Commentary Aristotle Physics
Lecture 9. Rules for the comparison of motions
Now, wherever there is a mover, its motion always acts upon something, is always in something, and always extends to something.
By “is always in something,” I mean that it occupies a time, and by “extends to something,” that it involves the traversing of a certain amount of distance.
For at any moment when a thing is causing motion, it also has caused motion, such that there must always be a certain amount of distance that has been traversed and in a certain amount of time.
If force A has moved B a distance C in a time D, then, in the same time, the same force A will move half B twice the distance C, and in half D it will move half B the whole distance for C. Thus the rules of analogy will be observed.
Again, if a given force move a given weight a certain distance in a certain time and half the distance in half the time, then half the motive power will move half the weight the same distance in the same time.
Let E represent half the motive power A, and Z half the weight B. Then the ratio between the motive power and the weight in the one case is similar and proportionate to the ratio in the other, such that each force will cause the same distance to be traversed in the same time.
But, if E moves Z a distance C in a time D, it does not necessarily follow that E can move twice Z half the distance C in the same time.
If, then, A moves B a distance C in a time D, it does not follow that E, being half of A, will in the time D or in any fraction of it cause B to traverse a part of C, the ratio between which and the whole of C is proportionate to that between A and E (whatever fraction E may be of A).
In fact, it might well be that it will cause no motion at all; for it does not follow that, if a given motive power causes a certain amount of motion, half that power will cause motion either of any particular amount or in any length of time.
Otherwise, one man might move a ship, since both the motive power of the ship haulers and the distance that they all cause the ship to traverse are divisible into as many parts as there are men.
Hence, Zeno’s reasoning is false when he argues that there is no part of the millet that does not make a sound; for there is no reason that any such part should not in any length of time fail to move the air that the whole bushel moves in falling. In fact, it does not of itself move even such a quantity of the air as it would move if this part were by itself, for no part even exists otherwise than potentially.
If, on the other hand, we have two forces, each of which separately moves one of two weights a given distance in a given time, then the forces in combination will move the combined weights an equal distance in an equal time. For in this case, the rules of proportion apply.
Then, does this hold good of alteration and of increase also? Surely it does, for in any given case, we have a definite thing that causes increase and a definite thing that suffers increase, and the one causes and the other suffers a certain amount of increase in a certain amount of time.
Similarly, we have a definite thing that causes alteration and a definite thing that undergoes alteration, and a certain amount, or rather degree, of alteration is completed in a certain amount of time.
Thus, in twice as much time, twice as much alteration will be completed, and conversely twice as much alteration will occupy twice as much time;
and the alteration of half of its object will occupy half as much time, and half will be moved in half or double will be moved in an equal.
But if what causes alteration or increase causes a certain amount of increase or alteration respectively in a certain amount of time, it does not necessarily follow that half the force will occupy half the time or that, in half the time, half the alteration or increase will be completed.
It may happen that there will be no alteration or increase at all, as is the case with weight.
956. After showing which motions are mutually comparable, the Philosopher now teaches how they are compared:
first, in local motions;
second, in other motions, at then, does this hold (250a28; [962]).
About the first, he does two things:
first, he mentions the aspects according to which local motions ought to be mutually compared;
second, he sets forth the rules of comparison in the light of the foregoing, at if force A (249b30; [957]).
He says first, therefore (249b27), that the mover in local motion always moves some mobile in some definite time and through some quantity of space. And this is required because, as was proved in book 6, something always moves and has moved simultaneously. For it was proved there that whatever is being moved has already been moved through some part of a distance and through some part of time. Hence, it follows that what is being moved is something quantitative and divisible, as are the distance and the time involved. However, not every mover is quantified, as will be proved in book 8; nevertheless, it is clear that some quantitative things are movers, and it is in respect to those that he proposes the following rules of comparison.
957. Then, at “if force A” (249b30), he lays down the rules of comparison.
First, according to division of the mobile;
second, when the mover is divided, at “again, if a given force” (250a4; [958]).
He says first (249b30) to let A be a mover, and B a mobile, and C the length of space traversed, and D the time in which A moves B through C. If, therefore, we take another moving power equal to the power of A, it will follow that it will move half of the mobile B through a distance twice C in the same time; but in half the time D it will move half of mobile B through the entire length C.
From these statements of the Philosopher, two general rules may be gathered.
The first is that, if some power moves a mobile through some certain distance in a given time, then it or an equal power will move half of that mobile through twice the distance in the same time or in an equal time.
The other rule is that an equal power will move half the mobile over the same distance in half the time.
The reason behind these rules is that the same analogy, or proportion, is being kept. For it is clear that the swiftness of a motion results from the triumph of the mover’s power over the mobile, for the weaker the mobile, the more the strength of the mover prevails over it and the more swiftly will it move the mobile. The swiftness of a motion cuts down on the time and increases the length traversed, for the swifter is what traverses a greater distance in an equal time or an equal distance in less time, as was proved in book 6. Therefore, according to the same proportion by which the mobile is diminished, either the time is diminished or the length traversed is increased—provided, of course, that the mover is the same or an equal mover.
958. Then, at “again, if a given force” (250a4), he teaches how motions are to be compared from the viewpoint of the mover: first, according to a division of the mover; second, according to the opposite, a union of movers, at “if, on the other hand” (250a25; [961]).
About the first, he does three things: first, he sets forth a true comparison; second, he rejects some false comparisons, at, but, if E moves Z (250a9; [959]);
third, from this, he answers an argument of Zeno, at “hence, Zeno’s” (250a19; [960]).
He says first (250a4) that, if a power moves the same mobile through a certain distance in a given time, it moves half the mobile the same distance in half the time, or it moves half the mobile through twice the distance in the given original time, as was said of an equal power.
Further, if the power be divided, half the power will move half the mobile through the same distance in the given time. However, this must be understood of a mover that is not destroyed by division, for he has been speaking in a general way without making application to the particular natures involved. And he gives an example. Let E be half of power A and let Z be half of mobile B, then, just as A moved B through C in time D, so E will move Z through the same distance in the same amount of time, because the same proportion of motive power to moved body mass is preserved. Hence, it follows that, in an equal time, the motion will traverse an equal distance, as was said.
959. Then, when he says, but, if E moves Z (250a9), he rejects two false comparisons.
The first consists in adding to the mobile without adding to the motive power. Hence, he says that, if E, which is half the motive power, moves Z, which is half the mobile, a distance C in a time D, it is not necessarily true that the halved power E will move a mobile twice Z through half the distance in the given time, for it could happen that the halved power cannot move the doubled mobile at all. But, if it can move it, the comparison will hold.
The second false comparison occurs when the mover is divided and the mobile is not divided. This he rejects at, if, then, A moves B (250a12), saying that, if the motive power A moves the mobile B through distance C in time D, it does not necessarily follow that half the motive power will move the entire mobile B in time D through a part of distance C such that this part of C is related to the entire distance C as A was related to Z in our other example. For when A was compared with Z, it was a suitable comparison, but in the present case, it is not, for it can happen that half the motive power will not move the whole mobile any distance. For if some whole power moves some whole mobile, it does not follow that half of it will move the same mobile any distance, no matter how much time is allowed. Otherwise, it would follow that a man by himself could move a whole ship a certain distance, if the combined power of the ship haulers is divided by the number of haulers and the distance they haul it be so divided.
960. Then, at “hence, Zeno’s reasoning” (250a19), he uses the foregoing to answer an argument of Zeno, who wished to prove that each grain of millet falling to the earth makes a sound, because an entire bushel of it, when poured to the earth, makes a sound. But Aristotle says that this argument of Zeno is not true that there is no part of the millet that does not make a sound, that is, that each grain of millet makes a sound when it falls to the earth. For there is no reason why any such part should in any length of time move the air to produce a sound as does the whole bushel in falling.
And, from this, we can conclude that it is not necessary that, if a part existing in a whole causes a motion, this part, now existing in isolation from the whole, can cause a motion. For in the whole, the part is not actual, but potential, especially in continua. For a thing is a being in the same way that it is one, and one is that which is undivided in itself and divided from others. But a part, precisely as existing in a whole, is not divided from it actually, but only potentially; hence, it is not actually one, but only potentially. For this reason, it is not the part that acts, but the whole.
961. Then, at “if, on the other hand” (250a25), he sets forth a comparison based on an aggregate of movers and says that, if there are two and each of them causes motion, and if each by itself is moving its own mobile a certain distance in a given time, then, when the two are united, they will move the mobiles—which are now joined together—through an equal distance in the same time, for even in this case, the same proportion is maintained.
962. Then, at “then, does this hold” (250a28), he sets forth the same rules of comparison for other motions.
About this, he does three things: first, he shows that the things according to which the comparison of motions must be judged are divisible; second, he sets forth the true comparison, at, thus, in twice as much (250b2; [963]); third, he rejects some false comparisons, at “but if what causes alteration” (250b4; [964]).
He first says, therefore (250a28), that there are three things involved in respect to growth: the cause of increase, the thing increased, and the time. These three have a certain quantity.
Also, there is a fourth thing to be considered: the quantity of increase produced by the cause and received by the growing thing. And these four things must be considered also in alteration: the cause of alteration, the thing altered, the amount or degree of alteration (which is present according to more and less), and the amount of time. These four, of course, are the same as are involved in local motion.
963. Then, at “thus, in twice as much” (250b2), he sets forth the true comparison and says that, if a power moves some amount in a given time according to these motions, then it will move twice the amount in twice the time; and if it moves twice the amount, it will be in twice the time. Likewise, the same power will move half the amount in half the time, or if it moves in half the time, then the motion will be half the amount. Or, if there is twice the power, it will move something twice the amount in an equal time.
964. Then, at “but if what causes alteration” (250b4), he dismisses a false comparison and says that, if what causes alteration or increase causes a certain amount of increase or alteration respectively in a certain amount of time, it does not necessarily follow that half the force will alter or increase half the object or some given amount in half the time, for it may happen that there will be no alteration or increase at all, as is the case with weight, that is, the case being the same as with the locally mobile that has weight.
It should be observed that, when Aristotle says that half will be moved in half or double will be moved in an equal, the words “double” and “half” (which are in the accusative case) refer not to the mobile, but to that in which there is motion—that is, the quality or the quantity, which are related to alteration and growth as length of distance is related to local motion. For in local motion, it was said that, if a certain power moves a certain mobile, half will move half the mobile, but here it is said that half might not move anything. But it must be understood that we are speaking of an integral mobile whole, which will not be moved by a halved motive power to any amount of quantity or degree of quality, much less to half.