Commentary Aristotle Physics
Lecture 5
He defines “contact,” “consecutiveness,” “continuity,” and other related things
Let us now proceed to define the terms “together” and “apart,” “in contact,” “between,” “consecutive,” “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 is “between” which a changing thing, if it changes continuously in a natural manner, naturally reaches before it reaches that to which it changes last. 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 thing—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.
The contrary in respect of place is what 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 “consecutive” 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 consecutive —for instance, 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 consecutive is consecutive to a particular thing and is something posterior: one is not consecutive to two, nor is the first day of the month to be second; in each case the latter is consecutive to the former. A thing that is consecutive and touches is contiguous.
Since all change is between opposites, while opposites include both the contrary and the contradictory, and there is no mean between contradictories, it is clear that there is a mean between contraries.
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 form a unity in virtue of their mutual contact. And in whatever way that which holds them together is one, so too will the whole be one, such as by a rivet or glue or contact or organic union.
It is obvious that, of these terms, “consecutive” is first in order of analysis, for that which touches is necessarily consecutive but not everything that is consecutive touches, and so consecutiveness is a property of things prior in definition, such as 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 generation, for the extremities must necessarily come into contact if they are to be naturally joined: things that are in contact are not all naturally joined, but where 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: points can touch, but units can only be consecutive.
Moreover, there can always be something between points (for all lines are intermediate between points), but 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,” “apart,” “contact,” “between,” “consecutive,” “contiguous,” and “continuous”; and we have shown in what circumstances each of these terms is applicable.
684. After dividing change and motion into its species, the Philosopher now begins to discuss the senses in which motion is said to be one and the senses in which motions are said to be contrary.
About this, he does two things: first, he establishes a background of preliminary notions that will be of use; second, he pursues his main objective, at “there are many senses” (227b3; [695]).
About the first, he does three things: first, he states his intention; second, he pursues it, at “things are said to be together in place” (226a21; [685]); third, he makes a summary, at “we have now defined” (227a32; [694]).
He says first that we must now define the terms “together,” “extraneous” or “separate,” “touching,” “intermediate,” “consecutive to,” “contiguous,” and “continuous,” and in what circumstances each of these terms is naturally applicable. The reason for positing these definitions now is that they will be used in later demonstrations, just as in the beginning of Euclid are posited definitions that serve as principles of later demonstrations.
685. Then, at “things are said to be together in place” (226a21), he carries out his plan.
First, he defines the terms he mentioned; second, he compares one to the other, at “it is obvious” (227a17; [692]).
About the first he does three things: first, he defines those that pertain to contact, that is, touching; second, those that pertain to consecutiveness, at that is “between” (226b23; [686]); third, those that pertain to continuum, at the “continuous” is a subdivision (227a10; [691]).
Since “together” occurs in the definition of “in contact,” the Philosopher defines it first (227a7) and says that those things are said to be together in place that are in one first place, meaning the first place of each one, that is, its own proper place. For things are said to be together not because they are in one common place, but in one proper place; otherwise, we should be able to say that all bodies are together, since they are all contained under the heavens.
He speaks of such things that are together in respect of place in distinction from those that are said to be together in time—a point we are not now discussing. Conversely, whatever things are in two different places are said to be separate or apart.
But “in contact” is said of things whose termini are together. The termini of bodies are surfaces, and of surfaces, lines, and of lines, points. Therefore, if two lines are in contact as to their termini, the two points of the two lines that are in contact will be contained under one point of the place containing them. From this, however, it does not follow that the thing in place is greater than the place, for point added to point does not make anything larger. And the same holds for the others.
686. Then, at that is “between” (226b23), he defines the things that pertain to consecutiveness.
About this, he does three things: first, he defines “between,” which is placed in the definition of “consecutive to”; second, he defines “consecutive to,” at a thing is “consecutive” (226b27; [689]); third, he draws a corollary, at “since all change” (226b34; [690]).
He says first (226b23) that “between” is what a naturally and uninterruptedly changing thing is apt to arrive at before it reaches the ultimate terminus of the motion into which it is being changed; for example, if something is changing from A to C through B, then, provided it is a continuous motion, it reaches B before C.
In some cases, there are a number of betweens to be traversed as you pass from one extreme to the other, as from black to white, between which there are many colors; but there must be at least three things involved, two of which are extremes, and one the between. Consequently, the between is what must be passed through before arriving at the terminus of a change, but the terminus of a change is a contrary; for it has already been stated that motion goes from contrary to contrary.
687. Because the definition of “between” made mention of continuity of motion, he now shows what continuous motion means.
Now, continuity of motion may be viewed from two aspects: first, from the time during which the motion occurs; second, from the thing through which the motion takes place—for example, the magnitude in local motion.
For a motion to be continuous, it is required that there be no interruptions in time, since even the slightest interruption of the motion as to time prevents the motion from being continuous.
But, on the side of the magnitude through which the motion passes, there can be slight variations without prejudice to the continuity of the motion. This is clear in crossings over streets, at which stones are placed slightly distant from each other, and over which a person passes from one side of the street to another without interrupting his motion. This, therefore, is what he says: continuity of motion is present if it leaves no gap or only the smallest possible gap in the thing, that is, when there is no interruption in the thing over which the motion passes or, if there is, it is very slight. But there cannot be the slightest interruption of time if the motion is to be continuous.
He explains how there can be a gap in continuous motion by adding that a motion will be continuous even if there is a gap in the material, as long as there is no time gap—for example, if, in playing the harp, one strikes the highest note immediately after having sounded the lowest and none of the intermediate ones. This is not a gap in time, but in the material in which the motion takes place. What has been said about the continuity of motion applies not only to local motion, but to all the others as well.
688. But, because it is not evident how the terminus of a local motion is a contrary, since one place does not seem to be contrary to another, he now gives an explanation. And he says that the contrary in respect of place is the greatest rectilinear distance, where greatest distance is taken in relation to the motion and the mobiles and the movers. For example, for the motion of heavy and light things, the distance from the center of the earth to the extremity of the sky is the greatest distance, while in regard to my motion and your motion, the greatest distance is the interval between where we start and where we intend to arrive.
What he means by the phrase in a straight line he explains by adding that the shortest line is definitely limited. To understand this, consider that the shortest distance between two points is a straight line, for between any two points, there is only one straight line. But there are any number of curved lines between two points—by “curved lines,” we mean the arcs of major or minor circles. Now, since every measure should be finite (otherwise, there would be no way of knowing the quantity of a thing, which is the purpose of measuring), the greatest distance between two objects is not measured by a curved line, but by a straight line, which is finite and determinate.
689. Then, at a thing is “consecutive” (226b27), he defines what is meant by “consecutive to” and a species of it, namely, “contiguous.”
And he says that two things are required in order that something be called consecutive to another.
One is that it be after the first and in a certain order: either according to position, as things that are in order in place; or according to species, as duality comes after unity; or in any way in which things can be in order, as according to virtue, according to dignity, according to knowledge, and so on.
The other requirement is that there not be anything of the same kind intervening between that which is consecutive and that to which it is consecutive. For example, one line is consecutive to another if there is no line between—likewise from one unit to another and one house to another. However, this does not forbid something else intervening. For example, an animal could be found between two houses.
He explains why he said, to which it is consecutive and after the beginning, by adding that whatever is said to be consecutive is so in respect to something else not as being prior to it, but as following it. For one is not said to be consecutive to two, or a new moon to a second new moon; rather, it is just the opposite.
Then he defines a certain species of consecutive called “contiguous.” And he says that not everything consecutive is also contiguous, but only when it is consecutive and in contact, such that there is nothing at all between—that is, nothing of the same genus or of any other genus.
690. Then, at “since all change” (226b34), he concludes from the foregoing that, since the between is that through which something is changed into what is final, and since every change is between opposites that are either contrary or contradictory, and there is no between in contradictories, it follows that it is between contraries that the between is found.
691. Then, at the “continuous” is a subdivision (227a10), he shows what a continuum is, and he says that it is a species of the contiguous. For when the terminus of two things in contact is one and the same, then something is continuous.
And the very word “continuum” denotes this. For “continuum” is derived from continere, meaning “to hold together”: when many parts are held together in a unit and, as it were, keep themselves together, then there is a continuum. But this cannot be while the endings are two, only when they are one.
From this, he further concludes that continuity can occur only in things from which a unity through contact is naturally apt to come about. For in whatever way a whole is naturally one and continuous, in the same way is a continuous unity formed from many things, whether by riveting, by gluing, or by any form of contact that makes one terminus for two parts, or even by being born of another, as fruit is born of a tree and forms a sort of continuum with it.
692. Then, at “it is obvious that” (227a17), he compares three of the foregoing with one another—namely, the consecutive to the continuous and the continuum.
About this, he does three things: first, he compares consecutiveness to contact; second, contact with continuum, at “and, if there is continuity” (227a21; [693]); third, he draws a corollary, at “hence, if, as some say” (227a27; [694]).
He says first (227a17) that it is clear why, among these three, consecutiveness is naturally first in the order of nature, for in the cases of contact, there is always consecutiveness, since there must be an order, at least of position, among things that are in contact. But not all cases of consecutiveness involve contact, for an order can exist among things in which there is no contact, as in substances separated from matter. Hence, consecutiveness is present in things that are prior in definition, for it is found in numbers, in which there is no contact (which is present only among continua). Numbers, however, are prior to continuous quantities in definition, for they are more simple and more abstract.
693. Then, at “and, if there is continuity” (227a21), he compares in contact with continuous and says that, for the same reason, in contact is prior to continuous, because a thing must be in contact if it is continuous, but it does not necessarily follow that, if it is in contact, it is continuous.
And he proves this from the definitions of the two. It is not necessary that the endings of things be one (which is implied in the account of continuum) if they are together (which is implied in the account of contact). But, on the other hand, if the endings are one, they must be together, for what is one is together unto itself.
However, if “together” implies a relationship between distinct things, then things that are together are not one; and, according to this, continua are not in contact. But they are if we do not speak so precisely. Hence, he concludes that natural junction—that is, continuity, in which one part is joined to another at one terminus—is last in coming to be, in the sense that what is specific comes to be after what is general, as animal comes to be before man. And, therefore, I say that natural junction is last, because things must mutually touch if their extremities are naturally joined, that is, naturally united. However, it is not necessary that all things that touch be naturally joined to one another. But, in regard to things that cannot touch, it is clear that junction, or continuity, is impossible.
694. Then, at “hence, if, as some say” (227a27), he draws a corollary from the preceding: if point and unit have an independent existence of their own, as some say who suppose a separated existence for mathematical objects, it follows that unity and the point are not the same.
And this is clear for two reasons.
First, points are present in things that are capable of mutual contact, and certain things touch at points; but, in units, contact is never found, but only consecutiveness.
Second, there must be something existing between two points, but there is not necessarily anything between two unities. For it is evident that, between the two unities that form two and the very first unity, which is one, there is nothing intermediate.
Finally, at “we have now defined” (227a32), he summarizes what has been said, and this is clear in the text.