LESSON 13

Concepts Related to Motion

  Chapter 12: 1068b 26-1069a 14

             1021. Things which are in one primary place are together in place, and those which are in different places are separate, and those whose extremities are together are in contact. And an intermediate is that at which something continuously changing according to its nature naturally arrives before it reaches the limit to which it is changing. That is contrary in place which is most distant in a straight line. That is subsequent which comes after a starting point (the order being determined by position or form or in some other way) and has nothing in the same genus between itself and that which it follows; for example, lines in the case of a line, and units in the case of a unit, or a house in the case of a house. But there is nothing to prevent something else from coming between. For that which follows something is subsequent and comes after something else; for one does not follow two, nor does [the first day of] the new moon follow the second. Again, what is subsequent and in contact is contiguous. And since every change is between opposites, and these are contraries and contradictories, and since there is no intermediate between contradictories, it is evident that an intermediate is between contraries. The continuous has something of the nature of the contiguous; and I call two things continuous when both have the same extremity in which they are in contact and are uninterrupted.

             1022. It is evident, then, that the continuous belongs to those things from which one thing results in virtue of their contact. And it is evident that the subsequent is the first of these; for things which are subsequent are not necessarily in contact, but what is in contact is subsequent. But if it is in contact it is not necessarily continuous. And in things in which there is no contact there is no natural coherence. The point, then, is not the same as the unit; for contact belongs to the former but not to the latter, but only successiveness, and there is an intermediate between the former but not between the latter.

COMMENTARY

             2404. He explains the terms which apply to motion, especially local motion. First (1021:C 2404), he explains them. Second (1022:C 2413), he draws a corollary from his remarks ("It is evident").

             He accordingly says, first (1021), that things which are "in one primary place," i.e., a proper place, are said to be together in place; for if some things are in one common place, they are not for this reason said to be together, for then all things which are contained in the circumference of the heavens would be said to be together.

             2405. Things which are in different places are said to be separate.

             2406. And those whose extremities are said to touch one another are said to be in contact; for example, two bodies whose surfaces are joined.

             2407. And an intermediate between two things is that at which it is natural for something that continuously changes to arrive before it reaches its limit; for example, if there is continuous motion from a to c, the thing being changed first arrives at b before it reaches c.

             2408. Again, that which is most distant in a straight line is contrary in place; for that which is most distant cannot be measured by a curved line, because an infinite number of unlike sections of circles can be drawn between two points, but there can be only one straight line between two points. Now a measure must be definite and fixed. And that which is most distant as to place admits of being above and below, which are the extremity and the center of the universe.

             2409. That is said to be subsequent which comes after some starting point, whether the order is determined by position or by form or in some other way; for example, two comes after one. And there must also be nothing of the same genus between that which is subsequent and that which it follows, as lines are subsequent to a line and units to a unit and a house to a house. But nothing prevents something of another genus from being an intermediate between two things one of which follows the other; for example, there may be one intermediate horse between two houses. In order to make the above distinction clear he adds that what is said to follow something must be subsequent and come after something. For one does not come after two, since it is first; nor does the first day of the new moon follow the second, but the other way around.

             2410. Then he says that the contiguous means what is subsequent and in contact with something else--for example, if two bodies are so related that one touches the other.

             2411. Then he says that, since every change is between opposites, and the opposites between which there is change are either contraries or contradictories, as has been shown (1008:C 2363), and since there is no intermediate between contradictories, it is evident that there is an intermediate only between contraries; for that which is intermediate is between the limits of a motion, as is clear from the definition given above. His introduction of this is timely; for since he said that those things are subsequent between which there is no intermediate, it was fitting that he should indicate between what things it is possible to have an intermediate.

             2412. Then he shows what the continuous is. He says that the continuous adds something to the continguous; for there is continuity when both of those things which are in contact and together have one and the same extremity, as the parts of a line are continuous in relation to a point.

             2413. It is evident (1022).

             Then he draws three corollaries from what has been said. The first is that continuity belongs to those things from which one thing naturally results in virtue of their contact; and this is because the continuous requires identical extremities.

             2414. The second corollary is that, of these three things--the subsequent, the contiguous and the continuous--the first and most common is the subsequent; for not everything that is subsequent is in contact, but everything which is in contact is subsequent or consecutive. For things which are in contact are arranged according to their position, and no one of them is an intermediate. Similarly, the contiguous is prior to and more common than the continuous, because, if a thing is continuous, there must be contact. For what is one must be together, unless perhaps plurality is understood in the phrase being together. For in that case the continuous would not involve being in contact. But the continuous must involve contact in the way in which something one is together. Yet if there is contact it does not follow that there is continuity; for example, if certain things are together it does not follow that they are one. But in things in which there is no contact "there is no natural coherence," i.e., natural union, which is a property of the continuous.

             2415. The third corollary is that the point and the unit are not the same, as the Platonists claimed when they said that the point is the unit having position. That they are not the same is evident for two reasons: first, because there is contact between points but not between units, which only follow each other; second, because there is always some intermediate between two points, as is proved in Book V of the Physics. But it is not necessary that there should be an intermediate between two units.

Book XII