motion can be nothing but rest, on the ground that such things
+in motion, themselves and their parts, will occupy the same position
+for a period of time, and that therefore they will be at once at rest
+and in motion. For in the first place the parts do not occupy the
+same position for any period of time: and in the second place the
+whole also is always changing to a different position: for if we take
+the orbit as described from a point A on a circumference, it will
+not be the same as the orbit as described from B or G or any other
+point on the same circumference except in an accidental sense, the
+sense that is to say in which a musical man is the same as a man.
+Thus one orbit is always changing into another, and the thing will
+never be at rest. And it is the same with the sphere and everything
+else whose motion is confined within the space that it occupies.
+
+Part 10
+
+Our next point is that that which is without parts cannot be in motion
+except accidentally: i.e. it can be in motion only in so far as the
+body or the magnitude is in motion and the partless is in motion by
+inclusion therein, just as that which is in a boat may be in motion
+in consequence of the locomotion of the boat, or a part may be in
+motion in virtue of the motion of the whole. (It must be remembered,
+however, that by 'that which is without parts' I mean that which is
+quantitatively indivisible (and that the case of the motion of a part
+is not exactly parallel): for parts have motions belonging essentially
+and severally to themselves distinct from the motion of the whole.
+The distinction may be seen most clearly in the case of a revolving
+sphere, in which the velocities of the parts near the centre and of
+those on the surface are different from one another and from that
+of the whole; this implies that there is not one motion but many).
+As we have said, then, that which is without parts can be in motion
+in the sense in which a man sitting in a boat is in motion when the
+boat is travelling, but it cannot be in motion of itself. For suppose
+that it is changing from AB to BG-either from one magnitude to another,
+or from one form to another, or from some state to its contradictory-and
+let D be the primary time in which it undergoes the change. Then in
+the time in which it is changing it must be either in AB or in BG
+or partly in one and partly in the other: for this, as we saw, is
+true of everything that is changing. Now it cannot be partly in each
+of the two: for then it would be divisible into parts. Nor again can
+it be in BG: for then it will have completed the change, whereas the
+assumption is that the change is in process. It remains, then, that
+in the time in which it is changing, it is in Ab. That being so, it
+will be at rest: for, as we saw, to be in the same condition for a
+period of time is to be at rest. So it is not possible for that which
+has no parts to be in motion or to change in any way: for only one
+condition could have made it possible for it to have motion, viz.
+that time should be composed of moments, in which case at any moment
+it would have completed a motion or a change, so that it would never
+be in motion, but would always have been in motion. But this we have
+already shown above to be impossible: time is not composed of moments,
+just as a line is not composed of points, and motion is not composed
+of starts: for this theory simply makes motion consist of indivisibles
+in exactly the same way as time is made to consist of moments or a
+length of points. 
+
+Again, it may be shown in the following way that there can be no motion
+of a point or of any other indivisible. That which is in motion can
+never traverse a space greater than itself without first traversing
+a space equal to or less than itself. That being so, it is evident
+that the point also must first traverse a space equal to or less than
+itself. But since it is indivisible, there can be no space less than
+itself for it to traverse first: so it will have to traverse a distance
+equal to itself. Thus the line will be composed of points, for the
+point, as it continually traverses a distance equal to itself, will
+be a measure of the whole line. But since this is impossible, it is
+likewise impossible for the indivisible to be in motion.
+
+Again, since motion is always in a period of time and never in a moment,
+and all time is divisible, for everything that is in motion there
+must be a time less than that in which it traverses a distance as
+great as itself. For that in which it is in motion will be a time,
+because all motion is in a period of time; and all time has been shown
+above to be divisible. Therefore, if a point is in motion, there must
+be a time less than that in which it has itself traversed any distance.
+But this is impossible, for in less time it must traverse less distance,
+and thus the indivisible will be divisible into something less than
+itself, just as the time is so divisible: the fact being that the
+only condition under which that which is without parts and indivisible
+could be in motion would have been the possibility of the infinitely
+small being in motion in a moment: for in the two questions-that of
+motion in a moment and that of motion of something indivisible-the
+same principle is involved. 
+
+Our next point is that no process of change is infinite: for every
+change, whether between contradictories or between contraries, is
+a change from something to something. Thus in contradictory changes
+the positive or the negative, as the case may be, is the limit, e.g.
+being is the limit of coming to be and not-being is the limit of ceasing
+to be: and in contrary changes the particular contraries are the limits,
+since these are the extreme points of any such process of change,
+and consequently of every process of alteration: for alteration is
+always dependent upon some contraries. Similarly contraries are the
+extreme points of processes of increase and decrease: the limit of
+increase is to be found in the complete magnitude proper to the peculiar
+nature of the thing that is increasing, while the limit of decrease
+is the complete loss of such magnitude. Locomotion, it is true, we
+cannot show to be finite in this way, since it is not always between
+contraries. But since that which cannot be cut (in the sense that
+it is inconceivable that it should be cut, the term 'cannot' being
+used in several senses)-since it is inconceivable that that which
+in this sense cannot be cut should be in process of being cut, and
+generally that that which cannot come to be should be in process of
+coming to be, it follows that it is inconceivable that that which
+cannot complete a change should be in process of changing to that
+to which it cannot complete a change. If, then, it is to be assumed
+that that which is in locomotion is in process of changing, it must
+be capable of completing the change. Consequently its motion is not
+infinite, and it will not be in locomotion over an infinite distance,
+for it cannot traverse such a distance. 
+
+It is evident, then, that a process of change cannot be infinite in
+the sense that it is not defined by limits. But it remains to be considered
+whether it is possible in the sense that one and the same process
+of change may be infinite in respect of the time which it occupies.
+If it is not one process, it would seem that there is nothing to prevent
+its being infinite in this sense; e.g. if a process of locomotion
+be succeeded by a process of alteration and that by a process of increase
+and that again by a process of coming to be: in this way there may
+be motion for ever so far as the time is concerned, but it will not
+be one motion, because all these motions do not compose one. If it
+is to be one process, no motion can be infinite in respect of the
+time that it occupies, with the single exception of rotatory locomotion.
+
+----------------------------------------------------------------------
+
+BOOK VII
+
+Part 1 
+
+Everything that is in motion must be moved by something. For if it
+has not the source of its motion in itself it is evident that it is
+moved by something other than itself, for there must be something
+else that moves it. If on the other hand it has the source of its
+motion in itself, let AB be taken to represent that which is in motion
+essentially of itself and not in virtue of the fact that something
+belonging to it is in motion. Now in the first place to assume that
+Ab, because it is in motion as a whole and is not moved by anything
+external to itself, is therefore moved by itself-this is just as if,
+supposing that KL is moving LM and is also itself in motion, we were
+to deny that KM is moved by anything on the ground that it is not
+evident which is the part that is moving it and which the part that
+is moved. In the second place that which is in motion without being
+moved by anything does not necessarily cease from its motion because
+something else is at rest, but a thing must be moved by something
+if the fact of something else having ceased from its motion causes
+it to be at rest. Thus, if this is accepted, everything that is in
+motion must be moved by something. For AB, which has been taken to
+represent that which is in motion, must be divisible since everything
+that is in motion is divisible. Let it be divided, then, at G. Now
+if GB is not in motion, then AB will not be in motion: for if it is,
+it is clear that AG would be in motion while BG is at rest, and thus
+AB cannot be in motion essentially and primarily. But ex hypothesi
+AB is in motion essentially and primarily. Therefore if GB is not
+in motion AB will be at rest. But we have agreed that that which is
+at rest if something else is not in motion must be moved by something.
+Consequently, everything that is in motion must be moved by something:
+for that which is in motion will always be divisible, and if a part
+of it is not in motion the whole must be at rest. 
+
+Since everything that is in motion must be moved by something, let
+us take the case in which a thing is in locomotion and is moved by
+something that is itself in motion, and that again is moved by something
+else that is in motion, and that by something else, and so on continually:
+then the series cannot go on to infinity, but there must be some first
+movent. For let us suppose that this is not so and take the series
+to be infinite. Let A then be moved by B, B by G, G by D, and so on,
+each member of the series being moved by that which comes next to
+it. Then since ex hypothesi the movent while causing motion is also
+itself in motion, and the motion of the moved and the motion of the
+movent must proceed simultaneously (for the movent is causing motion
+and the moved is being moved simultaneously) it is evident that the
+respective motions of A, B, G, and each of the other moved movents
+are simultaneous. Let us take the motion of each separately and let
+E be the motion of A, Z of B, and H and O respectively the motions
+of G and D: for though they are all moved severally one by another,
+yet we may still take the motion of each as numerically one, since
+every motion is from something to something and is not infinite in
+respect of its extreme points. By a motion that is numerically one
+I mean a motion that proceeds from something numerically one and the
+same to something numerically one and the same in a period of time
+numerically one and the same: for a motion may be the same generically,
+specifically, or numerically: it is generically the same if it belongs
+to the same category, e.g. substance or quality: it is specifically
+the same if it proceeds from something specifically the same to something
+specifically the same, e.g. from white to black or from good to bad,
+which is not of a kind specifically distinct: it is numerically the
+same if it proceeds from something numerically one to something numerically
+one in the same period of time, e.g. from a particular white to a
+particular black, or from a particular place to a particular place,
+in a particular period of time: for if the period of time were not
+one and the same, the motion would no longer be numerically one though
+it would still be specifically one. 
+
+We have dealt with this question above. Now let us further take the
+time in which A has completed its motion, and let it be represented
+by K. Then since the motion of A is finite the time will also be finite.
+But since the movents and the things moved are infinite, the motion
+EZHO, i.e. the motion that is composed of all the individual motions,
+must be infinite. For the motions of A, B, and the others may be equal,
+or the motions of the others may be greater: but assuming what is
+conceivable, we find that whether they are equal or some are greater,
+in both cases the whole motion is infinite. And since the motion of
+A and that of each of the others are simultaneous, the whole motion
+must occupy the same time as the motion of A: but the time occupied
+by the motion of A is finite: consequently the motion will be infinite
+in a finite time, which is impossible. 
+
+It might be thought that what we set out to prove has thus been shown,
+but our argument so far does not prove it, because it does not yet
+prove that anything impossible results from the contrary supposition:
+for in a finite time there may be an infinite motion, though not of
+one thing, but of many: and in the case that we are considering this
+is so: for each thing accomplishes its own motion, and there is no
+impossibility in many things being in motion simultaneously. But if
+(as we see to be universally the case) that which primarily is moved
+locally and corporeally must be either in contact with or continuous
+with that which moves it, the things moved and the movents must be
+continuous or in contact with one another, so that together they all
+form a single unity: whether this unity is finite or infinite makes
+no difference to our present argument; for in any case since the things
+in motion are infinite in number the whole motion will be infinite,
+if, as is theoretically possible, each motion is either equal to or
+greater than that which follows it in the series: for we shall take
+as actual that which is theoretically possible. If, then, A, B, G,
+D form an infinite magnitude that passes through the motion EZHO in
+the finite time K, this involves the conclusion that an infinite motion
+is passed through in a finite time: and whether the magnitude in question
+is finite or infinite this is in either case impossible. Therefore
+the series must come to an end, and there must be a first movent and
+a first moved: for the fact that this impossibility results only from
+the assumption of a particular case is immaterial, since the case
+assumed is theoretically possible, and the assumption of a theoretically
+possible case ought not to give rise to any impossible result.
+
+Part 2
+
+That which is the first movement of a thing-in the sense that it supplies
+not 'that for the sake of which' but the source of the motion-is always
+together with that which is moved by it by 'together' I mean that
+there is nothing intermediate between them). This is universally true
+wherever one thing is moved by another. And since there are three
+kinds of motion, local, qualitative, and quantitative, there must
+also be three kinds of movent, that which causes locomotion, that
+which causes alteration, and that which causes increase or decrease.
+
+Let us begin with locomotion, for this is the primary motion. Everything
+that is in locomotion is moved either by itself or by something else.
+In the case of things that are moved by themselves it is evident that
+the moved and the movent are together: for they contain within themselves
+their first movent, so that there is nothing in between. The motion
+of things that are moved by something else must proceed in one of
+four ways: for there are four kinds of locomotion caused by something
+other than that which is in motion, viz. pulling, pushing, carrying,
+and twirling. All forms of locomotion are reducible to these. Thus
+pushing on is a form of pushing in which that which is causing motion
+away from itself follows up that which it pushes and continues to
+push it: pushing off occurs when the movent does not follow up the
+thing that it has moved: throwing when the movent causes a motion
+away from itself more violent than the natural locomotion of the thing
+moved, which continues its course so long as it is controlled by the
+motion imparted to it. Again, pushing apart and pushing together are
+forms respectively of pushing off and pulling: pushing apart is pushing
+off, which may be a motion either away from the pusher or away from
+something else, while pushing together is pulling, which may be a
+motion towards something else as well as the puller. We may similarly
+classify all the varieties of these last two, e.g. packing and combing:
+the former is a form of pushing together, the latter a form of pushing
+apart. The same is true of the other processes of combination and
+separation (they will all be found to be forms of pushing apart or
+of pushing together), except such as are involved in the processes
+of becoming and perishing. (At same time it is evident that there
+is no other kind of motion but combination and separation: for they
+may all be apportioned to one or other of those already mentioned.)
+Again, inhaling is a form of pulling, exhaling a form of pushing:
+and the same is true of spitting and of all other motions that proceed
+through the body, whether secretive or assimilative, the assimilative
+being forms of pulling, the secretive of pushing off. All other kinds
+of locomotion must be similarly reduced, for they all fall under one
+or other of our four heads. And again, of these four, carrying and
+twirling are to pulling and pushing. For carrying always follows one
+of the other three methods, for that which is carried is in motion
+accidentally, because it is in or upon something that is in motion,
+and that which carries it is in doing so being either pulled or pushed
+or twirled; thus carrying belongs to all the other three kinds of
+motion in common. And twirling is a compound of pulling and pushing,
+for that which is twirling a thing must be pulling one part of the
+thing and pushing another part, since it impels one part away from
+itself and another part towards itself. If, therefore, it can be shown
+that that which is pushing and that which is pushing and pulling are
+adjacent respectively to that which is being pushed and that which
+is being pulled, it will be evident that in all locomotion there is
+nothing intermediate between moved and movent. But the former fact
+is clear even from the definitions of pushing and pulling, for pushing
+is motion to something else from oneself or from something else, and
+pulling is motion from something else to oneself or to something else,
+when the motion of that which is pulling is quicker than the motion
+that would separate from one another the two things that are continuous:
+for it is this that causes one thing to be pulled on along with the
+other. (It might indeed be thought that there is a form of pulling
+that arises in another way: that wood, e.g. pulls fire in a manner
+different from that described above. But it makes no difference whether
+that which pulls is in motion or is stationary when it is pulling:
+in the latter case it pulls to the place where it is, while in the
+former it pulls to the place where it was.) Now it is impossible to
+move anything either from oneself to something else or something else
+to oneself without being in contact with it: it is evident, therefore,
+that in all locomotion there is nothing intermediate between moved
+and movent. 
+
+Nor again is there anything intermediate between that which undergoes
+and that which causes alteration: this can be proved by induction:
+for in every case we find that the respective extremities of that
+which causes and that which undergoes alteration are adjacent. For
+our assumption is that things that are undergoing alteration are altered
+in virtue of their being affected in respect of their so-called affective
+qualities, since that which is of a certain quality is altered in
+so far as it is sensible, and the characteristics in which bodies
+differ from one another are sensible characteristics: for every body
+differs from another in possessing a greater or lesser number of sensible
+characteristics or in possessing the same sensible characteristics
+in a greater or lesser degree. But the alteration of that which undergoes
+alteration is also caused by the above-mentioned characteristics,
+which are affections of some particular underlying quality. Thus we
+say that a thing is altered by becoming hot or sweet or thick or dry
+or white: and we make these assertions alike of what is inanimate
+and of what is animate, and further, where animate things are in question,
+we make them both of the parts that have no power of sense-perception
+and of the senses themselves. For in a way even the senses undergo
+alteration, since the active sense is a motion through the body in
+the course of which the sense is affected in a certain way. We see,
+then, that the animate is capable of every kind of alteration of which
+the inanimate is capable: but the inanimate is not capable of every
+kind of alteration of which the animate is capable, since it is not
+capable of alteration in respect of the senses: moreover the inanimate
+is unconscious of being affected by alteration, whereas the animate
+is conscious of it, though there is nothing to prevent the animate
+also being unconscious of it when the process of the alteration does
+not concern the senses. Since, then, the alteration of that which
+undergoes alteration is caused by sensible things, in every case of
+such alteration it is evident that the respective extremities of that
+which causes and that which undergoes alteration are adjacent. Thus
+the air is continuous with that which causes the alteration, and the
+body that undergoes alteration is continuous with the air. Again,
+the colour is continuous with the light and the light with the sight.
+And the same is true of hearing and smelling: for the primary movent
+in respect to the moved is the air. Similarly, in the case of tasting,
+the flavour is adjacent to the sense of taste. And it is just the
+same in the case of things that are inanimate and incapable of sense-perception.
+Thus there can be nothing intermediate between that which undergoes
+and that which causes alteration. 
+
+Nor, again, can there be anything intermediate between that which
+suffers and that which causes increase: for the part of the latter
+that starts the increase does so by becoming attached in such a way
+to the former that the whole becomes one. Again, the decrease of that
+which suffers decrease is caused by a part of the thing becoming detached.
+So that which causes increase and that which causes decrease must
+be continuous with that which suffers increase and that which suffers
+decrease respectively: and if two things are continuous with one another
+there can be nothing intermediate between them. 
+
+It is evident, therefore, that between the extremities of the moved
+and the movent that are respectively first and last in reference to
+the moved there is nothing intermediate. 
+
+Part 3
+
+Everything, we say, that undergoes alteration is altered by sensible
+causes, and there is alteration only in things that are said to be
+essentially affected by sensible things. The truth of this is to be
+seen from the following considerations. Of all other things it would
+be most natural to suppose that there is alteration in figures and
+shapes, and in acquired states and in the processes of acquiring and
+losing these: but as a matter of fact in neither of these two classes
+of things is there alteration. 
+
+In the first place, when a particular formation of a thing is completed,
+we do not call it by the name of its material: e.g. we do not call
+the statue 'bronze' or the pyramid 'wax' or the bed 'wood', but we
+use a derived expression and call them 'of bronze', 'waxen', and 'wooden'
+respectively. But when a thing has been affected and altered in any
+way we still call it by the original name: thus we speak of the bronze
+or the wax being dry or fluid or hard or hot. 
+
+And not only so: we also speak of the particular fluid or hot substance
+as being bronze, giving the material the same name as that which we
+use to describe the affection. 
+
+Since, therefore, having regard to the figure or shape of a thing
+we no longer call that which has become of a certain figure by the
+name of the material that exhibits the figure, whereas having regard
+to a thing's affections or alterations we still call it by the name
+of its material, it is evident that becomings of the former kind cannot
+be alterations. 
+
+Moreover it would seem absurd even to speak in this way, to speak,
+that is to say, of a man or house or anything else that has come into
+existence as having been altered. Though it may be true that every
+such becoming is necessarily the result of something's being altered,
+the result, e.g. of the material's being condensed or rarefied or
+heated or cooled, nevertheless it is not the things that are coming
+into existence that are altered, and their becoming is not an alteration.
+
+Again, acquired states, whether of the body or of the soul, are not
+alterations. For some are excellences and others are defects, and
+neither excellence nor defect is an alteration: excellence is a perfection
+(for when anything acquires its proper excellence we call it perfect,
+since it is then if ever that we have a thing in its natural state:
+e.g. we have a perfect circle when we have one as good as possible),
+while defect is a perishing of or departure from this condition. So
+as when speaking of a house we do not call its arrival at perfection
+an alteration (for it would be absurd to suppose that the coping or
+the tiling is an alteration or that in receiving its coping or its
+tiling a house is altered and not perfected), the same also holds
+good in the case of excellences and defects and of the persons or
+things that possess or acquire them: for excellences are perfections
+of a thing's nature and defects are departures from it: consequently
+they are not alterations. 
+
+Further, we say that all excellences depend upon particular relations.
+Thus bodily excellences such as health and a good state of body we
+regard as consisting in a blending of hot and cold elements within
+the body in due proportion, in relation either to one another or to
+the surrounding atmosphere: and in like manner we regard beauty, strength,
+and all the other bodily excellences and defects. Each of them exists
+in virtue of a particular relation and puts that which possesses it
+in a good or bad condition with regard to its proper affections, where
+by 'proper' affections I mean those influences that from the natural
+constitution of a thing tend to promote or destroy its existence.
+Since then, relatives are neither themselves alterations nor the subjects
+of alteration or of becoming or in fact of any change whatever, it
+is evident that neither states nor the processes of losing and acquiring
+states are alterations, though it may be true that their becoming
+or perishing is necessarily, like the becoming or perishing of a specific
+character or form, the result of the alteration of certain other things,
+e.g. hot and cold or dry and wet elements or the elements, whatever
+they may be, on which the states primarily depend. For each several
+bodily defect or excellence involves a relation with those things
+from which the possessor of the defect or excellence is naturally
+subject to alteration: thus excellence disposes its possessor to be
+unaffected by these influences or to be affected by those of them
+that ought to be admitted, while defect disposes its possessor to
+be affected by them or to be unaffected by those of them that ought
+to be admitted. 
+
+And the case is similar in regard to the states of the soul, all of
+which (like those of body) exist in virtue of particular relations,
+the excellences being perfections of nature and the defects departures
+from it: moreover, excellence puts its possessor in good condition,
+while defect puts its possessor in a bad condition, to meet his proper
+affections. Consequently these cannot any more than the bodily states
+be alterations, nor can the processes of losing and acquiring them
+be so, though their becoming is necessarily the result of an alteration
+of the sensitive part of the soul, and this is altered by sensible
+objects: for all moral excellence is concerned with bodily pleasures
+and pains, which again depend either upon acting or upon remembering
+or upon anticipating. Now those that depend upon action are determined
+by sense-perception, i.e. they are stimulated by something sensible:
+and those that depend upon memory or anticipation are likewise to
+be traced to sense-perception, for in these cases pleasure is felt
+either in remembering what one has experienced or in anticipating
+what one is going to experience. Thus all pleasure of this kind must
+be produced by sensible things: and since the presence in any one
+of moral defect or excellence involves the presence in him of pleasure
+or pain (with which moral excellence and defect are always concerned),
+and these pleasures and pains are alterations of the sensitive part,
+it is evident that the loss and acquisition of these states no less
+than the loss and acquisition of the states of the body must be the
+result of the alteration of something else. Consequently, though their
+becoming is accompanied by an alteration, they are not themselves
+alterations. 
+
+Again, the states of the intellectual part of the soul are not alterations,
+nor is there any becoming of them. In the first place it is much more
+true of the possession of knowledge that it depends upon a particular
+relation. And further, it is evident that there is no becoming of
+these states. For that which is potentially possessed of knowledge
+becomes actually possessed of it not by being set in motion at all
+itself but by reason of the presence of something else: i.e. it is
+when it meets with the particular object that it knows in a manner
+the particular through its knowledge of the universal. (Again, there
+is no becoming of the actual use and activity of these states, unless
+it is thought that there is a becoming of vision and touching and
+that the activity in question is similar to these.) And the original
+acquisition of knowledge is not a becoming or an alteration: for the
+terms 'knowing' and 'understanding' imply that the intellect has reached
+a state of rest and come to a standstill, and there is no becoming
+that leads to a state of rest, since, as we have said above, change
+at all can have a becoming. Moreover, just as to say, when any one
+has passed from a state of intoxication or sleep or disease to the
+contrary state, that he has become possessed of knowledge again is
+incorrect in spite of the fact that he was previously incapable of
+using his knowledge, so, too, when any one originally acquires the
+state, it is incorrect to say that he becomes possessed of knowledge:
+for the possession of understanding and knowledge is produced by the
+soul's settling down out of the restlessness natural to it. Hence,
+too, in learning and in forming judgements on matters relating to
+their sense-perceptions children are inferior to adults owing to the
+great amount of restlessness and motion in their souls. Nature itself
+causes the soul to settle down and come to a state of rest for the
+performance of some of its functions, while for the performance of
+others other things do so: but in either case the result is brought
+about through the alteration of something in the body, as we see in
+the case of the use and activity of the intellect arising from a man's
+becoming sober or being awakened. It is evident, then, from the preceding
+argument that alteration and being altered occur in sensible things
+and in the sensitive part of the soul, and, except accidentally, in
+nothing else. 
+
+Part 4
+
+A difficulty may be raised as to whether every motion is commensurable
+with every other or not. Now if they are all commensurable and if
+two things to have the same velocity must accomplish an equal motion
+in an equal time, then we may have a circumference equal to a straight
+line, or, of course, the one may be greater or less than the other.
+Further, if one thing alters and another accomplishes a locomotion
+in an equal time, we may have an alteration and a locomotion equal
+to one another: thus an affection will be equal to a length, which
+is impossible. But is it not only when an equal motion is accomplished
+by two things in an equal time that the velocities of the two are
+equal? Now an affection cannot be equal to a length. Therefore there
+cannot be an alteration equal to or less than a locomotion: and consequently
+it is not the case that every motion is commensurable with every other.
+
+But how will our conclusion work out in the case of the circle and
+the straight line? It would be absurd to suppose that the motion of
+one in a circle and of another in a straight line cannot be similar,
+but that the one must inevitably move more quickly or more slowly
+than the other, just as if the course of one were downhill and of
+the other uphill. Moreover it does not as a matter of fact make any
+difference to the argument to say that the one motion must inevitably
+be quicker or slower than the other: for then the circumference can
+be greater or less than the straight line; and if so it is possible
+for the two to be equal. For if in the time A the quicker (B) passes
+over the distance B' and the slower (G) passes over the distance G',
+B' will be greater than G': for this is what we took 'quicker' to
+mean: and so quicker motion also implies that one thing traverses
+an equal distance in less time than another: consequently there will
+be a part of A in which B will pass over a part of the circle equal
+to G', while G will occupy the whole of A in passing over G'. None
+the less, if the two motions are commensurable, we are confronted
+with the consequence stated above, viz. that there may be a straight
+line equal to a circle. But these are not commensurable: and so the
+corresponding motions are not commensurable either. 
+
+But may we say that things are always commensurable if the same terms
+are applied to them without equivocation? e.g. a pen, a wine, and
+the highest note in a scale are not commensurable: we cannot say whether
+any one of them is sharper than any other: and why is this? they are
+incommensurable because it is only equivocally that the same term
+'sharp' is applied to them: whereas the highest note in a scale is
+commensurable with the leading-note, because the term 'sharp' has
+the same meaning as applied to both. Can it be, then, that the term
+'quick' has not the same meaning as applied to straight motion and
+to circular motion respectively? If so, far less will it have the
+same meaning as applied to alteration and to locomotion.
+
+Or shall we in the first place deny that things are always commensurable
+if the same terms are applied to them without equivocation? For the
+term 'much' has the same meaning whether applied to water or to air,
+yet water and air are not commensurable in respect of it: or, if this
+illustration is not considered satisfactory, 'double' at any rate
+would seem to have the same meaning as applied to each (denoting in
+each case the proportion of two to one), yet water and air are not
+commensurable in respect of it. But here again may we not take up
+the same position and say that the term 'much' is equivocal? In fact
+there are some terms of which even the definitions are equivocal;
+e.g. if 'much' were defined as 'so much and more','so much' would
+mean something different in different cases: 'equal' is similarly
+equivocal; and 'one' again is perhaps inevitably an equivocal term;
+and if 'one' is equivocal, so is 'two'. Otherwise why is it that some
+things are commensurable while others are not, if the nature of the
+attribute in the two cases is really one and the same? 
+
+Can it be that the incommensurability of two things in respect of
+any attribute is due to a difference in that which is primarily capable
+of carrying the attribute? Thus horse and dog are so commensurable
+that we may say which is the whiter, since that which primarily contains
+the whiteness is the same in both, viz. the surface: and similarly
+they are commensurable in respect of size. But water and speech are
+not commensurable in respect of clearness, since that which primarily
+contains the attribute is different in the two cases. It would seem,
+however that we must reject this solution, since clearly we could
+thus make all equivocal attributes univocal and say merely that that
+contains each of them is different in different cases: thus 'equality',
+'sweetness', and 'whiteness' will severally always be the same, though
+that which contains them is different in different cases. Moreover,
+it is not any casual thing that is capable of carrying any attribute:
+each single attribute can be carried primarily only by one single
+thing. 
+
+Must we then say that, if two things are to be commensurable in respect
+of any attribute, not only must the attribute in question be applicable
+to both without equivocation, but there must also be no specific differences
+either in the attribute itself or in that which contains the attribute-that
+these, I mean, must not be divisible in the way in which colour is
+divided into kinds? Thus in this respect one thing will not be commensurable
+with another, i.e. we cannot say that one is more coloured than the
+other where only colour in general and not any particular colour is
+meant; but they are commensurable in respect of whiteness.
+
+Similarly in the case of motion: two things are of the same velocity
+if they occupy an equal time in accomplishing a certain equal amount
+of motion. Suppose, then, that in a certain time an alteration is
+undergone by one half of a body's length and a locomotion is accomplished
+the other half: can be say that in this case the alteration is equal
+to the locomotion and of the same velocity? That would be absurd,
+and the reason is that there are different species of motion. And
+if in consequence of this we must say that two things are of equal
+velocity if they accomplish locomotion over an equal distance in an
+equal time, we have to admit the equality of a straight line and a
+circumference. What, then, is the reason of this? Is it that locomotion
+is a genus or that line is a genus? (We may leave the time out of
+account, since that is one and the same.) If the lines are specifically
+different, the locomotions also differ specifically from one another:
+for locomotion is specifically differentiated according to the specific
+differentiation of that over which it takes place. (It is also similarly
+differentiated, it would seem, accordingly as the instrument of the
+locomotion is different: thus if feet are the instrument, it is walking,
+if wings it is flying; but perhaps we should rather say that this
+is not so, and that in this case the differences in the locomotion
+are merely differences of posture in that which is in motion.) We
+may say, therefore, that things are of equal velocity in an equal
+time they traverse the same magnitude: and when I call it 'the same'
+I mean that it contains no specific difference and therefore no difference
+in the motion that takes place over it. So we have now to consider
+how motion is differentiated: and this discussion serves to show that
+the genus is not a unity but contains a plurality latent in it and
+distinct from it, and that in the case of equivocal terms sometimes
+the different senses in which they are used are far removed from one
+another, while sometimes there is a certain likeness between them,
+and sometimes again they are nearly related either generically or
+analogically, with the result that they seem not to be equivocal though
+they really are. 
+
+When, then, is there a difference of species? Is an attribute specifically
+different if the subject is different while the attribute is the same,
+or must the attribute itself be different as well? And how are we
+to define the limits of a species? What will enable us to decide that
+particular instances of whiteness or sweetness are the same or different?
+Is it enough that it appears different in one subject from what appears
+in another? Or must there be no sameness at all? And further, where
+alteration is in question, how is one alteration to be of equal velocity
+with another? One person may be cured quickly and another slowly,
+and cures may also be simultaneous: so that, recovery of health being
+an alteration, we have here alterations of equal velocity, since each
+alteration occupies an equal time. But what alteration? We cannot
+here speak of an 'equal' alteration: what corresponds in the category
+of quality to equality in the category of quantity is 'likeness'.
+However, let us say that there is equal velocity where the same change
+is accomplished in an equal time. Are we, then, to find the commensurability
+in the subject of the affection or in the affection itself? In the
+case that we have just been considering it is the fact that health
+is one and the same that enables us to arrive at the conclusion that
+the one alteration is neither more nor less than the other, but that
+both are alike. If on the other hand the affection is different in
+the two cases, e.g. when the alterations take the form of becoming
+white and becoming healthy respectively, here there is no sameness
+or equality or likeness inasmuch as the difference in the affections
+at once makes the alterations specifically different, and there is
+no unity of alteration any more than there would be unity of locomotion
+under like conditions. So we must find out how many species there
+are of alteration and of locomotion respectively. Now if the things
+that are in motion-that is to say, the things to which the motions
+belong essentially and not accidentally-differ specifically, then
+their respective motions will also differ specifically: if on the
+other hand they differ generically or numerically, the motions also
+will differ generically or numerically as the case may be. But there
+still remains the question whether, supposing that two alterations
+are of equal velocity, we ought to look for this equality in the sameness
+(or likeness) of the affections, or in the things altered, to see
+e.g. whether a certain quantity of each has become white. Or ought
+we not rather to look for it in both? That is to say, the alterations
+are the same or different according as the affections are the same
+or different, while they are equal or unequal according as the things
+altered are equal or unequal. 
+
+And now we must consider the same question in the case of becoming
+and perishing: how is one becoming of equal velocity with another?
+They are of equal velocity if in an equal time there are produced
+two things that are the same and specifically inseparable, e.g. two
+men (not merely generically inseparable as e.g. two animals). Similarly
+one is quicker than the other if in an equal time the product is different
+in the two cases. I state it thus because we have no pair of terms
+that will convey this 'difference' in the way in which unlikeness
+is conveyed. If we adopt the theory that it is number that constitutes
+being, we may indeed speak of a 'greater number' and a 'lesser number'
+within the same species, but there is no common term that will include
+both relations, nor are there terms to express each of them separately
+in the same way as we indicate a higher degree or preponderance of
+an affection by 'more', of a quantity by 'greater.' 
+
+Part 5
+
+Now since wherever there is a movent, its motion always acts upon
+something, is always in something, and always extends to something
+(by 'is always in something' I mean that it occupies a time: and by
+'extends to something' I mean that it involves the traversing of a
+certain amount of distance: for at any moment when a thing is causing
+motion, it also has caused motion, so that there must always be a
+certain amount of distance that has been traversed and a certain amount
+of time that has been occupied). then, A the movement have moved B
+a distance G in a time D, then in the same time the same force A will
+move 1/2B twice the distance G, and in 1/2D it will move 1/2B the
+whole distance for G: thus the rules of proportion will be observed.
+Again if a given force move a given weight a certain distance in a
+certain time and half the distance in half the time, half the motive
+power will move half the weight the same distance in the same time.
+Let E represent half the motive power A and Z half the weight B: then
+the ratio between the motive power and the weight in the one case
+is similar and proportionate to the ratio in the other, so that each
+force will cause the same distance to be traversed in the same time.
+But if E move Z a distance G in a time D, it does not necessarily
+follow that E can move twice Z half the distance G in the same time.
+If, then, A move B a distance G in a time D, it does not follow that
+E, being half of A, will in the time D or in any fraction of it cause
+B to traverse a part of G the ratio between which and the whole of
+G is proportionate to that between A and E (whatever fraction of AE
+may be): in fact it might well be that it will cause no motion at
+all; for it does not follow that, if a given motive power causes a
+certain amount of motion, half that power will cause motion either
+of any particular amount or in any length of time: otherwise one man
+might move a ship, since both the motive power of the ship-haulers
+and the distance that they all cause the ship to traverse are divisible
+into as many parts as there are men. Hence Zeno's reasoning is false
+when he argues that there is no part of the millet that does not make
+a sound: for there is no reason why any such part should not in any
+length of time fail to move the air that the whole bushel moves in
+falling. In fact it does not of itself move even such a quantity of
+the air as it would move if this part were by itself: for no part
+even exists otherwise than potentially. 
+
+If on the other hand we have two forces each of which separately moves
+one of two weights a given distance in a given time, then the forces
+in combination will move the combined weights an equal distance in
+an equal time: for in this case the rules of proportion apply.
+
+Then does this hold good of alteration and of increase also? Surely
+it does, for in any given case we have a definite thing that cause
+increase and a definite thing that suffers increase, and the one causes
+and the other suffers a certain amount of increase in a certain amount
+of time. Similarly we have a definite thing that causes alteration
+and a definite thing that undergoes alteration, and a certain amount,
+or rather degree, of alteration is completed in a certain amount of
+time: thus in twice as much time twice as much alteration will be
+completed and conversely twice as much alteration will occupy twice
+as much time: and the alteration of half of its object will occupy
+half as much time and in half as much time half of the object will
+be altered: or again, in the same amount of time it will be altered
+twice as much. 
+
+On the other hand if that which causes alteration or increase causes
+a certain amount of increase or alteration respectively in a certain
+amount of time, it does not necessarily follow that half the force
+will occupy twice the time in altering or increasing the object, or
+that in twice the time the alteration or increase will be completed
+by it: it may happen that there will be no alteration or increase
+at all, the case being the same as with the weight. 
+
+----------------------------------------------------------------------
+
+BOOK VIII
+
+Part 1 
+
+It remains to consider the following question. Was there ever a becoming
+of motion before which it had no being, and is it perishing again
+so as to leave nothing in motion? Or are we to say that it never had
+any becoming and is not perishing, but always was and always will
+be? Is it in fact an immortal never-failing property of things that
+are, a sort of life as it were to all naturally constituted things?
+
+Now the existence of motion is asserted by all who have anything to
+say about nature, because they all concern themselves with the construction
+of the world and study the question of becoming and perishing, which
+processes could not come about without the existence of motion. But
+those who say that there is an infinite number of worlds, some of
+which are in process of becoming while others are in process of perishing,
+assert that there is always motion (for these processes of becoming
+and perishing of the worlds necessarily involve motion), whereas those
+who hold that there is only one world, whether everlasting or not,
+make corresponding assumptions in regard to motion. If then it is
+possible that at any time nothing should be in motion, this must come
+about in one of two ways: either in the manner described by Anaxagoras,
+who says that all things were together and at rest for an infinite
+period of time, and that then Mind introduced motion and separated
+them; or in the manner described by Empedocles, according to whom
+the universe is alternately in motion and at rest-in motion, when
+Love is making the one out of many, or Strife is making many out of
+one, and at rest in the intermediate periods of time-his account being
+as follows: 
+
+'Since One hath learned to spring from Manifold, And One disjoined
+makes manifold arise, Thus they Become, nor stable is their life:
+But since their motion must alternate be, Thus have they ever Rest
+upon their round': for we must suppose that he means by this that
+they alternate from the one motion to the other. We must consider,
+then, how this matter stands, for the discovery of the truth about
+it is of importance, not only for the study of nature, but also for
+the investigation of the First Principle. 
+
+Let us take our start from what we have already laid down in our course
+on Physics. Motion, we say, is the fulfilment of the movable in so
+far as it is movable. Each kind of motion, therefore, necessarily
+involves the presence of the things that are capable of that motion.
+In fact, even apart from the definition of motion, every one would
+admit that in each kind of motion it is that which is capable of that
+motion that is in motion: thus it is that which is capable of alteration
+that is altered, and that which is capable of local change that is
+in locomotion: and so there must be something capable of being burned
+before there can be a process of being burned, and something capable
+of burning before there can be a process of burning. Moreover, these
+things also must either have a beginning before which they had no
+being, or they must be eternal. Now if there was a becoming of every
+movable thing, it follows that before the motion in question another
+change or motion must have taken place in which that which was capable
+of being moved or of causing motion had its becoming. To suppose,
+on the other hand, that these things were in being throughout all
+previous time without there being any motion appears unreasonable
+on a moment's thought, and still more unreasonable, we shall find,
+on further consideration. For if we are to say that, while there are
+on the one hand things that are movable, and on the other hand things
+that are motive, there is a time when there is a first movent and
+a first moved, and another time when there is no such thing but only
+something that is at rest, then this thing that is at rest must previously
+have been in process of change: for there must have been some cause
+of its rest, rest being the privation of motion. Therefore, before
+this first change there will be a previous change. For some things
+cause motion in only one way, while others can produce either of two
+contrary motions: thus fire causes heating but not cooling, whereas
+it would seem that knowledge may be directed to two contrary ends
+while remaining one and the same. Even in the former class, however,
+there seems to be something similar, for a cold thing in a sense causes
+heating by turning away and retiring, just as one possessed of knowledge
+voluntarily makes an error when he uses his knowledge in the reverse
+way. But at any rate all things that are capable respectively of affecting
+and being affected, or of causing motion and being moved, are capable
+of it not under all conditions, but only when they are in a particular
+condition and approach one another: so it is on the approach of one
+thing to another that the one causes motion and the other is moved,
+and when they are present under such conditions as rendered the one
+motive and the other movable. So if the motion was not always in process,
+it is clear that they must have been in a condition not such as to
+render them capable respectively of being moved and of causing motion,
+and one or other of them must have been in process of change: for
+in what is relative this is a necessary consequence: e.g. if one thing
+is double another when before it was not so, one or other of them,
+if not both, must have been in process of change. It follows then,
+that there will be a process of change previous to the first.
+
+(Further, how can there be any 'before' and 'after' without the existence
+of time? Or how can there be any time without the existence of motion?
+If, then, time is the number of motion or itself a kind of motion,
+it follows that, if there is always time, motion must also be eternal.
+But so far as time is concerned we see that all with one exception
+are in agreement in saying that it is uncreated: in fact, it is just
+this that enables Democritus to show that all things cannot have had
+a becoming: for time, he says, is uncreated. Plato alone asserts the
+creation of time, saying that it had a becoming together with the
+universe, the universe according to him having had a becoming. Now
+since time cannot exist and is unthinkable apart from the moment,
+and the moment a kind of middle-point, uniting as it does in itself
+both a beginning and an end, a beginning of future time and an end
+of past time, it follows that there must always be time: for the extremity
+of the last period of time that we take must be found in some moment,
+since time contains no point of contact for us except the moment.
+Therefore, since the moment is both a beginning and an end, there
+must always be time on both sides of it. But if this is true of time,
+it is evident that it must also be true of motion, time being a kind
+of affection of motion.) 
+
+The same reasoning will also serve to show the imperishability of
+motion: just as a becoming of motion would involve, as we saw, the
+existence of a process of change previous to the first, in the same
+way a perishing of motion would involve the existence of a process
+of change subsequent to the last: for when a thing ceases to be moved,
+it does not therefore at the same time cease to be movable-e.g. the
+cessation of the process of being burned does not involve the cessation
+of the capacity of being burned, since a thing may be capable of being
+burned without being in process of being burned-nor, when a thing
+ceases to be movent, does it therefore at the same time cease to a
+be motive. Again, the destructive agent will have to be destroyed,
+after what it destroys has been destroyed, and then that which has
+the capacity of destroying it will have to be destroyed afterwards,
+(so that there will be a process of change subsequent to the last,)
+for being destroyed also is a kind of change. If, then, view which
+we are criticizing involves these impossible consequences, it is clear
+that motion is eternal and cannot have existed at one time and not
+at another: in fact such a view can hardly be described as anythling
+else than fantastic. 
+
+And much the same may be said of the view that such is the ordinance
+of nature and that this must be regarded as a principle, as would
+seem to be the view of Empedocles when he says that the constitution
+of the world is of necessity such that Love and Strife alternately
+predominate and cause motion, while in the intermediate period of
+time there is a state of rest. Probably also those who like like Anaxagoras,
+assert a single principle (of motion) would hold this view. But that
+which is produced or directed by nature can never be anything disorderly:
+for nature is everywhere the cause of order. Moreover, there is no
+ratio in the relation of the infinite to the infinite, whereas order
+always means ratio. But if we say that there is first a state of rest
+for an infinite time, and then motion is started at some moment, and
+that the fact that it is this rather than a previous moment is of
+no importance, and involves no order, then we can no longer say that
+it is nature's work: for if anything is of a certain character naturally,
+it either is so invariably and is not sometimes of this and sometimes
+of another character (e.g. fire, which travels upwards naturally,
+does not sometimes do so and sometimes not) or there is a ratio in
+the variation. It would be better, therefore, to say with Empedocles
+and any one else who may have maintained such a theory as his that
+the universe is alternately at rest and in motion: for in a system
+of this kind we have at once a certain order. But even here the holder
+of the theory ought not only to assert the fact: he ought to explain
+the cause of it: i.e. he should not make any mere assumption or lay
+down any gratuitous axiom, but should employ either inductive or demonstrative
+reasoning. The Love and Strife postulated by Empedocles are not in
+themselves causes of the fact in question, nor is it of the essence
+of either that it should be so, the essential function of the former
+being to unite, of the latter to separate. If he is to go on to explain
+this alternate predominance, he should adduce cases where such a state
+of things exists, as he points to the fact that among mankind we have
+something that unites men, namely Love, while on the other hand enemies
+avoid one another: thus from the observed fact that this occurs in
+certain cases comes the assumption that it occurs also in the universe.
+Then, again, some argument is needed to explain why the predominance
+of each of the two forces lasts for an equal period of time. But it
+is a wrong assumption to suppose universally that we have an adequate
+first principle in virtue of the fact that something always is so
+or always happens so. Thus Democritus reduces the causes that explain
+nature to the fact that things happened in the past in the same way
+as they happen now: but he does not think fit to seek for a first
+principle to explain this 'always': so, while his theory is right
+in so far as it is applied to certain individual cases, he is wrong
+in making it of universal application. Thus, a triangle always has
+its angles equal to two right angles, but there is nevertheless an
+ulterior cause of the eternity of this truth, whereas first principles
+are eternal and have no ulterior cause. Let this conclude what we
+have to say in support of our contention that there never was a time
+when there was not motion, and never will be a time when there will
+not be motion. 
+
+Part 2
+
+The arguments that may be advanced against this position are not difficult
+to dispose of. The chief considerations that might be thought to indicate
+that motion may exist though at one time it had not existed at all
+are the following: 
+
+First, it may be said that no process of change is eternal: for the
+nature of all change is such that it proceeds from something to something,
+so that every process of change must be bounded by the contraries
+that mark its course, and no motion can go on to infinity.
+
+Secondly, we see that a thing that neither is in motion nor contains
+any motion within itself can be set in motion; e.g. inanimate things
+that are (whether the whole or some part is in question) not in motion
+but at rest, are at some moment set in motion: whereas, if motion
+cannot have a becoming before which it had no being, these things
+ought to be either always or never in motion. 
+
+Thirdly, the fact is evident above all in the case of animate beings:
+for it sometimes happens that there is no motion in us and we are
+quite still, and that nevertheless we are then at some moment set
+in motion, that is to say it sometimes happens that we produce a beginning
+of motion in ourselves spontaneously without anything having set us
+in motion from without. We see nothing like this in the case of inanimate
+things, which are always set in motion by something else from without:
+the animal, on the other hand, we say, moves itself: therefore, if
+an animal is ever in a state of absolute rest, we have a motionless
+thing in which motion can be produced from the thing itself, and not
+from without. Now if this can occur in an animal, why should not the
+same be true also of the universe as a whole? If it can occur in a
+small world it could also occur in a great one: and if it can occur
+in the world, it could also occur in the infinite; that is, if the
+infinite could as a whole possibly be in motion or at rest.
+
+Of these objections, then, the first-mentioned motion to opposites
+is not always the same and numerically one a correct statement; in
+fact, this may be said to be a necessary conclusion, provided that
+it is possible for the motion of that which is one and the same to
+be not always one and the same. (I mean that e.g. we may question
+whether the note given by a single string is one and the same, or
+is different each time the string is struck, although the string is
+in the same condition and is moved in the same way.) But still, however
+this may be, there is nothing to prevent there being a motion that
+is the same in virtue of being continuous and eternal: we shall have
+something to say later that will make this point clearer.
+
+As regards the second objection, no absurdity is involved in the fact
+that something not in motion may be set in motion, that which caused
+the motion from without being at one time present, and at another
+absent. Nevertheless, how this can be so remains matter for inquiry;
+how it comes about, I mean, that the same motive force at one time
+causes a thing to be in motion, and at another does not do so: for
+the difficulty raised by our objector really amounts to this-why is
+it that some things are not always at rest, and the rest always in
+motion? 
+
+The third objection may be thought to present more difficulty than
+the others, namely, that which alleges that motion arises in things
+in which it did not exist before, and adduces in proof the case of
+animate things: thus an animal is first at rest and afterwards walks,
+not having been set in motion apparently by anything from without.
+This, however, is false: for we observe that there is always some
+part of the animal's organism in motion, and the cause of the motion
+of this part is not the animal itself, but, it may be, its environment.
+Moreover, we say that the animal itself originates not all of its
+motions but its locomotion. So it may well be the case-or rather we
+may perhaps say that it must necessarily be the case-that many motions
+are produced in the body by its environment, and some of these set
+in motion the intellect or the appetite, and this again then sets
+the whole animal in motion: this is what happens when animals are
+asleep: though there is then no perceptive motion in them, there is
+some motion that causes them to wake up again. But we will leave this
+point also to be elucidated at a later stage in our discussion.
+
+Part 3
+
+Our enquiry will resolve itself at the outset into a consideration
+of the above-mentioned problem-what can be the reason why some things
+in the world at one time are in motion and at another are at rest
+again? Now one of three things must be true: either all things are
+always at rest, or all things are always in motion, or some things
+are in motion and others at rest: and in this last case again either
+the things that are in motion are always in motion and the things
+that are at rest are always at rest, or they are all constituted so
+as to be capable alike of motion and of rest; or there is yet a third
+possibility remaining-it may be that some things in the world are
+always motionless, others always in motion, while others again admit
+of both conditions. This last is the account of the matter that we
+must give: for herein lies the solution of all the difficulties raised
+and the conclusion of the investigation upon which we are engaged.
+
+To maintain that all things are at rest, and to disregard sense-perception
+in an attempt to show the theory to be reasonable, would be an instance
+of intellectual weakness: it would call in question a whole system,
+not a particular detail: moreover, it would be an attack not only
+on the physicist but on almost all sciences and all received opinions,
+since motion plays a part in all of them. Further, just as in arguments
+about mathematics objections that involve first principles do not
+affect the mathematician-and the other sciences are in similar case-so,
+too, objections involving the point that we have just raised do not
+affect the physicist: for it is a fundamental assumption with him
+that motion is ultimately referable to nature herself. 
+
+The assertion that all things are in motion we may fairly regard as
+equally false, though it is less subversive of physical science: for
+though in our course on physics it was laid down that rest no less
+than motion is ultimately referable to nature herself, nevertheless
+motion is the characteristic fact of nature: moreover, the view is
+actually held by some that not merely some things but all things in
+the world are in motion and always in motion, though we cannot apprehend
+the fact by sense-perception. Although the supporters of this theory
+do not state clearly what kind of motion they mean, or whether they
+mean all kinds, it is no hard matter to reply to them: thus we may
+point out that there cannot be a continuous process either of increase
+or of decrease: that which comes between the two has to be included.
+The theory resembles that about the stone being worn away by the drop
+of water or split by plants growing out of it: if so much has been
+extruded or removed by the drop, it does not follow that half the
+amount has previously been extruded or removed in half the time: the
+case of the hauled ship is exactly comparable: here we have so many
+drops setting so much in motion, but a part of them will not set as
+much in motion in any period of time. The amount removed is, it is
+true, divisible into a number of parts, but no one of these was set
+in motion separately: they were all set in motion together. It is
+evident, then, that from the fact that the decrease is divisible into
+an infinite number of parts it does not follow that some part must
+always be passing away: it all passes away at a particular moment.
+Similarly, too, in the case of any alteration whatever if that which
+suffers alteration is infinitely divisible it does not follow from
+this that the same is true of the alteration itself, which often occurs
+all at once, as in freezing. Again, when any one has fallen ill, there
+must follow a period of time in which his restoration to health is
+in the future: the process of change cannot take place in an instant:
+yet the change cannot be a change to anything else but health. The
+assertion. therefore, that alteration is continuous is an extravagant
+calling into question of the obvious: for alteration is a change from
+one contrary to another. Moreover, we notice that a stone becomes
+neither harder nor softer. Again, in the matter of locomotion, it
+would be a strange thing if a stone could be falling or resting on
+the ground without our being able to perceive the fact. Further, it
+is a law of nature that earth and all other bodies should remain in
+their proper places and be moved from them only by violence: from
+the fact then that some of them are in their proper places it follows
+that in respect of place also all things cannot be in motion. These
+and other similar arguments, then, should convince us that it is impossible
+either that all things are always in motion or that all things are
+always at rest. 
+
+Nor again can it be that some things are always at rest, others always
+in motion, and nothing sometimes at rest and sometimes in motion.
+This theory must be pronounced impossible on the same grounds as those
+previously mentioned: viz. that we see the above-mentioned changes
+occurring in the case of the same things. We may further point out
+that the defender of this position is fighting against the obvious,
+for on this theory there can be no such thing as increase: nor can
+there be any such thing as compulsory motion, if it is impossible
+that a thing can be at rest before being set in motion unnaturally.
+This theory, then, does away with becoming and perishing. Moreover,
+motion, it would seem, is generally thought to be a sort of becoming
+and perishing, for that to which a thing changes comes to be, or occupancy
+of it comes to be, and that from which a thing changes ceases to be,
+or there ceases to be occupancy of it. It is clear, therefore, that
+there are cases of occasional motion and occasional rest.
+
+We have now to take the assertion that all things are sometimes at
+rest and sometimes in motion and to confront it with the arguments
+previously advanced. We must take our start as before from the possibilities
+that we distinguished just above. Either all things are at rest, or
+all things are in motion, or some things are at rest and others in
+motion. And if some things are at rest and others in motion, then
+it must be that either all things are sometimes at rest and sometimes
+in motion, or some things are always at rest and the remainder always
+in motion, or some of the things are always at rest and others always
+in motion while others again are sometimes at rest and sometimes in
+motion. Now we have said before that it is impossible that all things
+should be at rest: nevertheless we may now repeat that assertion.
+We may point out that, even if it is really the case, as certain persons
+assert, that the existent is infinite and motionless, it certainly
+does not appear to be so if we follow sense-perception: many things
+that exist appear to be in motion. Now if there is such a thing as
+false opinion or opinion at all, there is also motion; and similarly
+if there is such a thing as imagination, or if it is the case that
+anything seems to be different at different times: for imagination
+and opinion are thought to be motions of a kind. But to investigate
+this question at all-to seek a reasoned justification of a belief
+with regard to which we are too well off to require reasoned justification-implies
+bad judgement of what is better and what is worse, what commends itself
+to belief and what does not, what is ultimate and what is not. It
+is likewise impossible that all things should be in motion or that
+some things should be always in motion and the remainder always at
+rest. We have sufficient ground for rejecting all these theories in
+the single fact that we see some things that are sometimes in motion
+and sometimes at rest. It is evident, therefore, that it is no less
+impossible that some things should be always in motion and the remainder
+always at rest than that all things should be at rest or that all
+things should be in motion continuously. It remains, then, to consider
+whether all things are so constituted as to be capable both of being
+in motion and of being at rest, or whether, while some things are
+so constituted, some are always at rest and some are always in motion:
+for it is this last view that we have to show to be true.
+
+Part 4
+
+Now of things that cause motion or suffer motion, to some the motion
+is accidental, to others essential: thus it is accidental to what
+merely belongs to or contains as a part a thing that causes motion
+or suffers motion, essential to a thing that causes motion or suffers
+motion not merely by belonging to such a thing or containing it as
+a part. 
+
+Of things to which the motion is essential some derive their motion
+from themselves, others from something else: and in some cases their
+motion is natural, in others violent and unnatural. Thus in things
+that derive their motion from themselves, e.g. all animals, the motion
+is natural (for when an animal is in motion its motion is derived
+from itself): and whenever the source of the motion of a thing is
+in the thing itself we say that the motion of that thing is natural.
+Therefore the animal as a whole moves itself naturally: but the body
+of the animal may be in motion unnaturally as well as naturally: it
+depends upon the kind of motion that it may chance to be suffering
+and the kind of element of which it is composed. And the motion of
+things that derive their motion from something else is in some cases
+natural, in other unnatural: e.g. upward motion of earthy things and
+downward motion of fire are unnatural. Moreover the parts of animals
+are often in motion in an unnatural way, their positions and the character
+of the motion being abnormal. The fact that a thing that is in motion
+derives its motion from something is most evident in things that are
+in motion unnaturally, because in such cases it is clear that the
+motion is derived from something other than the thing itself. Next
+to things that are in motion unnaturally those whose motion while
+natural is derived from themselves-e.g. animals-make this fact clear:
+for here the uncertainty is not as to whether the motion is derived
+from something but as to how we ought to distinguish in the thing
+between the movent and the moved. It would seem that in animals, just
+as in ships and things not naturally organized, that which causes
+motion is separate from that which suffers motion, and that it is
+only in this sense that the animal as a whole causes its own motion.
+
+The greatest difficulty, however, is presented by the remaining case
+of those that we last distinguished. Where things derive their motion
+from something else we distinguished the cases in which the motion
+is unnatural: we are left with those that are to be contrasted with
+the others by reason of the fact that the motion is natural. It is
+in these cases that difficulty would be experienced in deciding whence
+the motion is derived, e.g. in the case of light and heavy things.
+When these things are in motion to positions the reverse of those
+they would properly occupy, their motion is violent: when they are
+in motion to their proper positions-the light thing up and the heavy
+thing down-their motion is natural; but in this latter case it is
+no longer evident, as it is when the motion is unnatural, whence their
+motion is derived. It is impossible to say that their motion is derived
+from themselves: this is a characteristic of life and peculiar to
+living things. Further, if it were, it would have been in their power
+to stop themselves (I mean that if e.g. a thing can cause itself to
+walk it can also cause itself not to walk), and so, since on this
+supposition fire itself possesses the power of upward locomotion,
+it is clear that it should also possess the power of downward locomotion.
+Moreover if things move themselves, it would be unreasonable to suppose
+that in only one kind of motion is their motion derived from themselves.
+Again, how can anything of continuous and naturally connected substance
+move itself? In so far as a thing is one and continuous not merely
+in virtue of contact, it is impassive: it is only in so far as a thing
+is divided that one part of it is by nature active and another passive.
+Therefore none of the things that we are now considering move themselves
+(for they are of naturally connected substance), nor does anything
+else that is continuous: in each case the movent must be separate
+from the moved, as we see to be the case with inanimate things when
+an animate thing moves them. It is the fact that these things also
+always derive their motion from something: what it is would become
+evident if we were to distinguish the different kinds of cause.
+
+The above-mentioned distinctions can also be made in the case of things
+that cause motion: some of them are capable of causing motion unnaturally
+(e.g. the lever is not naturally capable of moving the weight), others
+naturally (e.g. what is actually hot is naturally capable of moving
+what is potentially hot): and similarly in the case of all other things
+of this kind. 
+
+In the same way, too, what is potentially of a certain quality or
+of a certain quantity in a certain place is naturally movable when
+it contains the corresponding principle in itself and not accidentally
+(for the same thing may be both of a certain quality and of a certain
+quantity, but the one is an accidental, not an essential property
+of the other). So when fire or earth is moved by something the motion
+is violent when it is unnatural, and natural when it brings to actuality
+the proper activities that they potentially possess. But the fact
+that the term 'potentially' is used in more than one sense is the
+reason why it is not evident whence such motions as the upward motion
+of fire and the downward motion of earth are derived. One who is learning
+a science potentially knows it in a different sense from one who while
+already possessing the knowledge is not actually exercising it. Wherever
+we have something capable of acting and something capable of being
+correspondingly acted on, in the event of any such pair being in contact
+what is potential becomes at times actual: e.g. the learner becomes
+from one potential something another potential something: for one
+who possesses knowledge of a science but is not actually exercising
+it knows the science potentially in a sense, though not in the same
+sense as he knew it potentially before he learnt it. And when he is
+in this condition, if something does not prevent him, he actively
+exercises his knowledge: otherwise he would be in the contradictory
+state of not knowing. In regard to natural bodies also the case is
+similar. Thus what is cold is potentially hot: then a change takes
+place and it is fire, and it burns, unless something prevents and
+hinders it. So, too, with heavy and light: light is generated from
+heavy, e.g. air from water (for water is the first thing that is potentially
+light), and air is actually light, and will at once realize its proper
+activity as such unless something prevents it. The activity of lightness
+consists in the light thing being in a certain situation, namely high
+up: when it is in the contrary situation, it is being prevented from
+rising. The case is similar also in regard to quantity and quality.
+But, be it noted, this is the question we are trying to answer-how
+can we account for the motion of light things and heavy things to
+their proper situations? The reason for it is that they have a natural
+tendency respectively towards a certain position: and this constitutes
+the essence of lightness and heaviness, the former being determined
+by an upward, the latter by a downward, tendency. As we have said,
+a thing may be potentially light or heavy in more senses than one.
+Thus not only when a thing is water is it in a sense potentially light,
+but when it has become air it may be still potentially light: for
+it may be that through some hindrance it does not occupy an upper
+position, whereas, if what hinders it is removed, it realizes its
+activity and continues to rise higher. The process whereby what is
+of a certain quality changes to a condition of active existence is
+similar: thus the exercise of knowledge follows at once upon the possession
+of it unless something prevents it. So, too, what is of a certain
+quantity extends itself over a certain space unless something prevents
+it. The thing in a sense is and in a sense is not moved by one who
+moves what is obstructing and preventing its motion (e.g. one who
+pulls away a pillar from under a roof or one who removes a stone from
+a wineskin in the water is the accidental cause of motion): and in
+the same way the real cause of the motion of a ball rebounding from
+a wall is not the wall but the thrower. So it is clear that in all
+these cases the thing does not move itself, but it contains within
+itself the source of motion-not of moving something or of causing
+motion, but of suffering it. 
+
+If then the motion of all things that are in motion is either natural
+or unnatural and violent, and all things whose motion is violent and
+unnatural are moved by something, and something other than themselves,
+and again all things whose motion is natural are moved by something-both
+those that are moved by themselves and those that are not moved by
+themselves (e.g. light things and heavy things, which are moved either
+by that which brought the thing into existence as such and made it
+light and heavy, or by that which released what was hindering and
+preventing it); then all things that are in motion must be moved by
+something. 
+
+Part 5
+
+Now this may come about in either of two ways. Either the movent is
+not itself responsible for the motion, which is to be referred to
+something else which moves the movent, or the movent is itself responsible
+for the motion. Further, in the latter case, either the movent immediately
+precedes the last thing in the series, or there may be one or more
+intermediate links: e.g. the stick moves the stone and is moved by
+the hand, which again is moved by the man: in the man, however, we
+have reached a movent that is not so in virtue of being moved by something
+else. Now we say that the thing is moved both by the last and by the
+first movent in the series, but more strictly by the first, since
+the first movent moves the last, whereas the last does not move the
+first, and the first will move the thing without the last, but the
+last will not move it without the first: e.g. the stick will not move
+anything unless it is itself moved by the man. If then everything
+that is in motion must be moved by something, and the movent must
+either itself be moved by something else or not, and in the former
+case there must be some first movent that is not itself moved by anything
+else, while in the case of the immediate movent being of this kind
+there is no need of an intermediate movent that is also moved (for
+it is impossible that there should be an infinite series of movents,
+each of which is itself moved by something else, since in an infinite
+series there is no first term)-if then everything that is in motion
+is moved by something, and the first movent is moved but not by anything
+else, it much be moved by itself. 
+
+This same argument may also be stated in another way as follows. Every
+movent moves something and moves it with something, either with itself
+or with something else: e.g. a man moves a thing either himself or
+with a stick, and a thing is knocked down either by the wind itself
+or by a stone propelled by the wind. But it is impossible for that
+with which a thing is moved to move it without being moved by that
+which imparts motion by its own agency: on the other hand, if a thing
+imparts motion by its own agency, it is not necessary that there should
+be anything else with which it imparts motion, whereas if there is
+a different thing with which it imparts motion, there must be something
+that imparts motion not with something else but with itself, or else
+there will be an infinite series. If, then, anything is a movent while
+being itself moved, the series must stop somewhere and not be infinite.
+Thus, if the stick moves something in virtue of being moved by the
+hand, the hand moves the stick: and if something else moves with the
+hand, the hand also is moved by something different from itself. So
+when motion by means of an instrument is at each stage caused by something
+different from the instrument, this must always be preceded by something
+else which imparts motion with itself. Therefore, if this last movent
+is in motion and there is nothing else that moves it, it must move
+itself. So this reasoning also shows that when a thing is moved, if
+it is not moved immediately by something that moves itself, the series
+brings us at some time or other to a movent of this kind.
+
+And if we consider the matter in yet a third wa Ly we shall get this
+same result as follows. If everything that is in motion is moved by
+something that is in motion, ether this being in motion is an accidental
+attribute of the movents in question, so that each of them moves something
+while being itself in motion, but not always because it is itself
+in motion, or it is not accidental but an essential attribute. Let
+us consider the former alternative. If then it is an accidental attribute,
+it is not necessary that that is in motion should be in motion: and
+if this is so it is clear that there may be a time when nothing that
+exists is in motion, since the accidental is not necessary but contingent.
+Now if we assume the existence of a possibility, any conclusion that
+we thereby reach will not be an impossibility though it may be contrary
+to fact. But the nonexistence of motion is an impossibility: for we
+have shown above that there must always be motion. 
+
+Moreover, the conclusion to which we have been led is a reasonable
+one. For there must be three things-the moved, the movent, and the
+instrument of motion. Now the moved must be in motion, but it need
+not move anything else: the instrument of motion must both move something
+else and be itself in motion (for it changes together with the moved,
+with which it is in contact and continuous, as is clear in the case
+of things that move other things locally, in which case the two things
+must up to a certain point be in contact): and the movent-that is
+to say, that which causes motion in such a manner that it is not merely
+the instrument of motion-must be unmoved. Now we have visual experience
+of the last term in this series, namely that which has the capacity
+of being in motion, but does not contain a motive principle, and also
+of that which is in motion but is moved by itself and not by anything
+else: it is reasonable, therefore, not to say necessary, to suppose
+the existence of the third term also, that which causes motion but
+is itself unmoved. So, too, Anaxagoras is right when he says that
+Mind is impassive and unmixed, since he makes it the principle of
+motion: for it could cause motion in this sense only by being itself
+unmoved, and have supreme control only by being unmixed.
+
+We will now take the second alternative. If the movement is not accidentally
+but necessarily in motion-so that, if it were not in motion, it would
+not move anything-then the movent, in so far as it is in motion, must
+be in motion in one of two ways: it is moved either as that is which
+is moved with the same kind of motion, or with a different kind-either
+that which is heating, I mean, is itself in process of becoming hot,
+that which is making healthy in process of becoming healthy, and that
+which is causing locomotion in process of locomotion, or else that
+which is making healthy is, let us say, in process of locomotion,
+and that which is causing locomotion in process of, say, increase.
+But it is evident that this is impossible. For if we adopt the first
+assumption we have to make it apply within each of the very lowest
+species into which motion can be divided: e.g. we must say that if
+some one is teaching some lesson in geometry, he is also in process
+of being taught that same lesson in geometry, and that if he is throwing
+he is in process of being thrown in just the same manner. Or if we
+reject this assumption we must say that one kind of motion is derived
+from another; e.g. that that which is causing locomotion is in process
+of increase, that which is causing this increase is in process of
+being altered by something else, and that which is causing this alteration
+is in process of suffering some different kind of motion. But the
+series must stop somewhere, since the kinds of motion are limited;
+and if we say that the process is reversible, and that that which
+is causing alteration is in process of locomotion, we do no more than
+if we had said at the outset that that which is causing locomotion
+is in process of locomotion, and that one who is teaching is in process
+of being taught: for it is clear that everything that is moved is
+moved by the movent that is further back in the series as well as
+by that which immediately moves it: in fact the earlier movent is
+that which more strictly moves it. But this is of course impossible:
+for it involves the consequence that one who is teaching is in process
+of learning what he is teaching, whereas teaching necessarily implies
+possessing knowledge, and learning not possessing it. Still more unreasonable
+is the consequence involved that, since everything that is moved is
+moved by something that is itself moved by something else, everything
+that has a capacity for causing motion has as such a corresponding
+capacity for being moved: i.e. it will have a capacity for being moved
+in the sense in which one might say that everything that has a capacity
+for making healthy, and exercises that capacity, has as such a capacity
+for being made healthy, and that which has a capacity for building
+has as such a capacity for being built. It will have the capacity
+for being thus moved either immediately or through one or more links
+(as it will if, while everything that has a capacity for causing motion
+has as such a capacity for being moved by something else, the motion
+that it has the capacity for suffering is not that with which it affects
+what is next to it, but a motion of a different kind; e.g. that which
+has a capacity for making healthy might as such have a capacity for
+learn. the series, however, could be traced back, as we said before,
+until at some time or other we arrived at the same kind of motion).
+Now the first alternative is impossible, and the second is fantastic:
+it is absurd that that which has a capacity for causing alteration
+should as such necessarily have a capacity, let us say, for increase.
+It is not necessary, therefore, that that which is moved should always
+be moved by something else that is itself moved by something else:
+so there will be an end to the series. Consequently the first thing
+that is in motion will derive its motion either from something that
+is at rest or from itself. But if there were any need to consider
+which of the two, that which moves itself or that which is moved by
+something else, is the cause and principle of motion, every one would
+decide the former: for that which is itself independently a cause
+is always prior as a cause to that which is so only in virtue of being
+itself dependent upon something else that makes it so. 
+
+We must therefore make a fresh start and consider the question; if
+a thing moves itself, in what sense and in what manner does it do
+so? Now everything that is in motion must be infinitely divisible,
+for it has been shown already in our general course on Physics, that
+everything that is essentially in motion is continuous. Now it is
+impossible that that which moves itself should in its entirety move
+itself: for then, while being specifically one and indivisible, it
+would as a Whole both undergo and cause the same locomotion or alteration:
+thus it would at the same time be both teaching and being taught (the
+same thing), or both restoring to and being restored to the same health.
+Moreover, we have established the fact that it is the movable that
+is moved; and this is potentially, not actually, in motion, but the
+potential is in process to actuality, and motion is an incomplete
+actuality of the movable. The movent on the other hand is already
+in activity: e.g. it is that which is hot that produces heat: in fact,
+that which produces the form is always something that possesses it.
+Consequently (if a thing can move itself as a whole), the same thing
+in respect of the same thing may be at the same time both hot and
+not hot. So, too, in every other case where the movent must be described
+by the same name in the same sense as the moved. Therefore when a
+thing moves itself it is one part of it that is the movent and another
+part that is moved. But it is not self-moving in the sense that each
+of the two parts is moved by the other part: the following considerations
+make this evident. In the first place, if each of the two parts is
+to move the other, there will be no first movent. If a thing is moved
+by a series of movents, that which is earlier in the series is more
+the cause of its being moved than that which comes next, and will
+be more truly the movent: for we found that there are two kinds of
+movent, that which is itself moved by something else and that which
+derives its motion from itself: and that which is further from the
+thing that is moved is nearer to the principle of motion than that
+which is intermediate. In the second place, there is no necessity
+for the movent part to be moved by anything but itself: so it can
+only be accidentally that the other part moves it in return. I take
+then the possible case of its not moving it: then there will be a
+part that is moved and a part that is an unmoved movent. In the third
+place, there is no necessity for the movent to be moved in return:
+on the contrary the necessity that there should always be motion makes
+it necessary that there should be some movent that is either unmoved
+or moved by itself. In the fourth place we should then have a thing
+undergoing the same motion that it is causing-that which is producing
+heat, therefore, being heated. But as a matter of fact that which
+primarily moves itself cannot contain either a single part that moves
+itself or a number of parts each of which moves itself. For, if the
+whole is moved by itself, it must be moved either by some part of
+itself or as a whole by itself as a whole. If, then, it is moved in
+virtue of some part of it being moved by that part itself, it is this
+part that will be the primary self-movent, since, if this part is
+separated from the whole, the part will still move itself, but the
+whole will do so no longer. If on the other hand the whole is moved
+by itself as a whole, it must be accidentally that the parts move
+themselves: and therefore, their self-motion not being necessary,
+we may take the case of their not being moved by themselves. Therefore
+in the whole of the thing we may distinguish that which imparts motion
+without itself being moved and that which is moved: for only in this
+way is it possible for a thing to be self-moved. Further, if the whole
+moves itself we may distinguish in it that which imparts the motion
+and that which is moved: so while we say that AB is moved by itself,
+we may also say that it is moved by A. And since that which imparts
+motion may be either a thing that is moved by something else or a
+thing that is unmoved, and that which is moved may be either a thing
+that imparts motion to something else or a thing that does not, that
+which moves itself must be composed of something that is unmoved but
+imparts motion and also of something that is moved but does not necessarily
+impart motion but may or may not do so. Thus let A be something that
+imparts motion but is unmoved, B something that is moved by A and
+moves G, G something that is moved by B but moves nothing (granted
+that we eventually arrive at G we may take it that there is only one
+intermediate term, though there may be more). Then the whole ABG moves
+itself. But if I take away G, AB will move itself, A imparting motion
+and B being moved, whereas G will not move itself or in fact be moved
+at all. Nor again will BG move itself apart from A: for B imparts
+motion only through being moved by something else, not through being
+moved by any part of itself. So only AB moves itself. That which moves
+itself, therefore, must comprise something that imparts motion but
+is unmoved and something that is moved but does not necessarily move
+anything else: and each of these two things, or at any rate one of
+them, must be in contact with the other. If, then, that which imparts
+motion is a continuous substance-that which is moved must of course
+be so-it is clear that it is not through some part of the whole being
+of such a nature as to be capable of moving itself that the whole
+moves itself: it moves itself as a whole, both being moved and imparting
+motion through containing a part that imparts motion and a part that
+is moved. It does not impart motion as a whole nor is it moved as
+a whole: it is A alone that imparts motion and B alone that is moved.
+It is not true, further, that G is moved by A, which is impossible.
+
+Here a difficulty arises: if something is taken away from A (supposing
+that that which imparts motion but is unmoved is a continuous substance),
+or from B the part that is moved, will the remainder of A continue
+to impart motion or the remainder of B continue to be moved? If so,
+it will not be AB primarily that is moved by itself, since, when something
+is taken away from AB, the remainder of AB will still continue to
+move itself. Perhaps we may state the case thus: there is nothing
+to prevent each of the two parts, or at any rate one of them, that
+which is moved, being divisible though actually undivided, so that
+if it is divided it will not continue in the possession of the same
+capacity: and so there is nothing to prevent self-motion residing
+primarily in things that are potentially divisible. 
+
+From what has been said, then, it is evident that that which primarily
+imparts motion is unmoved: for, whether the series is closed at once
+by that which is in motion but moved by something else deriving its
+motion directly from the first unmoved, or whether the motion is derived
+from what is in motion but moves itself and stops its own motion,
+on both suppositions we have the result that in all cases of things
+being in motion that which primarily imparts motion is unmoved.
+
+Part 6
+
+Since there must always be motion without intermission, there must
+necessarily be something, one thing or it may be a plurality, that
+first imparts motion, and this first movent must be unmoved. Now the
+question whether each of the things that are unmoved but impart motion
+is eternal is irrelevant to our present argument: but the following
+considerations will make it clear that there must necessarily be some
+such thing, which, while it has the capacity of moving something else,
+is itself unmoved and exempt from all change, which can affect it
+neither in an unqualified nor in an accidental sense. Let us suppose,
+if any one likes, that in the case of certain things it is possible
+for them at different times to be and not to be, without any process
+of becoming and perishing (in fact it would seem to be necessary,
+if a thing that has not parts at one time is and at another time is
+not, that any such thing should without undergoing any process of
+change at one time be and at another time not be). And let us further
+suppose it possible that some principles that are unmoved but capable
+of imparting motion at one time are and at another time are not. Even
+so, this cannot be true of all such principles, since there must clearly
+be something that causes things that move themselves at one time to
+be and at another not to be. For, since nothing that has not parts
+can be in motion, that which moves itself must as a whole have magnitude,
+though nothing that we have said makes this necessarily true of every
+movent. So the fact that some things become and others perish, and
+that this is so continuously, cannot be caused by any one of those
+things that, though they are unmoved, do not always exist: nor again
+can it be caused by any of those which move certain particular things,
+while others move other things. The eternity and continuity of the
+process cannot be caused either by any one of them singly or by the
+sum of them, because this causal relation must be eternal and necessary,
+whereas the sum of these movents is infinite and they do not all exist
+together. It is clear, then, that though there may be countless instances
+of the perishing of some principles that are unmoved but impart motion,
+and though many things that move themselves perish and are succeeded
+by others that come into being, and though one thing that is unmoved
+moves one thing while another moves another, nevertheless there is
+something that comprehends them all, and that as something apart from
+each one of them, and this it is that is the cause of the fact that
+some things are and others are not and of the continuous process of
+change: and this causes the motion of the other movents, while they
+are the causes of the motion of other things. Motion, then, being
+eternal, the first movent, if there is but one, will be eternal also:
+if there are more than one, there will be a plurality of such eternal
+movents. We ought, however, to suppose that there is one rather than
+many, and a finite rather than an infinite number. When the consequences
+of either assumption are the same, we should always assume that things
+are finite rather than infinite in number, since in things constituted
+by nature that which is finite and that which is better ought, if
+possible, to be present rather than the reverse: and here it is sufficient
+to assume only one movent, the first of unmoved things, which being
+eternal will be the principle of motion to everything else.
+
+The following argument also makes it evident that the first movent
+must be something that is one and eternal. We have shown that there
+must always be motion. That being so, motion must also be continuous,
+because what is always is continuous, whereas what is merely in succession
+is not continuous. But further, if motion is continuous, it is one:
+and it is one only if the movent and the moved that constitute it
+are each of them one, since in the event of a thing's being moved
+now by one thing and now by another the whole motion will not be continuous
+but successive. 
+
+Moreover a conviction that there is a first unmoved something may
+be reached not only from the foregoing arguments, but also by considering
+again the principles operative in movents. Now it is evident that
+among existing things there are some that are sometimes in motion
+and sometimes at rest. This fact has served above to make it clear
+that it is not true either that all things are in motion or that all
+things are at rest or that some things are always at rest and the
+remainder always in motion: on this matter proof is supplied by things
+that fluctuate between the two and have the capacity of being sometimes
+in motion and sometimes at rest. The existence of things of this kind
+is clear to all: but we wish to explain also the nature of each of
+the other two kinds and show that there are some things that are always
+unmoved and some things that are always in motion. In the course of
+our argument directed to this end we established the fact that everything
+that is in motion is moved by something, and that the movent is either
+unmoved or in motion, and that, if it is in motion, it is moved either
+by itself or by something else and so on throughout the series: and
+so we proceeded to the position that the first principle that directly
+causes things that are in motion to be moved is that which moves itself,
+and the first principle of the whole series is the unmoved. Further
+it is evident from actual observation that there are things that have
+the characteristic of moving themselves, e.g. the animal kingdom and
+the whole class of living things. This being so, then, the view was
+suggested that perhaps it may be possible for motion to come to be
+in a thing without having been in existence at all before, because
+we see this actually occurring in animals: they are unmoved at one
+time and then again they are in motion, as it seems. We must grasp
+the fact, therefore, that animals move themselves only with one kind
+of motion, and that this is not strictly originated by them. The cause
+of it is not derived from the animal itself: it is connected with
+other natural motions in animals, which they do not experience through
+their own instrumentality, e.g. increase, decrease, and respiration:
+these are experienced by every animal while it is at rest and not
+in motion in respect of the motion set up by its own agency: here
+the motion is caused by the atmosphere and by many things that enter
+into the animal: thus in some cases the cause is nourishment: when
+it is being digested animals sleep, and when it is being distributed
+through the system they awake and move themselves, the first principle
+of this motion being thus originally derived from outside. Therefore
+animals are not always in continuous motion by their own agency: it
+is something else that moves them, itself being in motion and changing
+as it comes into relation with each several thing that moves itself.
+(Moreover in all these self-moving things the first movent and cause
+of their self-motion is itself moved by itself, though in an accidental
+sense: that is to say, the body changes its place, so that that which
+is in the body changes its place also and is a self-movent through
+its exercise of leverage.) Hence we may confidently conclude that
+if a thing belongs to the class of unmoved movents that are also themselves
+moved accidentally, it is impossible that it should cause continuous
+motion. So the necessity that there should be motion continuously
+requires that there should be a first movent that is unmoved even
+accidentally, if, as we have said, there is to be in the world of
+things an unceasing and undying motion, and the world is to remain
+permanently self-contained and within the same limits: for if the
+first principle is permanent, the universe must also be permanent,
+since it is continuous with the first principle. (We must distinguish,
+however, between accidental motion of a thing by itself and such motion
+by something else, the former being confined to perishable things,
+whereas the latter belongs also to certain first principles of heavenly
+bodies, of all those, that is to say, that experience more than one
+locomotion.) 
+
+And further, if there is always something of this nature, a movent
+that is itself unmoved and eternal, then that which is first moved
+by it must be eternal. Indeed this is clear also from the consideration
+that there would otherwise be no becoming and perishing and no change
+of any kind in other things, which require something that is in motion
+to move them: for the motion imparted by the unmoved will always be
+imparted in the same way and be one and the same, since the unmoved
+does not itself change in relation to that which is moved by it. But
+that which is moved by something that, though it is in motion, is
+moved directly by the unmoved stands in varying relations to the things
+that it moves, so that the motion that it causes will not be always
+the same: by reason of the fact that it occupies contrary positions
+or assumes contrary forms at different times it will produce contrary
+motions in each several thing that it moves and will cause it to be
+at one time at rest and at another time in motion. 
+
+The foregoing argument, then, has served to clear up the point about
+which we raised a difficulty at the outset-why is it that instead
+of all things being either in motion or at rest, or some things being
+always in motion and the remainder always at rest, there are things
+that are sometimes in motion and sometimes not? The cause of this
+is now plain: it is because, while some things are moved by an eternal
+unmoved movent and are therefore always in motion, other things are
+moved by a movent that is in motion and changing, so that they too
+must change. But the unmoved movent, as has been said, since it remains
+permanently simple and unvarying and in the same state, will cause
+motion that is one and simple. 
+
+Part 7
+
+This matter will be made clearer, however, if we start afresh from
+another point. We must consider whether it is or is not possible that
+there should be a continuous motion, and, if it is possible, which
+this motion is, and which is the primary motion: for it is plain that
+if there must always be motion, and a particular motion is primary
+and continuous, then it is this motion that is imparted by the first
+movent, and so it is necessarily one and the same and continuous and
+primary. 
+
+Now of the three kinds of motion that there are-motion in respect
+of magnitude, motion in respect of affection, and motion in respect
+of place-it is this last, which we call locomotion, that must be primary.
+This may be shown as follows. It is impossible that there should be
+increase without the previous occurrence of alteration: for that which
+is increased, although in a sense it is increased by what is like
+itself, is in a sense increased by what is unlike itself: thus it
+is said that contrary is nourishment to contrary: but growth is effected
+only by things becoming like to like. There must be alteration, then,
+in that there is this change from contrary to contrary. But the fact
+that a thing is altered requires that there should be something that
+alters it, something e.g. that makes the potentially hot into the
+actually hot: so it is plain that the movent does not maintain a uniform
+relation to it but is at one time nearer to and at another farther
+from that which is altered: and we cannot have this without locomotion.
+If, therefore, there must always be motion, there must also always
+be locomotion as the primary motion, and, if there is a primary as
+distinguished from a secondary form of locomotion, it must be the
+primary form. Again, all affections have their origin in condensation
+and rarefaction: thus heavy and light, soft and hard, hot and cold,
+are considered to be forms of density and rarity. But condensation
+and rarefaction are nothing more than combination and separation,
+processes in accordance with which substances are said to become and
+perish: and in being combined and separated things must change in
+respect of place. And further, when a thing is increased or decreased
+its magnitude changes in respect of place. 
+
+Again, there is another point of view from which it will be clearly
+seen that locomotion is primary. As in the case of other things so
+too in the case of motion the word 'primary' may be used in several
+senses. A thing is said to be prior to other things when, if it does
+not exist, the others will not exist, whereas it can exist without
+the others: and there is also priority in time and priority in perfection
+of existence. Let us begin, then, with the first sense. Now there
+must be motion continuously, and there may be continuously either
+continuous motion or successive motion, the former, however, in a
+higher degree than the latter: moreover it is better that it should
+be continuous rather than successive motion, and we always assume
+the presence in nature of the better, if it be possible: since, then,
+continuous motion is possible (this will be proved later: for the
+present let us take it for granted), and no other motion can be continuous
+except locomotion, locomotion must be primary. For there is no necessity
+for the subject of locomotion to be the subject either of increase
+or of alteration, nor need it become or perish: on the other hand
+there cannot be any one of these processes without the existence of
+the continuous motion imparted by the first movent. 
+
+Secondly, locomotion must be primary in time: for this is the only
+motion possible for things. It is true indeed that, in the case of
+any individual thing that has a becoming, locomotion must be the last
+of its motions: for after its becoming it first experiences alteration
+and increase, and locomotion is a motion that belongs to such things
+only when they are perfected. But there must previously be something
+else that is in process of locomotion to be the cause even of the
+becoming of things that become, without itself being in process of
+becoming, as e.g. the begotten is preceded by what begot it: otherwise
+becoming might be thought to be the primary motion on the ground that
+the thing must first become. But though this is so in the case of
+any individual thing that becomes, nevertheless before anything becomes,
+something else must be in motion, not itself becoming but being, and
+before this there must again be something else. And since becoming
+cannot be primary-for, if it were, everything that is in motion would
+be perishable-it is plain that no one of the motions next in order
+can be prior to locomotion. By the motions next in order I mean increase
+and then alteration, decrease, and perishing. All these are posterior
+to becoming: consequently, if not even becoming is prior to locomotion,
+then no one of the other processes of change is so either.
+
+Thirdly, that which is in process of becoming appears universally
+as something imperfect and proceeding to a first principle: and so
+what is posterior in the order of becoming is prior in the order of
+nature. Now all things that go through the process of becoming acquire
+locomotion last. It is this that accounts for the fact that some living
+things, e.g. plants and many kinds of animals, owing to lack of the
+requisite organ, are entirely without motion, whereas others acquire
+it in the course of their being perfected. Therefore, if the degree
+in which things possess locomotion corresponds to the degree in which
+they have realized their natural development, then this motion must
+be prior to all others in respect of perfection of existence: and
+not only for this reason but also because a thing that is in motion
+loses its essential character less in the process of locomotion than
+in any other kind of motion: it is the only motion that does not involve
+a change of being in the sense in which there is a change in quality
+when a thing is altered and a change in quantity when a thing is increased
+or decreased. Above all it is plain that this motion, motion in respect
+of place, is what is in the strictest sense produced by that which
+moves itself; but it is the self-movent that we declare to be the
+first principle of things that are moved and impart motion and the
+primary source to which things that are in motion are to be referred.
+
+It is clear, then, from the foregoing arguments that locomotion is
+the primary motion. We have now to show which kind of locomotion is
+primary. The same process of reasoning will also make clear at the
+same time the truth of the assumption we have made both now and at
+a previous stage that it is possible that there should be a motion
+that is continuous and eternal. Now it is clear from the following
+considerations that no other than locomotion can be continuous. Every
+other motion and change is from an opposite to an opposite: thus for
+the processes of becoming and perishing the limits are the existent
+and the non-existent, for alteration the various pairs of contrary
+affections, and for increase and decrease either greatness and smallness
+or perfection and imperfection of magnitude: and changes to the respective
+contraries are contrary changes. Now a thing that is undergoing any
+particular kind of motion, but though previously existent has not
+always undergone it, must previously have been at rest so far as that
+motion is concerned. It is clear, then, that for the changing thing
+the contraries will be states of rest. And we have a similar result
+in the case of changes that are not motions: for becoming and perishing,
+whether regarded simply as such without qualification or as affecting
+something in particular, are opposites: therefore provided it is impossible
+for a thing to undergo opposite changes at the same time, the change
+will not be continuous, but a period of time will intervene between
+the opposite processes. The question whether these contradictory changes
+are contraries or not makes no difference, provided only it is impossible
+for them both to be present to the same thing at the same time: the
+point is of no importance to the argument. Nor does it matter if the
+thing need not rest in the contradictory state, or if there is no
+state of rest as a contrary to the process of change: it may be true
+that the non-existent is not at rest, and that perishing is a process
+to the non-existent. All that matters is the intervention of a time:
+it is this that prevents the change from being continuous: so, too,
+in our previous instances the important thing was not the relation
+of contrariety but the impossibility of the two processes being present
+to a thing at the same time. And there is no need to be disturbed
+by the fact that on this showing there may be more than one contrary
+to the same thing, that a particular motion will be contrary both
+to rest and to motion in the contrary direction. We have only to grasp
+the fact that a particular motion is in a sense the opposite both
+of a state of rest and of the contrary motion, in the same way as
+that which is of equal or standard measure is the opposite both of
+that which surpasses it and of that which it surpasses, and that it
+is impossible for the opposite motions or changes to be present to
+a thing at the same time. Furthermore, in the case of becoming and
+perishing it would seem to be an utterly absurd thing if as soon as
+anything has become it must necessarily perish and cannot continue
+to exist for any time: and, if this is true of becoming and perishing,
+we have fair grounds for inferring the same to be true of the other
+kinds of change, since it would be in the natural order of things
+that they should be uniform in this respect. 
+
+Part 8
+
+Let us now proceed to maintain that it is possible that there should
+be an infinite motion that is single and continuous, and that this
+motion is rotatory motion. The motion of everything that is in process
+of locomotion is either rotatory or rectilinear or a compound of the
+two: consequently, if one of the former two is not continuous, that
+which is composed of them both cannot be continuous either. Now it
+is plain that if the locomotion of a thing is rectilinear and finite
+it is not continuous locomotion: for the thing must turn back, and
+that which turns back in a straight line undergoes two contrary locomotions,
+since, so far as motion in respect of place is concerned, upward motion
+is the contrary of downward motion, forward motion of backward motion,
+and motion to the left of motion to the right, these being the pairs
+of contraries in the sphere of place. But we have already defined
+single and continuous motion to be motion of a single thing in a single
+period of time and operating within a sphere admitting of no further
+specific differentiation (for we have three things to consider, first
+that which is in motion, e.g. a man or a god, secondly the 'when'
+of the motion, that is to say, the time, and thirdly the sphere within
+which it operates, which may be either place or affection or essential
+form or magnitude): and contraries are specifically not one and the
+same but distinct: and within the sphere of place we have the above-mentioned
+distinctions. Moreover we have an indication that motion from A to
+B is the contrary of motion from B to A in the fact that, if they
+occur at the same time, they arrest and stop each other. And the same
+is true in the case of a circle: the motion from A towards B is the
+contrary of the motion from A towards G: for even if they are continuous
+and there is no turning back they arrest each other, because contraries
+annihilate or obstruct one another. On the other hand lateral motion
+is not the contrary of upward motion. But what shows most clearly
+that rectilinear motion cannot be continuous is the fact that turning
+back necessarily implies coming to a stand, not only when it is a
+straight line that is traversed, but also in the case of locomotion
+in a circle (which is not the same thing as rotatory locomotion: for,
+when a thing merely traverses a circle, it may either proceed on its
+course without a break or turn back again when it has reached the
+same point from which it started). We may assure ourselves of the
+necessity of this coming to a stand not only on the strength of observation,
+but also on theoretical grounds. We may start as follows: we have
+three points, starting-point, middle-point, and finishing-point, of
+which the middle-point in virtue of the relations in which it stands
+severally to the other two is both a starting-point and a finishing-point,
+and though numerically one is theoretically two. We have further the
+distinction between the potential and the actual. So in the straight
+line in question any one of the points lying between the two extremes
+is potentially a middle-point: but it is not actually so unless that
+which is in motion divides the line by coming to a stand at that point
+and beginning its motion again: thus the middle-point becomes both
+a starting-point and a goal, the starting-point of the latter part
+and the finishing-point of the first part of the motion. This is the
+case e.g. when A in the course of its locomotion comes to a stand
+at B and starts again towards G: but when its motion is continuous
+A cannot either have come to be or have ceased to be at the point
+B: it can only have been there at the moment of passing, its passage
+not being contained within any period of time except the whole of
+which the particular moment is a dividing-point. To maintain that
+it has come to be and ceased to be there will involve the consequence
+that A in the course of its locomotion will always be coming to a
+stand: for it is impossible that A should simultaneously have come
+to be at B and ceased to be there, so that the two things must have
+happened at different points of time, and therefore there will be
+the intervening period of time: consequently A will be in a state
+of rest at B, and similarly at all other points, since the same reasoning
+holds good in every case. When to A, that which is in process of locomotion,
+B, the middle-point, serves both as a finishing-point and as a starting-point
+for its motion, A must come to a stand at B, because it makes it two
+just as one might do in thought. However, the point A is the real
+starting-point at which the moving body has ceased to be, and it is
+at G that it has really come to be when its course is finished and
+it comes to a stand. So this is how we must meet the difficulty that
+then arises, which is as follows. Suppose the line E is equal to the
+line Z, that A proceeds in continuous locomotion from the extreme
+point of E to G, and that, at the moment when A is at the point B,
+D is proceeding in uniform locomotion and with the same velocity as
+A from the extremity of Z to H: then, says the argument, D will have
+reached H before A has reached G for that which makes an earlier start
+and departure must make an earlier arrival: the reason, then, for
+the late arrival of A is that it has not simultaneously come to be
+and ceased to be at B: otherwise it will not arrive later: for this
+to happen it will be necessary that it should come to a stand there.
+Therefore we must not hold that there was a moment when A came to
+be at B and that at the same moment D was in motion from the extremity
+of Z: for the fact of A's having come to be at B will involve the
+fact of its also ceasing to be there, and the two events will not
+be simultaneous, whereas the truth is that A is at B at a sectional
+point of time and does not occupy time there. In this case, therefore,
+where the motion of a thing is continuous, it is impossible to use
+this form of expression. On the other hand in the case of a thing
+that turns back in its course we must do so. For suppose H in the
+course of its locomotion proceeds to D and then turns back and proceeds
+downwards again: then the extreme point D has served as finishing-point
+and as starting-point for it, one point thus serving as two: therefore
+H must have come to a stand there: it cannot have come to be at D
+and departed from D simultaneously, for in that case it would simultaneously
+be there and not be there at the same moment. And here we cannot apply
+the argument used to solve the difficulty stated above: we cannot
+argue that H is at D at a sectional point of time and has not come
+to be or ceased to be there. For here the goal that is reached is
+necessarily one that is actually, not potentially, existent. Now the
+point in the middle is potential: but this one is actual, and regarded
+from below it is a finishing-point, while regarded from above it is
+a starting-point, so that it stands in these same two respective relations
+to the two motions. Therefore that which turns back in traversing
+a rectilinear course must in so doing come to a stand. Consequently
+there cannot be a continuous rectilinear motion that is eternal.
+
+The same method should also be adopted in replying to those who ask,
+in the terms of Zeno's argument, whether we admit that before any
+distance can be traversed half the distance must be traversed, that
+these half-distances are infinite in number, and that it is impossible
+to traverse distances infinite in number-or some on the lines of this
+same argument put the questions in another form, and would have us
+grant that in the time during which a motion is in progress it should
+be possible to reckon a half-motion before the whole for every half-distance
+that we get, so that we have the result that when the whole distance
+is traversed we have reckoned an infinite number, which is admittedly
+impossible. Now when we first discussed the question of motion we
+put forward a solution of this difficulty turning on the fact that
+the period of time occupied in traversing the distance contains within
+itself an infinite number of units: there is no absurdity, we said,
+in supposing the traversing of infinite distances in infinite time,
+and the element of infinity is present in the time no less than in
+the distance. But, although this solution is adequate as a reply to
+the questioner (the question asked being whether it is possible in
+a finite time to traverse or reckon an infinite number of units),
+nevertheless as an account of the fact and explanation of its true
+nature it is inadequate. For suppose the distance to be left out of
+account and the question asked to be no longer whether it is possible
+in a finite time to traverse an infinite number of distances, and
+suppose that the inquiry is made to refer to the time taken by itself
+(for the time contains an infinite number of divisions): then this
+solution will no longer be adequate, and we must apply the truth that
+we enunciated in our recent discussion, stating it in the following
+way. In the act of dividing the continuous distance into two halves
+one point is treated as two, since we make it a starting-point and
+a finishing-point: and this same result is also produced by the act
+of reckoning halves as well as by the act of dividing into halves.
+But if divisions are made in this way, neither the distance nor the
+motion will be continuous: for motion if it is to be continuous must
+relate to what is continuous: and though what is continuous contains
+an infinite number of halves, they are not actual but potential halves.
+If the halves are made actual, we shall get not a continuous but an
+intermittent motion. In the case of reckoning the halves, it is clear
+that this result follows: for then one point must be reckoned as two:
+it will be the finishing-point of the one half and the starting-point
+of the other, if we reckon not the one continuous whole but the two
+halves. Therefore to the question whether it is possible to pass through
+an infinite number of units either of time or of distance we must
+reply that in a sense it is and in a sense it is not. If the units
+are actual, it is not possible: if they are potential, it is possible.
+For in the course of a continuous motion the traveller has traversed
+an infinite number of units in an accidental sense but not in an unqualified
+sense: for though it is an accidental characteristic of the distance
+to be an infinite number of half-distances, this is not its real and
+essential character. It is also plain that unless we hold that the
+point of time that divides earlier from later always belongs only
+to the later so far as the thing is concerned, we shall be involved
+in the consequence that the same thing is at the same moment existent
+and not existent, and that a thing is not existent at the moment when
+it has become. It is true that the point is common to both times,
+the earlier as well as the later, and that, while numerically one
+and the same, it is theoretically not so, being the finishing-point
+of the one and the starting-point of the other: but so far as the
+thing is concerned it belongs to the later stage of what happens to
+it. Let us suppose a time ABG and a thing D, D being white in the
+time A and not-white in the time B. Then D is at the moment G white
+and not-white: for if we were right in saying that it is white during
+the whole time A, it is true to call it white at any moment of A,
+and not-white in B, and G is in both A and B. We must not allow, therefore,
+that it is white in the whole of A, but must say that it is so in
+all of it except the last moment G. G belongs already to the later
+period, and if in the whole of A not-white was in process of becoming
+and white of perishing, at G the process is complete. And so G is
+the first moment at which it is true to call the thing white or not
+white respectively. Otherwise a thing may be non-existent at the moment
+when it has become and existent at the moment when it has perished:
+or else it must be possible for a thing at the same time to be white
+and not white and in fact to be existent and non-existent. Further,
+if anything that exists after having been previously non-existent
+must become existent and does not exist when it is becoming, time
+cannot be divisible into time-atoms. For suppose that D was becoming
+white in the time A and that at another time B, a time-atom consecutive
+with the last atom of A, D has already become white and so is white
+at that moment: then, inasmuch as in the time A it was becoming white
+and so was not white and at the moment B it is white, there must have
+been a becoming between A and B and therefore also a time in which
+the becoming took place. On the other hand, those who deny atoms of
+time (as we do) are not affected by this argument: according to them
+D has become and so is white at the last point of the actual time
+in which it was becoming white: and this point has no other point
+consecutive with or in succession to it, whereas time-atoms are conceived
+as successive. Moreover it is clear that if D was becoming white in
+the whole time A, the time occupied by it in having become white in
+addition to having been in process of becoming white is no more than
+all that it occupied in the mere process of becoming white.
+
+These and such-like, then, are the arguments for our conclusion that
+derive cogency from the fact that they have a special bearing on the
+point at issue. If we look at the question from the point of view
+of general theory, the same result would also appear to be indicated
+by the following arguments. Everything whose motion is continuous
+must, on arriving at any point in the course of its locomotion, have
+been previously also in process of locomotion to that point, if it
+is not forced out of its path by anything: e.g. on arriving at B a
+thing must also have been in process of locomotion to B, and that
+not merely when it was near to B, but from the moment of its starting
+on its course, since there can be, no reason for its being so at any
+particular stage rather than at an earlier one. So, too, in the case
+of the other kinds of motion. Now we are to suppose that a thing proceeds
+in locomotion from A to G and that at the moment of its arrival at
+G the continuity of its motion is unbroken and will remain so until
+it has arrived back at A. Then when it is undergoing locomotion from
+A to G it is at the same time undergoing also its locomotion to A
+from G: consequently it is simultaneously undergoing two contrary
+motions, since the two motions that follow the same straight line
+are contrary to each other. With this consequence there also follows
+another: we have a thing that is in process of change from a position
+in which it has not yet been: so, inasmuch as this is impossible,
+the thing must come to a stand at G. Therefore the motion is not a
+single motion, since motion that is interrupted by stationariness
+is not single. 
+
+Further, the following argument will serve better to make this point
+clear universally in respect of every kind of motion. If the motion
+undergone by that which is in motion is always one of those already
+enumerated, and the state of rest that it undergoes is one of those
+that are the opposites of the motions (for we found that there could
+be no other besides these), and moreover that which is undergoing
+but does not always undergo a particular motion (by this I mean one
+of the various specifically distinct motions, not some particular
+part of the whole motion) must have been previously undergoing the
+state of rest that is the opposite of the motion, the state of rest
+being privation of motion; then, inasmuch as the two motions that
+follow the same straight line are contrary motions, and it is impossible
+for a thing to undergo simultaneously two contrary motions, that which
+is undergoing locomotion from A to G cannot also simultaneously be
+undergoing locomotion from G to A: and since the latter locomotion
+is not simultaneous with the former but is still to be undergone,
+before it is undergone there must occur a state of rest at G: for
+this, as we found, is the state of rest that is the opposite of the
+motion from G. The foregoing argument, then, makes it plain that the
+motion in question is not continuous. 
+
+Our next argument has a more special bearing than the foregoing on
+the point at issue. We will suppose that there has occurred in something
+simultaneously a perishing of not-white and a becoming of white. Then
+if the alteration to white and from white is a continuous process
+and the white does not remain any time, there must have occurred simultaneously
+a perishing of not-white, a becoming of white, and a becoming of not-white:
+for the time of the three will be the same. 
+
+Again, from the continuity of the time in which the motion takes place
+we cannot infer continuity in the motion, but only successiveness:
+in fact, how could contraries, e.g. whiteness and blackness, meet
+in the same extreme point? 
+
+On the other hand, in motion on a circular line we shall find singleness
+and continuity: for here we are met by no impossible consequence:
+that which is in motion from A will in virtue of the same direction
+of energy be simultaneously in motion to A (since it is in motion
+to the point at which it will finally arrive), and yet will not be
+undergoing two contrary or opposite motions: for a motion to a point
+and a motion from that point are not always contraries or opposites:
+they are contraries only if they are on the same straight line (for
+then they are contrary to one another in respect of place, as e.g.
+the two motions along the diameter of the circle, since the ends of
+this are at the greatest possible distance from one another), and
+they are opposites only if they are along the same line. Therefore
+in the case we are now considering there is nothing to prevent the
+motion being continuous and free from all intermission: for rotatory
+motion is motion of a thing from its place to its place, whereas rectilinear
+motion is motion from its place to another place. 
+
+Moreover the progress of rotatory motion is never localized within
+certain fixed limits, whereas that of rectilinear motion repeatedly
+is so. Now a motion that is always shifting its ground from moment
+to moment can be continuous: but a motion that is repeatedly localized
+within certain fixed limits cannot be so, since then the same thing
+would have to undergo simultaneously two opposite motions. So, too,
+there cannot be continuous motion in a semicircle or in any other
+arc of a circle, since here also the same ground must be traversed
+repeatedly and two contrary processes of change must occur. The reason
+is that in these motions the starting-point and the termination do
+not coincide, whereas in motion over a circle they do coincide, and
+so this is the only perfect motion. 
+
+This differentiation also provides another means of showing that the
+other kinds of motion cannot be continuous either: for in all of them
+we find that there is the same ground to be traversed repeatedly;
+thus in alteration there are the intermediate stages of the process,
+and in quantitative change there are the intervening degrees of magnitude:
+and in becoming and perishing the same thing is true. It makes no
+difference whether we take the intermediate stages of the process
+to be few or many, or whether we add or subtract one: for in either
+case we find that there is still the same ground to be traversed repeatedly.
+Moreover it is plain from what has been said that those physicists
+who assert that all sensible things are always in motion are wrong:
+for their motion must be one or other of the motions just mentioned:
+in fact they mostly conceive it as alteration (things are always in
+flux and decay, they say), and they go so far as to speak even of
+becoming and perishing as a process of alteration. On the other hand,
+our argument has enabled us to assert the fact, applying universally
+to all motions, that no motion admits of continuity except rotatory
+motion: consequently neither alteration nor increase admits of continuity.
+We need now say no more in support of the position that there is no
+process of change that admits of infinity or continuity except rotatory
+locomotion. 
+
+Part 9
+
+It can now be shown plainly that rotation is the primary locomotion.
+Every locomotion, as we said before, is either rotatory or rectilinear
+or a compound of the two: and the two former must be prior to the
+last, since they are the elements of which the latter consists. Moreover
+rotatory locomotion is prior to rectilinear locomotion, because it
+is more simple and complete, which may be shown as follows. The straight
+line traversed in rectilinear motion cannot be infinite: for there
+is no such thing as an infinite straight line; and even if there were,
+it would not be traversed by anything in motion: for the impossible
+does not happen and it is impossible to traverse an infinite distance.
+On the other hand rectilinear motion on a finite straight line is
+if it turns back a composite motion, in fact two motions, while if
+it does not turn back it is incomplete and perishable: and in the
+order of nature, of definition, and of time alike the complete is
+prior to the incomplete and the imperishable to the perishable. Again,
+a motion that admits of being eternal is prior to one that does not.
+Now rotatory motion can be eternal: but no other motion, whether locomotion
+or motion of any other kind, can be so, since in all of them rest
+must occur and with the occurrence of rest the motion has perished.
+Moreover the result at which we have arrived, that rotatory motion
+is single and continuous, and rectilinear motion is not, is a reasonable
+one. In rectilinear motion we have a definite starting-point, finishing-point,
+middle-point, which all have their place in it in such a way that
+there is a point from which that which is in motion can be said to
+start and a point at which it can be said to finish its course (for
+when anything is at the limits of its course, whether at the starting-point
+or at the finishing-point, it must be in a state of rest). On the
+other hand in circular motion there are no such definite points: for
+why should any one point on the line be a limit rather than any other?
+Any one point as much as any other is alike starting-point, middle-point,
+and finishing-point, so that we can say of certain things both that
+they are always and that they never are at a starting-point and at
+a finishing-point (so that a revolving sphere, while it is in motion,
+is also in a sense at rest, for it continues to occupy the same place).
+The reason of this is that in this case all these characteristics
+belong to the centre: that is to say, the centre is alike starting-point,
+middle-point, and finishing-point of the space traversed; consequently
+since this point is not a point on the circular line, there is no
+point at which that which is in process of locomotion can be in a
+state of rest as having traversed its course, because in its locomotion
+it is proceeding always about a central point and not to an extreme
+point: therefore it remains still, and the whole is in a sense always
+at rest as well as continuously in motion. Our next point gives a
+convertible result: on the one hand, because rotation is the measure
+of motions it must be the primary motion (for all things are measured
+by what is primary): on the other hand, because rotation is the primary
+motion it is the measure of all other motions. Again, rotatory motion
+is also the only motion that admits of being regular. In rectilinear
+locomotion the motion of things in leaving the starting-point is not
+uniform with their motion in approaching the finishing-point, since
+the velocity of a thing always increases proportionately as it removes
+itself farther from its position of rest: on the other hand rotatory
+motion is the only motion whose course is naturally such that it has
+no starting-point or finishing-point in itself but is determined from
+elsewhere. 
+
+As to locomotion being the primary motion, this is a truth that is
+attested by all who have ever made mention of motion in their theories:
+they all assign their first principles of motion to things that impart
+motion of this kind. Thus 'separation' and 'combination' are motions
+in respect of place, and the motion imparted by 'Love' and 'Strife'
+takes these forms, the latter 'separating' and the former 'combining'.
+Anaxagoras, too, says that 'Mind', his first movent, 'separates'.
+Similarly those who assert no cause of this kind but say that 'void'
+accounts for motion-they also hold that the motion of natural substance
+is motion in respect of place: for their motion that is accounted
+for by 'void' is locomotion, and its sphere of operation may be said
+to be place. Moreover they are of opinion that the primary substances
+are not subject to any of the other motions, though the things that
+are compounds of these substances are so subject: the processes of
+increase and decrease and alteration, they say, are effects of the
+'combination' and 'separation' of atoms. It is the same, too, with
+those who make out that the becoming or perishing of a thing is accounted
+for by 'density' or 'rarity': for it is by 'combination' and 'separation'
+that the place of these things in their systems is determined. Moreover
+to these we may add those who make Soul the cause of motion: for they
+say that things that undergo motion have as their first principle
+'that which moves itself': and when animals and all living things
+move themselves, the motion is motion in respect of place. Finally
+it is to be noted that we say that a thing 'is in motion' in the strict
+sense of the term only when its motion is motion in respect of place:
+if a thing is in process of increase or decrease or is undergoing
+some alteration while remaining at rest in the same place, we say
+that it is in motion in some particular respect: we do not say that
+it 'is in motion' without qualification. 
+
+Our present position, then, is this: We have argued that there always
+was motion and always will be motion throughout all time, and we have
+explained what is the first principle of this eternal motion: we have
+explained further which is the primary motion and which is the only
+motion that can be eternal: and we have pronounced the first movent
+to be unmoved. 
+
+Part 10
+
+We have now to assert that the first movent must be without parts
+and without magnitude, beginning with the establishment of the premisses
+on which this conclusion depends. 
+
+One of these premisses is that nothing finite can cause motion during
+an infinite time. We have three things, the movent, the moved, and
+thirdly that in which the motion takes place, namely the time: and
+these are either all infinite or all finite or partly-that is to say
+two of them or one of them-finite and partly infinite. Let A be the
+movement, B the moved, and G the infinite time. Now let us suppose
+that D moves E, a part of B. Then the time occupied by this motion
+cannot be equal to G: for the greater the amount moved, the longer
+the time occupied. It follows that the time Z is not infinite. Now
+we see that by continuing to add to D, I shall use up A and by continuing
+to add to E, I shall use up B: but I shall not use up the time by
+continually subtracting a corresponding amount from it, because it
+is infinite. Consequently the duration of the part of G which is occupied
+by all A in moving the whole of B, will be finite. Therefore a finite
+thing cannot impart to anything an infinite motion. It is clear, then,
+that it is impossible for the finite to cause motion during an infinite
+time. 
+
+It has now to be shown that in no case is it possible for an infinite
+force to reside in a finite magnitude. This can be shown as follows:
+we take it for granted that the greater force is always that which
+in less time than another does an equal amount of work when engaged
+in any activity-in heating, for example, or sweetening or throwing;
+in fact, in causing any kind of motion. Then that on which the forces
+act must be affected to some extent by our supposed finite magnitude
+possessing an infinite force as well as by anything else, in fact
+to a greater extent than by anything else, since the infinite force
+is greater than any other. But then there cannot be any time in which
+its action could take place. Suppose that A is the time occupied by
+the infinite power in the performance of an act of heating or pushing,
+and that AB is the time occupied by a finite power in the performance
+of the same act: then by adding to the latter another finite power
+and continually increasing the magnitude of the power so added I shall
+at some time or other reach a point at which the finite power has
+completed the motive act in the time A: for by continual addition
+to a finite magnitude I must arrive at a magnitude that exceeds any
+assigned limit, and in the same way by continual subtraction I must
+arrive at one that falls short of any assigned limit. So we get the
+result that the finite force will occupy the same amount of time in
+performing the motive act as the infinite force. But this is impossible.
+Therefore nothing finite can possess an infinite force. So it is also
+impossible for a finite force to reside in an infinite magnitude.
+It is true that a greater force can reside in a lesser magnitude:
+but the superiority of any such greater force can be still greater
+if the magnitude in which it resides is greater. Now let AB be an
+infinite magnitude. Then BG possesses a certain force that occupies
+a certain time, let us say the time Z in moving D. Now if I take a
+magnitude twice as great at BG, the time occupied by this magnitude
+in moving D will be half of EZ (assuming this to be the proportion):
+so we may call this time ZH. That being so, by continually taking
+a greater magnitude in this way I shall never arrive at the full AB,
+whereas I shall always be getting a lesser fraction of the time given.
+Therefore the force must be infinite, since it exceeds any finite
+force. Moreover the time occupied by the action of any finite force
+must also be finite: for if a given force moves something in a certain
+time, a greater force will do so in a lesser time, but still a definite
+time, in inverse proportion. But a force must always be infinite-just
+as a number or a magnitude is-if it exceeds all definite limits. This
+point may also be proved in another way-by taking a finite magnitude
+in which there resides a force the same in kind as that which resides
+in the infinite magnitude, so that this force will be a measure of
+the finite force residing in the infinite magnitude. 
+
+It is plain, then, from the foregoing arguments that it is impossible
+for an infinite force to reside in a finite magnitude or for a finite
+force to reside in an infinite magnitude. But before proceeding to
+our conclusion it will be well to discuss a difficulty that arises
+in connexion with locomotion. If everything that is in motion with
+the exception of things that move themselves is moved by something
+else, how is it that some things, e.g. things thrown, continue to
+be in motion when their movent is no longer in contact with them?
+If we say that the movent in such cases moves something else at the
+same time, that the thrower e.g. also moves the air, and that this
+in being moved is also a movent, then it would be no more possible
+for this second thing than for the original thing to be in motion
+when the original movent is not in contact with it or moving it: all
+the things moved would have to be in motion simultaneously and also
+to have ceased simultaneously to be in motion when the original movent
+ceases to move them, even if, like the magnet, it makes that which
+it has moved capable of being a movent. Therefore, while we must accept
+this explanation to the extent of saying that the original movent
+gives the power of being a movent either to air or to water or to
+something else of the kind, naturally adapted for imparting and undergoing
+motion, we must say further that this thing does not cease simultaneously
+to impart motion and to undergo motion: it ceases to be in motion
+at the moment when its movent ceases to move it, but it still remains
+a movent, and so it causes something else consecutive with it to be
+in motion, and of this again the same may be said. The motion begins
+to cease when the motive force produced in one member of the consecutive
+series is at each stage less than that possessed by the preceding
+member, and it finally ceases when one member no longer causes the
+next member to be a movent but only causes it to be in motion. The
+motion of these last two-of the one as movent and of the other as
+moved-must cease simultaneously, and with this the whole motion ceases.
+Now the things in which this motion is produced are things that admit
+of being sometimes in motion and sometimes at rest, and the motion
+is not continuous but only appears so: for it is motion of things
+that are either successive or in contact, there being not one movent
+but a number of movents consecutive with one another: and so motion
+of this kind takes place in air and water. Some say that it is 'mutual
+replacement': but we must recognize that the difficulty raised cannot
+be solved otherwise than in the way we have described. So far as they
+are affected by 'mutual replacement', all the members of the series
+are moved and impart motion simultaneously, so that their motions
+also cease simultaneously: but our present problem concerns the appearance
+of continuous motion in a single thing, and therefore, since it cannot
+be moved throughout its motion by the same movent, the question is,
+what moves it? 
+
+Resuming our main argument, we proceed from the positions that there
+must be continuous motion in the world of things, that this is a single
+motion, that a single motion must be a motion of a magnitude (for
+that which is without magnitude cannot be in motion), and that the
+magnitude must be a single magnitude moved by a single movent (for
+otherwise there will not be continuous motion but a consecutive series
+of separate motions), and that if the movement is a single thing,
+it is either itself in motion or itself unmoved: if, then, it is in
+motion, it will have to be subject to the same conditions as that
+which it moves, that is to say it will itself be in process of change
+and in being so will also have to be moved by something: so we have
+a series that must come to an end, and a point will be reached at
+which motion is imparted by something that is unmoved. Thus we have
+a movent that has no need to change along with that which it moves
+but will be able to cause motion always (for the causing of motion
+under these conditions involves no effort): and this motion alone
+is regular, or at least it is so in a higher degree than any other,
+since the movent is never subject to any change. So, too, in order
+that the motion may continue to be of the same character, the moved
+must not be subject to change in respect of its relation to the movent.
+Moreover the movent must occupy either the centre or the circumference,
+since these are the first principles from which a sphere is derived.
+But the things nearest the movent are those whose motion is quickest,
+and in this case it is the motion of the circumference that is the
+quickest: therefore the movent occupies the circumference.
+
+There is a further difficulty in supposing it to be possible for anything
+that is in motion to cause motion continuously and not merely in the
+way in which it is caused by something repeatedly pushing (in which
+case the continuity amounts to no more than successiveness). Such
+a movent must either itself continue to push or pull or perform both
+these actions, or else the action must be taken up by something else
+and be passed on from one movent to another (the process that we described
+before as occurring in the case of things thrown, since the air or
+the water, being divisible, is a movent only in virtue of the fact
+that different parts of the air are moved one after another): and
+in either case the motion cannot be a single motion, but only a consecutive
+series of motions. The only continuous motion, then, is that which
+is caused by the unmoved movent: and this motion is continuous because
+the movent remains always invariable, so that its relation to that
+which it moves remains also invariable and continuous. 
+
+Now that these points are settled, it is clear that the first unmoved
+movent cannot have any magnitude. For if it has magnitude, this must
+be either a finite or an infinite magnitude. Now we have already'proved
+in our course on Physics that there cannot be an infinite magnitude:
+and we have now proved that it is impossible for a finite magnitude
+to have an infinite force, and also that it is impossible for a thing
+to be moved by a finite magnitude during an infinite time. But the
+first movent causes a motion that is eternal and does cause it during
+an infinite time. It is clear, therefore, that the first movent is
+indivisible and is without parts and without magnitude. 
+
+THE END
+
+----------------------------------------------------------------------
+
+Copyright statement:
+The Internet Classics Archive by Daniel C. Stevenson, Web Atomics.
+World Wide Web presentation is copyright (C) 1994-2000, Daniel
+C. Stevenson, Web Atomics.
+All rights reserved under international and pan-American copyright
+conventions, including the right of reproduction in whole or in part
+in any form. Direct permission requests to classics@classics.mit.edu.
+Translation of "The Deeds of the Divine Augustus" by Augustus is
+copyright (C) Thomas Bushnell, BSG.
\ No newline at end of file