LECTURE 5 (226 b 19-227 b 2)

HE DEFINES 'CONTACT', 'SUCCESSION', 'CONTINUITY', AND OTHER RELATED THINGS

             684. After the Philosopher has divided mutation and motion into their species, he proceeds here to determine the unity and contrariety of motion in its species.

             Concerning this he makes two points. First he sets forth certain things which are required by what follows. Secondly, where he says, 'There are many senses . . .' (227 b 3), he takes up the main problem.

             Concerning the first part he makes three points. First he states his intention. Secondly, where he says, 'Things are said to be together in place . . .' (226 b 21), he works out the problem. Thirdly, where he says, 'We have now defined . . .' (227 a 34), he summarizes.

             He says, therefore, first that we must next state what is meant by 'together', 'apart' or 'separated', 'contact', 'middle', 'succession', 'contiguity', 'continuity', and in what way such things are natural.

             He sets these definitions down here because he uses them in the demonstrations which follow throughout the whole book; just as in the beginning of Euclid definitions are given which are the principles of the subsequent demonstrations.

             685. Next where he says, 'Things are said to be together in place . . .' (226 b 21), he works out the problem.

             First he defines the things which were proposed. Secondly, where he says, 'It is obvious . . .' (227 a 17), he compares them to each other.

             Concerning the first part he makes three points. First he defines those things which pertain to 'contact'. Secondly, where he says, 'That which a changing thing . . .' (226 b 24), he defines those things which pertain to 'succession'. Thirdly, where he says, 'The "continuous" is a subdivision . . .' (227 a 10), he defines those things which pertain to 'continuity'.

             Since 'together' is included in the definition of 'contact', he first defines 'together'. He says that things are said to be together in place when they are in one primary place. By the primary place of anything he means its proper place. For things are said to be together because they are in one proper place, but not because they are in one common place. Otherwise one could say that all bodies are together because they are all contained under the heavens.

             Furthermore he is talking here of things being together in place rather than things being together in time. This latter point is not presently under discussion.

             Contrariwise, things are said to be 'separate' or 'apart' when they are in different places.

             Furthermore, things are said to be in 'contact' when their extremities are together. The extremities of bodies are surfaces, the extremities of surfaces are lines, and the extremities of lines are points. Therefore, if there be two lines touching in their extremities, then two points of the two touching lines will be contained under one point of the containing place. But because of this it does not follow that that which is located is greater than its place. For a point added to a point does not produce anything greater. And the same argument holds in the other cases.

             686. Next where he says, 'That which a changing thing . . .' (226 b 24), he defines those things which pertain to 'succession'.

             Concerning this he makes three points. First he defines 'middle', which is included in the definition of 'succession'. Secondly, where he says, 'A thing is "in succession" . . .' (226 b 34), he defines 'succession'. Thirdly, where he says, 'Now every change implies . . .' (227 a 7), he infers a corollary.

             He says, therefore, first that 'middle' is that which is naturally arrived at first when something is changed continuously by nature. It is not the ultimate terminus of motion to which something is being moved. For example, if something is changed from A to C through B, then, provided the motion is continuous, it arrives first at B rather than C.

             There can be a middle in many ways. For between two extremes there can be many intermediates. Thus between black and white there are many intermediate colours. But there must be at least three things: two extremes and one intermediate. Therefore, the middle is that through which one arrives at the terminus of a mutation; but the terminus of the mutation is a contrary. For it was said above that motion is from contrary to contrary.

             687. Since the continuity of motion is included in the definition of 'middle', he next explains what it means to be moved continuously.

             The continuity of motion can be understood in two ways--either in respect to the time in which a thing is moved or in respect to that through which a thing is moved, for example, magnitude in local motion.

             Hence, in order for motion to be continuous, it is necessary that there be no interruption in time. For if the motion be interrupted in time, no matter how briefly, it is not continuous.

             But in respect to the magnitude through which a motion occurs, there can be a small interruption without violating the continuity of motion. This is clear in the case of crossing a road in which the stones are placed a small distance from each other. By means of these stones a man crosses from one part of the road to another with a continuous motion.

             Therefore, he says that a thing is moved continuously when there is either no interruption or only a very small one. That is, in respect to that through which a thing is moved there is no interruption; or, if there is one, it is very small. But if motion is continuous, there cannot be even the slightest interruption of time.

             Furthermore he explains how a thing can lack continuous motion by adding that there is nothing to prevent a thing from being moved continuously if there is a break in the thing--but not in time. Thus when one plays the cithara, one sounds the highest note immediately after the lowest note, leaving out the intermediate notes. This is not a break in time but in the thing in which there is motion.

             What has been said about the continuity of motion must be understood of both local motion and the other motions.

             688. But it is not clear how the extreme in local motion is a contrary. For place does not seem to be contrary to place. Hence he also clarifies this point.

             He says that that which is at a maximum distance in respect to a straight line is a contrary in respect to place. This maximum distance must be understood in relation to the motion, the mobile object, and the mover. Thus, as related to the motion of heavy and light things, the maximum distance in place is the centre and extremity of the heavens in respect to us. But in respect to your motion or mine, that place to which we intend to go is at a maximum distance from that place from which we began to be moved.

             He explains what he means by 'straight line' by adding, 'The shortest line is definitely limited, and that which is definitely limited constitutes a measure' (226 b 32-33).

             To understand this we must consider that the minimum distance between any two designated points is a straight line. There can be only one such line between two points. But since two curved lines are the arcs of major or minor circles, the curved lines between two points can be multiplied to infinity. Furthermore, every measure must be finite (otherwise it could not designate a quantity, which is the function of a measure). Hence, the maximum distance between two things cannot be measured by a curved line, but only by a straight line, which is finite and determinate.

             689. Next where he says, 'A thing is "in succession" . . .' (226 b 34), he defines 'succession' and one of its species, namely, contiguity.

             He says that two things are required for a thing to be in succession to another.

             The first is that it be after the beginning in some order; either in respect to position, as in things which have an order in place, or in respect to species, as duality is after unity, or in respect to any other kind of determinate order, like power, or dignity, or knowledge, etc.

             The second requirement is that, between that which succeeds and that which is succeeded, there be no intermediate of the same genus. Thus one line succeeds another if there is no other line between them. And likewise a unity succeeds a unity, and a house a house. But when something succeeds another, there is nothing to prevent there being an intermediate of another genus between them. Thus an animal might be in the middle between two houses.

             Furthermore, he explains why he says, '. . . after the beginning . . .' (226 b 34), and '. . . that to which it is in succession . . .' (227 a 2). For he adds that whatever is in succession is in succession in respect to something, not as prior, but as posterior. For we do not say that one succeeds two, nor that the new moon succeeds the second, but vice versa.

             Next he defines a species of 'succession' which is called 'contiguity'. He says that not everything which is in succession is contiguous. Rather this occurs only when two things in succession touch, so that there is no intermediate either of the same genus or of any other genus.

             690. Next where he says, 'Now every change implies . . .' (227 a 7), he concludes from the above that every middle is between contraries in some way. For a middle is that through which something is moved to an extremity. And every mutation is between opposites which are either contraries or contradictories. However there is no middle between contradictories.

             691. Next where he says, 'The "continuous" is a subdivision . . .' (227 a 10), he explains what 'continuity' is.

             He says that 'continuity' is a species of 'contiguity'. For when two things which touch have one and the same terminus, they are said to be continuous. This indeed is what the word 'continuous' signifies. For 'continuous' is derived from 'containing'. When, therefore, many parts are contained in one, and are held together as it were, then there is a continuum. This cannot occur when there are two extremities, but only when there is one.

             From this he concludes further that there can be continuity only in those things which naturally become one by contact.

             For the same reason a whole is in itself one and continuous because one continuum results from many things. This occurs either by being nailed together; or by being brought into each other; or by any kind of touching such that two termini become one; or even by something being born as naturally joined to another, as fruit is born to a tree and in some way is continuous with it.

             692. Next where he says, 'It is obvious that . . .' (227 a 17), he compares to each other the three main points discussed above; namely, succession, contact, and continuity.

             Concerning this he makes three points. First he compares succession to contact. Secondly, where he says, 'And if there is continuity . . .' (227 a 21), he compares contact to continuity. Thirdly, where he says, 'Hence, if as some say . . .' (227 a 27), he infers a corollary.

             He says, therefore, first that in the order of nature succession is first among these three. It is called prior because it does not follow upon the other two. For everything that is in contact must be in succession, because between things which touch each other there must be some order, at least in position. But it is not necessary that everything which is in succession be touching. For there can be order in things which do not touch, as in things separated from matter. Hence succession is found in things which are prior according to reason [ratio]. For it is found in numbers in which there is no touching. This latter is found only in continuous things. Moreover, according to reason [ratio] numbers are prior to continuous quantities as being more simple and more abstract.

             693. Next where he says, 'And if there is continuity . . .' (227 a 21), he compares contact to continuity.

             He says that for the same reason contact is prior to continuity. For if something is continuous, it is necessary that it is touching. But if something is touching, it is not necessary that it is continuous.

             He proves this by the definition [ratio] of each of these things. For it is not necessary that the extremities of two things are one (which is the definition [ratio] of continuity) if they are together (which is the definition [ratio] of contact). But conversely, it is necessary that, if the extremities are one, they are together. The reason [ratio] for this is that that which is one is together with itself.

             However, if one uses 'together' to mean a relation of distinct things, then such things cannot be one. And accordingly things which are continuous cannot be in contact, but are taken in common. Hence he concludes that a continuity in which one part is included in another in one terminus is the ultimate result in the order of generation. Thus the special is posterior to the common, as animal is generated before man. Hence I call this the 'ultimate inclusion' because it is necessary that things touch each other if their extremities are naturally united. But it is not necessary that all things which touch each other be naturally united to each other. And it is clear that in things in which there cannot be contact there cannot be continuity.

             694. Next where he says, 'Hence, if as some say . . .' (227 a 27), he draws a corollary from the above. If unity and 'point' are separated, as some say who hold that mathematicals have a separated existence, it follows that unity and 'point' are not the same.

             He shows this with two arguments. The first is that points are in things whose nature it is to touch, and these things touch each other in respect to points. However in unities there is not found any contact, but only succession. The second argument is that between two points there is a middle. For every line is a middle between two points. But between two unities it is not necessary that there be any middle. For it is clear that there is no middle between the two unities which constitute duality and the very first unity.

             Finally, where he says, 'We have now defined . . .' (227 a 34), he summarizes what has been said. The meaning of this text is clear.