Commentary Aristotle Physics
Lecture 3
The divisibility of time and of magnitude follow on one another
And, if length and motion are thus indivisible, it is neither more nor less necessary that time also be similarly indivisible—that is to say, be composed of indivisible moments.
For if the whole distance is divisible and an equal velocity will cause a thing to pass through less of it in less time, the time must also be divisible; conversely, if the time in which a thing is carried over the section A is divisible, this section A must also be divisible.
And, since every magnitude is divisible into magnitudes—for we have shown that it is impossible for anything continuous to be composed of atoms, and every magnitude is continuous—it necessarily follows that the quicker of two things traverses a greater magnitude in an equal time, an equal magnitude in less time, and a greater magnitude in less time, in conformity with the definition sometimes given of “the quicker.”
Suppose that A is quicker than B. Now, since of two things that which changes sooner is quicker, in the time ZI, in which A has changed from C to D, B will not yet have arrived at D, but will be short of it, such that, in an equal time, the quicker will pass over a greater magnitude.
More than this, it will pass over a greater magnitude in less time: in the time in which A has arrived at D, the slower B has arrived, let us say, at E. Then, since A has occupied the whole time ZI in arriving at D, it will have arrived at T in less time than this, say ZK. Now, the magnitude CT that A has passed over is greater than the magnitude CE, and the time ZK is less than the whole time ZI, such that the quicker will pass over a greater magnitude in less time.
And from this, it is also clear that the quicker will pass over an equal magnitude in less time than the slower.
For since it passes over the greater magnitude in less time than the slower, and (regarded by itself) passes over LM the greater in more time than LX the lesser, the time PR, in which it passes over LM, will be more than the time PS, in which it passes over LX.
Thus, the time PR being less than the time PH, in which the slower passes over LX, the time PS will also be less than the time PH, for it is less than the time PR, and that which is less than something else that is less than a thing is also itself less than that thing.
Hence, it follows that the quicker will traverse an equal magnitude in less time than the slower.
Again, since the motion of anything must always occupy either an equal time or less or more time in comparison with that of another thing, and since a thing is slower if its motion occupies more time and of equal velocity if its motion occupies an equal time, but the quicker is neither of equal velocity nor slower, it follows that the motion of the quicker can occupy neither an equal time nor more time.
It can only be, then, that it occupies less time, and thus we get the necessary consequence that the quicker will pass over an equal magnitude (as well as a greater) in less time than the slower.
And, since every motion is in time and a motion may occupy any time, and the motion of everything that is in motion may be either quicker or slower, both quicker motion and slower motion may occupy any time.
And this being so, it necessarily follows that time also is a continuum. By “continuum,” I mean that which is divisible into divisibles that are infinitely divisible, and if we take this as the definition of “continuum,” it follows necessarily that time is a continuum.
For since it has been shown that the quicker will pass over an equal magnitude in less time than the slower, suppose that A is quicker and B slower, and that the slower has traversed the magnitude CD in the time ZI.
Now, it is clear that the quicker will traverse the same magnitude in less time than this—let us say in the time ZT. Again, since the quicker has passed over the whole CD in the time ZT, the slower will in the same time pass over CK, say, which is less than CD. And since B, the slower, has passed over CK in the time ZT, the quicker will pass over it in less time, such that the time ZT will again be divided. And if this is divided, the magnitude CK will also be divided, just as CD was; and again, if the magnitude is divided, the time will also be divided.
And we can carry on this process forever, taking the slower after the quicker and the quicker after the slower alternately, and using what has been demonstrated at each stage as a new point of departure, since the quicker will divide the time and the slower will divide the length. If, then, this alternation always holds good, and involves a division at every turn, it is evident that all time must be continuous. And, at the same time, it is clear that all magnitude is also continuous, for the divisions of which time and magnitude respectively are susceptible are the same and equal.
Moreover, the current popular arguments make it plain that, if time is continuous, magnitude is continuous also, inasmuch as a thing passes over half a given magnitude in half the time taken to cover the whole—in fact, without qualification it passes over a less magnitude in less time, for the divisions of time and of magnitude will be the same.
766. After showing that it is for the same reason that a magnitude and a motion traversing it would be composed of indivisibles, the Philosopher shows the same for time and magnitude.
And the treatment falls into two parts:
in the first, he shows that division of time follows upon division of magnitude, and vice versa;
in the second, that the infinity of one follows upon the infinity of the other, at “and, if either is infinite” (233a17; [777]).
About the first, he does two things:
first, he states his proposition;
second, he demonstrates it, at “for if the whole” (232a19; [767]).
He says first (232a18) that time, too, is divisible and indivisible, and composed of indivisibles, just as length and motion are.
767. Then, at “for if the whole” (232a19), he proves his proposition, giving three reasons:
the first of which is based on things equally fast;
the second is based on the faster and the slower, at “and, since every magnitude” (232a23; [768]);
the third uses one and the same mobile, at “moreover, the current popular” (233a10; [776]).
He says first (232a19) that a mobile that is as fast as another traverses a smaller magnitude in less time. Therefore, let us take a divisible magnitude that a mobile traverses in a given time. It follows that an equally fast mobile traverses part of that magnitude in less time. Consequently, the given time must be divisible. Conversely, if the time is given as divisible and a given mobile is in motion over a given magnitude, it follows that a mobile equally fast traverses a smaller magnitude in less time, which is part of the whole time. Consequently, the magnitude A is divisible.
768. Then, at “and, since every magnitude” (232a23), he proves the same thing with two mobiles, one of which is faster and the other slower.
First, he lays down some presuppositions to be used in proving his proposition;
second, he proves the proposition, at “and, since every motion” (232b20; [774]).
About the first, he does two things:
first, he explains how the faster and the slower compare with regard to being moved over a larger magnitude;
second, he shows how they compare with regard to being moved over an equal magnitude, at “and from this” (232b5; [772]).
About the first, he does two things:
first, he states his proposition, repeating something mentioned previously but needed for the demonstrations that follow;
second, he demonstrates his proposition, at “suppose that A” (232a27; [770]).
769. He thus repeats (232a23) that every magnitude is divisible into magnitudes. And this is evident from the previous conclusion that it is impossible for a continuum to be composed of atoms, that is, indivisibles; and every magnitude is a kind of continuum. From these, it follows that a faster body is moved through a greater magnitude in equal time and even in less time. Indeed, that is the way in which some have defined the faster, that it is moved more in equal and even in less time.
770. Then, at “suppose that A” (232a27), he proves his two presuppositions:
first, that a faster thing is moved a greater distance in equal time;
second, that it is moved a greater distance in less time, at “more than this” (232a31; [771]).
He first says, therefore (232a27), to let A and B be two mobiles, of which A is faster than B, and let CD be the magnitude traversed by A in time ZI. Now, let B, which is slower, and A, which is faster, pass over the same magnitude, and let them start together.
Therefore, under these conditions, the following argument is given. The faster is the one moved more in equal time; but A is faster than B. Therefore, when A shall have arrived at D, B will not have arrived at D (which is the terminus of the magnitude), but will be short of it; yet it will have covered part of the magnitude. Now, since every part is less than the whole, what remains is that A is moved through a greater distance in time ZI than B, which has traversed part of the magnitude in the same time. Consequently, the faster traverses more distance in equal time.
771. Then, at “more than this” (232a31), he shows that the faster traverses more space in less time. It was said that, at the time when A arrived at D, B, which is slower, was still distant from D. Let us grant, therefore, that B arrived at E when A arrived at D. Now, since every magnitude is divisible, let us divide the remaining magnitude ED (which is how much the faster exceeds the slower) at the point T.
It is evident that the magnitude CT is less than CD. But one and the same mobile traverses a smaller magnitude in less time. Therefore, since A arrived at D in the total time ZI, it arrived at T in less time. Let that less time be ZK.
Then the argument continues: the magnitude CT, which A traversed, is greater than the magnitude CE, which B traversed. But the time ZK, in which A traversed CT, is less than the whole time ZI, in which the slower B traversed CE. Therefore, it follows that the faster traverses a larger space in less time.
772. Then, at “and from this” (232b5), he shows how the faster compares with the slower in regard to being moved through an equal magnitude.
First, he states his intention;
second, he proves his proposition, at “for since it passes” (232b6).
He says first (232b5) that, from the foregoing, it could be clear that a faster thing traverses an equal space in less time.
Second, at “for since it passes” (232b6), he proves this with two arguments, to the first of which he prefaces two facts.
One of these has already been proved: a faster thing traverses a greater magnitude in less time than a slower.
The second is self-evident: one and the same mobile traverses a greater magnitude in a greater time than it does in a shorter time.
For let the mobile A, which is faster, traverse the magnitude LM in time PR, and the part LX of the magnitude in less time PS, which is less than PR, in which it traverses LM, just as LX is less than LM.
From the first supposition, he takes it that the whole time PR, in which A traverses the entire magnitude LM, is less than time H, in which B (which is slower) traverses the smaller magnitude LX. For it was said that a faster object traverses a greater magnitude in less time.
With this background, he proceeds to his argument. The time PR is less than time H, in which B (the slower) traverses magnitude LX; moreover, time PS is less than time PR. Therefore, it follows that time PS is less than time H, for what is less than the lesser is less than the greater. Therefore, since it was granted that the faster traverses magnitude LX in the time PS and the slower traverses the same LX in time H, it follows that the faster traverses an equal magnitude in less time.
773. Then, after these preliminaries, he gives his second argument, at “again, since the motion” (232b14), which is this. A thing that traverses an equal magnitude along with another mobile is moved through that magnitude either in equal time or less or more. If it is moved through that equal magnitude in greater time, it is slower, as was proved above; if it is moved in equal time through the equal magnitude, it is equally fast, as is self-evident. Therefore, since what is faster is neither slower nor equally fast, it follows that it is moved through an equal magnitude neither in more time nor in equal time. Therefore, it is moved through it in less time.
Thus, we have proved that, necessarily, the faster traverses an equal magnitude in less time.
774. Then, at “and, since every motion” (232b20), he proves the proposition that the same reasoning proves that both time and magnitude are always divided into divisibles, or are composed of indivisibles.
About this, he does three things:
first, he lays down premises to be used in the proof;
second, he states his proposition, at “and this being so” (232b23; [775]);
third, he proves it, at “for since it has been” (232b26; [775]).
Therefore (232b20), he lays down the premises that every motion exists in time, which was proved in book 4, and that motion is possible in any time, which is evident from the definition of time given in book 4.
The second is that whatever is being moved can be moved quicker or slower—that is, among mobiles, some are moved more quickly and some more slowly. But this statement seems false, because the speeds of motions are fixed in nature; for there is one motion so fast that none could be faster, namely, the motion of the first mobile.
In reply, it must be said that we can speak of the nature of anything in two ways: either according to its general account, or insofar as it is applied to its proper matter.
Now, there is nothing to forbid something that is possible in the light of a thing’s general definition to be prevented from happening when application is made to some definite matter; for example, it is not the general definition of the sun that precludes many suns, but the fact that the total matter of this nature is contained under one sun. Likewise, it is not the general nature of motion that prevents the existence of a speed greater than any given speed; rather, it is the particular powers of the mobiles and movers.
Now, Aristotle is here discussing motion from the viewpoint of its general account, without application to particular movers and mobiles. Indeed, he frequently uses such propositions in this sixth book, and they are true if you limit yourself to a general consideration of motion, but they are not necessarily true if applied to particular mobiles.
Likewise, it is not contrary to the account of magnitude that every magnitude be divisible into smaller ones. Therefore, in the present book, he goes on the assumption that it is possible to take a magnitude smaller than any given magnitude, even though, in every particular nature, there is always a minimum magnitude, since each nature has limits of largeness and smallness, as was mentioned even in book 1.
From these two premises, he infers a third one: in any given time, motions faster and slower than a given motion are possible.
775. Then, at “and this being so” (232b23), from the foregoing, he argues for his proposition. And he says that, since the foregoing are true, time must be a continuum, and thus divisible into parts that are further divisible. For if that is the definition of a continuum, then, if a magnitude is a continuum, time must be continuous, because the division of time follows upon division of magnitude, and vice versa.
Then, at “for since it has been” (232b26), he proves the proposition, namely, that time and magnitude are divided in a similar way. For since we have shown that a faster thing traverses an equal space in less time, let A be the faster and B the slower, and let B be moved more slowly through magnitude CD in time ZI.
It is plain that A, which is faster, traverses the same magnitude in less time, ZT.
But again, since A, which is faster, has traversed the entire magnitude CD in time ZT, then B, the slower, traversed a smaller magnitude CK in the same time. And, because B traversed the magnitude CK in time ZT, A traversed the same magnitude in even less time. Thus, the time ZT will be further divided. And when it is, the magnitude CK will also be divided, because the slower traverses less space in part of that time. And, if the magnitude is divided, the time also is divided, because the faster will cover that part of the magnitude in less time. So, we continue in this manner, taking a slower mobile after the motion of the faster, and after the slower taking the faster, and making use of the statements already proved, namely, that the faster traverses an equal space in less time and that the slower traverses a smaller magnitude in equal time. For by thus taking what is faster, we will divide the time, and by taking what is slower, we will divide the magnitude.
Therefore, it is true that such a conversion can be made by going from the faster to the slower and from the slower to the faster. And if such switching causes the magnitude and then the time to be divided, then it will be clear that time is a continuum, and thus divisible into times that are further divisible, and the same for magnitude, since both time and magnitude will receive the same and equal divisions, as we have already shown.
776. Then, at “moreover, the current” (233a10), he gives a third reason to show that magnitude and time are correspondingly divided. But this time we shall consider one and the same mobile. And he says that it is clear from the reasons customarily given that, if time is a continuum—that is, divisible into parts that are further divisible—then a magnitude is likewise divisible, because one and the same mobile in uniform motion, since it traverses the whole magnitude in a given time, will traverse half in half the time, and a smaller part in less than half the time. And the reason that this happens is that time is divided as magnitude is.