LECTURE 1 (231 a 21-b 18)

NO CONTINUUM IS COMPOSED OF INDIVISIBLE PARTS

             750. The Philosopher has already treated the division of motion into its species and the unity and contrariety of motions and states of rest. Here in Book VI he intends to treat those things which pertain to the division of motion in respect to its quantitative parts.

             This discussion is divided into two parts. First he shows that motion, like all other continua, is divisible. Secondly, where he says, 'The present also is . . .' (233 b 33), he explains how motion is divided.

             The first part is divided into two parts. First he shows that no continuum is composed of indivisible parts. Secondly, where he says, 'It is evident, then . . .' (233 b 15), he shows that no continuum is indivisible.

             The first part is divided into two parts. First he shows that no continuum is composed of indivisible parts. Secondly (since these proofs seem to pertain more to magnitude) he shows that the same argument [ratio] applies to magnitude, motion, and time, where he says, 'The same reasoning applies . . .' (231 b 19).

             Concerning the first part he makes two points. First he summarizes certain definitions given above which he uses here to demonstrate his position. Secondly, where he says, 'For the extremities of . . .' (231 a 26), he proves his position.

             751. He says, therefore, first that if the definitions given above of continuity, contiguity, and succession are correct (namely, the continuous is that whose extremities are one; the contiguous is that whose extremities are together; and succession is in things between which there is no middle of the same genus), from these definitions it follows that it is impossible for a continuum to be composed of indivisible parts; for example, that a line be composed of points, the line being a continuum and the point indivisible. He adds these latter remarks lest anyone use the words 'point' and 'line' differently.

             752. Next where he says, 'For the extremities of . . .' (231 a 26), he proves his position.

             First he introduces two arguments to prove his position. Secondly, where he says, 'Nor can there be . . .' (231 b 12), he brings out certain things which can be questioned in these arguments.

             Concerning the first argument he makes two points. First he shows that a continuum is not composed of indivisible parts either by means of continuity or by contiguity. Secondly, where he says, 'Nor, again, can a point . . .' (231 b 6), he shows that a continuum is not composed of indivisible parts in succession.

             Concerning the first part he gives two arguments. The first is as follows.

             The parts which compose one thing, either by continuity or by contiguity, must have extremities which either are one or are together.

             But the extremities of points cannot be one. For an extremity is determined in respect to some other part. But in an indivisible thing there is not something which is an extremity and something else which is some other part.

             Likewise it cannot be said that the extremities of points are together. For there cannot be an extremity of an indivisible thing. An extremity is always other than that of which it is an extremity. But there cannot be two different things in that which is indivisible.

             Therefore it follows that a line cannot be composed of points, either by continuity or by contiguity.

             753. He gives the second argument where he says, 'Moreover if that which . . .' (231 a 29). The argument is as follows.

             If a continuum is composed of points, the points must be either continuous or contiguous to each other. And this same argument, namely, that a continuum is not composed of indivisible parts, also applies to all other indivisible things.

             The first argument is sufficient to prove that indivisible things cannot be continuous with each other.

             But he brings in another argument to prove that they cannot be contiguous. The argument is as follows. Whenever anything touches another, either the whole of one touches the whole of the other, or part of one touches part of the other, or part of one touches the whole of the other. But since that which is indivisible has no parts, it cannot be said that part of one touches part of the other, or that part of one touches the whole of the other. And thus, if two points touch, it is necessary that the whole touches the whole. But a continuum cannot be composed of two parts of which the whole of one touches the whole of the other. For every continuum has distinct parts, one of which is here and the other there. And in those things that have position, the continuum is divided into diverse and distinct positions. But things which touch in respect to the whole are not distinct in place or position. Therefore it follows that a line cannot be composed of points in contact.

             754. Next where he says, 'Nor, again, can a point . . .' (231 b 6), he proves that a continuum is not composed of indivisible parts in succession.

             A point is not successively related to another point in such a way that they can constitute a line. Nor is one 'now' successively related to another 'now' in such a way that they can constitute time. For one thing is in succession to another when there is no middle of the same genus between them, as was explained above. But the middle between two points is always a line. And if a line were composed of points, as was assumed, it would follow that the middle between two points is always another point. And likewise the middle between two 'nows' is time. Therefore, a line is not composed of points in succession, nor is time composed of 'nows' in succession.

             755. He gives his second main argument where he says, 'Again, if length . . .' (231 b 10). This argument is based on the other definition of continuity, which was given above at the beginning of Book III; namely, the continuous is that which is infinitely divisible. The argument is as follows.

             A line or a time is divided into that from which it is composed. Therefore, if each of these is composed of indivisible parts, it follows that they are divided into indivisible parts. But this is false. For no continuum is divisible into indivisible parts, for then it would not be infinitely divisible. Therefore, no continuum is composed of indivisible parts.

             756. Next where he says, 'Now can there be . . .' (231 b 12), he clarifies two things which were said above.

             The first point is that the middle between two points is a line, and that the middle between two 'nows' is time. He clarifies this as follows.

             If there are two points, they must differ in site [situs], otherwise they would not be two, but one. Moreover, they cannot touch each other, as was shown above. Hence it follows that they are at a distance and that there is some middle between them. But no other middle is possible except a line between points and a time between 'nows'.

             He proves this as follows. If between points there be some middle other than a line, it is clear that that middle is either divisible or indivisible. If it is indivisible, it must be distinct in site [situs] from each of the points. And since it does not touch, there must again be some other middle between that indivisible middle and the extremities. This goes on to infinity unless there be a divisible middle. On the other hand, if the middle between the two points is divisible, then it will be divisible into either indivisible or infinitely divisible parts. But it cannot be said that it is divided into indivisible parts. For then the same difficulty of how the divisible can be composed of indivisible parts will be repeated. Therefore it follows that the middle is divisible into infinitely divisible parts. But this is the definition [ratio] of continuity. Hence, that middle is a continuum. Moreover, there can be no other middle between two points except a line. Thus the middle between any two points is a line.

             For the same reason time is the middle between any two 'nows'. And it is the same in other cases of continuity.

             757. Next where he says, 'Moreover, it is plain . . .' (231 b 15), he clarifies the second point given above; namely, that every continuum is divisible into divisible parts.

             If it be granted that a continuum is divisible into indivisible parts, it would follow that two indivisible things touch each other so that they can constitute the continuum. For continuous things must have one extremity, as is clear from the definition of continuity. And the parts of a continuum must touch each other. For if the extremities are one, it follows that they are together, as was said in Book V. Therefore, since it is impossible for two indivisible things to touch each other, it is impossible for a continuum to be divided into indivisible parts.