¹The infinite is either (a) that which cannot be traversed because it is not its nature to be traversed (just as sound is by nature invisible); or (b) that which admits of an endless traverse; or (c) scarcely admits of traverse; or (d) which, though it would naturally admit of traverse or limit, does not do so. 1066bFurther, it may be infinite in respect of addition or of subtraction or of both. That the infinite should be a separate independent entity,² and yet imperceptible, is impossible. For if it is neither magnitude nor plurality, but infinity itself is the essence of it, and not merely an accident, it must be indivisible; because that which is divisible is either magnitude or plurality. And if it is indivisible it cannot be infinite, except in the same way as sound is invisible. But this is not what people mean by infinite; and it is not the infinite in this sense that we are investigating, but the infinite in the sense of the untraversable. Again, how can the infinite exist independently unless number and magnitude, of which infinity is an attribute, also exist independently?³ And further, if the infinite is accidental, it cannot, qua infinite, be an element of things; just as the invisible is not an element of speech, although sound is invisible. It is clear also that the infinite cannot exist actually. Otherwise any part of it which we might take would be infinite; for infinity and the infinite are the same, if the infinite is substance and is not predicated of a subject. Therefore it is either indivisible, or if it is partible, the parts into which it is divisible are infinite. But the same thing cannot be many infinites; for just as a part of air is air, so a part of the infinite will be infinite, if the infinite is a substance and principle. Therefore it is impartible and indivisible. But this is impossible of the actually infinite, because it must be some quantity. Therefore infinity is an accidental attribute. But if so, as we have said, it cannot be it that is a principle, but that of which it is an accident: air⁴ or “the even.” ⁵ The foregoing inquiry is general; but what follows will show that the infinite does not exist in sensible things. If the definition of a body is “that which is bounded by surfaces,” then no body, whether sensible or intelligible, can be infinite nor can there be any separate and infinite number, since number or that which involves number is numerable. This is clearly shown by the following concrete argument. The infinite can neither be composite nor simple. For (a) it cannot be a composite body if the elements are limited in number⁶; for the contraries must be equal, and no one of them must be infinite; for if the potency of one of the two corporeal elements is in any way inferior, the finite element will be destroyed by the infinite. And every element cannot be infinite, because body is that which has extension in all directions, and the infinite is that which is extended without limit; so that if the infinite is corporeal it will be infinite in all directions.⁷ Nor (b) can the infinite be any simple body; neither, as some⁸ hold, something which is apart from the elements and from which they suppose the elements to be generated (for there is no such body apart from the elements; everything can be resolved into that of which it consists, but we do not see things resolved into anything apart from the simple bodies), 1067anor fire nor any other element. Apart from the question of how any of them could be infinite, the All, even if it is finite, cannot be or become any one of the elements, as Heraclitus says⁹ all things at certain times become fire. The same argument applies as to the One which the physicists posit besides the elements; for all change proceeds from the contrary, e.g. from hot to cold.¹⁰ Again, a sensible body is in some region, and the region of the whole and of the part (e.g. of the earth) is the same.¹¹ Therefore if the infinite body is homogeneous, it will be immovable or will always be in motion¹²; but this is impossible, for why should there be rest or motion below rather than above or in any other region? E.g., if there were a clod, in what region would it move or be at rest? The region proper to the body which is homogeneous with the clod is infinite. Then will the clod occupy the whole of that region? How can it? Then what of its rest or motion? It will either rest everywhere—in which case it cannot move—or move everywhere; in which case it cannot rest.¹³ And if the whole is not alike throughout, the regions proper to its parts are unlike also; and (a) the body of the whole is not one, except in virtue of contact; (b) the parts will be either finite or infinite in kind. Finite they cannot be, for then those of one kind would be infinite¹⁴ and those of another would not (if the whole is infinite); e.g., fire or water would be infinite. But such a condition would involve the destruction of the contraries. But if the parts are infinite¹⁵ and simple, the regions proper to them are infinite and the elements will be infinite. And since this is impossible,¹⁶ the regions are finite¹⁷ and the whole must be finite. In general, there cannot be an infinite body and a place for bodies if every body which is sensible has either weight or lightness; for it will have to move either towards the center or upwards, and the infinite—either the whole or the half—cannot do either; for how can you divide it? How can the infinite be part up and part down, or part extreme and part center? Further, every sensible body is in some place, and of place there are six kinds,¹⁸ but these cannot exist in an infinite body. In general, if an infinite place is impossible, so is an infinite body; because that which is in a place is somewhere, and this means either up or down or one of the other kinds of place, and each of these is a limit. The infinite is not the same in the sense that it is one nature whether it applies to magnitude or to motion or to time; the posterior is derived from the prior sense, e.g. motion is called infinite in virtue of the magnitude involved when a thing is moved or changed or increased, and time is so called on account of motion.¹⁹

1067bThat which changes either changes accidentally, as when “the cultured” walks; or is said to change in general because something in it changes, as in the case of things which change in their parts; the body becomes healthy because the eye does. But there is something which is moved directly per se, i.e. the essentially movable. The same applies to that which moves, for it moves sometimes accidentally, sometimes partially, and sometimes per se. There is something that moves directly, and something that is moved; and also a time in which, and something from which, and something into which it is moved. But the forms and modifications and place into which moving things are moved are immovable; e.g. knowledge and warmth. It is not warmth that is motion, but the process of warming. Non-accidental change is not found in all things, but only between contraries and intermediates and contradictories. We can convince ourselves of this by means of induction. That which changes changes either from positive into positive, or from negative into negative, or from positive into negative, or from negative into positive. By “positive” I mean that which is denoted by an affirmation. Thus there must be three forms of change; for that which is from negative into negative is not change, because they are neither contraries nor contradictories, since they entail no opposition. The change from the negative into its contradictory positive is generation—absolute change absolute generation, and qualified change qualified generation; and the change from the positive to the negative is destruction—absolute change absolute destruction, and qualified change qualified destruction.²⁰ Now if “what is not” has several meanings, and neither that which implies a combination or separation of terms,²¹ nor that which relates to potentiality and is opposed to unqualified Being, admits of motion (“not-white” or “not-good,” however, admits of motion accidentally, because “not-white” may be a man; but that which is “not so-and-so” in an absolute sense does not admit of it at all), then “what is not” cannot be moved. If this is so, generation cannot be motion; for it is “what is not” that is generated. For even if the generation is in the highest degree accidental, still it is true to say that not-being is predicable of that which is generated absolutely. And the argument applies similarly to rest. Thus not only do these difficult conclusions follow, but also that everything which is moved is in a place, whereas “what is not” is not in a place; for then it would be somewhere. Nor is destruction motion; for the contrary of motion is motion or rest, but the contrary of destruction is generation. 1068aAnd since every motion is a kind of change, and the three kinds of change are those which we have described,²² and of these those which relate to generation and destruction are not motions, and these are the changes between contradictories, the change from positive to positive must alone be motion. The subjects are either contraries or intermediates (for privative terms may also be regarded as contraries) and are denoted by a positive term—e.g. “naked” or “toothless” or “black.”