LECTURE 9 (249 b 26-250 b 9)

RULES FOR THE COMPARISON OF MOTIONS

             956. After the Philosopher has shown what motions are mutually comparable, he here explains how they are compared.

             He does this first with respect to local motion, and secondly with respect to the other motions, where he says, 'Then does this hold . . .' (250 a 28).

             Concerning the first part he makes two points. First he states the things which must govern the mutual comparison of local motions. Secondly, where he says, 'If, then, A the movent . . .' (249 b 30), he establishes from the above the rules for comparison.

             He says, therefore, first that that which moves locally always moves some mobile object in some time and through some quantity of space. This must be so, because as was proven in Book VI to move something and to have moved something are always together. For it was proven there that whatever is being moved has already been moved through some part of space and through some part of time. From this it follows that that which is moved is quantified and divisible, as is that through which it is moved and also the time in which it is moved. However, not every mover is quantified, as will be proven in Book VIII. Nevertheless, it is clear that some movers are quantified. And he here gives the rules of comparison in regard to this type of mover.

             957. Next where he says, 'If, then, A the movent . . .' (249 b 30), he gives the rules of comparison. He does this first in regard to the division of the mobile object, and secondly in regard to the division of the mover, where he says, 'Again if a given force . . .' (250 a 4).

             He says, therefore, first that A is given as a mover. And let B be the mobile object, let C be the length of space which is crossed, and let D be the time in which A moves B through C. If, therefore, there is assumed some other moving power which is equal to the power of A, it follows that that power will move half of the mobile object B in the same time through a length which is twice that of C. And it will move half of the mobile object through the whole length C in half the time D.

             From these words of the Philosopher two general rules can be established.

             The first is that if some power moves some mobile object through some space in some time, then half of that mobile object will be moved through twice the space either by an equal power in the same time or by the same power in another equal time.

             The other rule is that an equal power will move half of the mobile object through the same space in half the time. The reason for this is that the same analogy, that is, the same proportion, is thus preserved. For it is clear that the velocity of motion arises from the victory of the power of the mover over the mobile object. To the extent that the mobile object is less, so much will the power of the mover exceed it more and thus move it faster. Moreover, the velocity of motion lessens the time and increases the length of space. For that is faster which in the same time crosses a greater magnitude, or an equal magnitude in less time, as was proven in Book VI. Therefore, one must subtract from the time or add to the length of space in the same proportion in which one subtracts from the mobile object, provided that the mover is the same or equal.

             958. Next where he says, 'Again if a given force . . .' (250 a 4), he treats the comparison of motions in regard to the mover. He does this first in regard to the division of the mover, and secondly in regard to the opposite unification, where he says, 'If on the other hand . . .' (250 a 25).

             Concerning the first part he makes three points. First he gives the true comparison. Secondly, where he says, 'But if E move F . . .' (250 a 10), he rejects a false comparison. Thirdly, where he says, 'Hence Zeno's . . .' (250 a 20), he answers Zeno's argument.

             He says, therefore, first that if a power moves the same mobile object in the same time through so much space, then this same power moves half of the mobile object in half the time through the same space, or in the same time it moves half of the mobile object through twice the space, as was said of an equal power. And further, if the power is divided, half of the power will move half of the mobile object through the same space in an equal time. But it must be understood that the power is of such a kind that it is not corrupted by division. For he is speaking in terms of a common consideration and is not yet applying his remarks to some special nature, just as he has done in all that precedes. And he gives an example. Let E be half of the power A. And let F be half of the mobile object B. Now just as A moved B through C in time D, so E will move F through the same space in an equal time, for here also is preserved the same proportion of the motive power to the heavy body which is moved. From this it follows that in an equal time there will be motion through an equal space, as was said.

             959. Next where he says, 'But if E move F . . .' (250 a 10), he rejects two false comparisons. The first is that there may be an addition to a mobile object but not an addition to the moving power. Hence he says that if E, which is half of the motive power, moves F, which is half of the mobile object, in the time D through the space C, then it is not necessary that this halved power, which is E, move a mobile object twice as large as F in an equal time through half of the space C. For it could be that it is not possible for a halved power to move a doubled mobile object. But if it could move, this comparison would hold.

             The second false comparison arises when the mover is divided and the mobile object is not. He rejects this where he says, 'If, then, A move B . . .' (250 a 12). He says that if the moving power A moves the mobile object B in the time D through the space C, then it is not necessary that half of the mover move the whole mobile object B in the time D, not even through some part of the space C, which part is proportioned to the whole space C as we conversely compared A to F, that is, as we compared the whole motive power to part of the mobile object. For that was an appropriate comparison, but this is not. For it can happen that half of a mover will not move a whole mobile object through any space. If a whole power moves a whole mobile object, it does not follow that half of that power will move the whole mobile object either through some space or in some time. Otherwise it would follow that one single man could move a ship through some space if the power of those who pull is divided with respect to the number of those who pull and with respect to the length of space through which all together pull the ship.

             960. Next where he says, 'Hence, Zeno's . . .' (250 a 20), he answers according to the foregoing the argument of Zeno, who wished to prove that any grain of millet makes a sound when thrown to the ground because a whole measure of millet makes a sound when thrown to the ground. Aristotle says that this argument of Zeno is not true, that is, it is not true that any grain of millet will make a sound when it falls to the ground. For there is nothing to prevent one from saying that a grain of millet at no time moves enough air to make a sound, which air a whole falling measure does move to make a sound.

             And from this we can conclude that, if some small part which exists in a whole moves, it is not necessary that that part can move when it exists separately and per se. For a part is in a whole not in act, but in potency, especially in continuous things. For as a thing is a being, it is also one. And a one is that which is undivided in itself and divided from others. But a part existing in a whole is not divided in act but only in potency. Hence it is neither a being nor a one in act but only in potency. And for this reason it is not the part which acts but the whole.

             961. Next where he says, 'If on the other hand . . .' (250 a 25), he gives a comparison with respect to the unification of movers. He says that if there are two movers, each of which in itself moves a certain mobile object in a certain time through a certain space, then when the powers of these two movers are joined, they will move that which is composed of these moved weights through an equal space in an equal time. For here also the same proportion is preserved.

             962. Next where he says, 'Then does this hold . . .' (250 a 28), he gives the same rules of comparison for other motions.

             Concerning this he makes three points. First he explains the divisibility of those things to which the comparison of motions pertains. Secondly, where he says, '. . . thus in twice as much . . .' (250 b 1), he gives the true comparisons. Thirdly, where he says, 'On the other hand . . .' (250 b 4), he rejects false comparisons.

             He says, therefore, first in regard to increase that three things are involved, namely, the increaser, that which is increased, and the time. These three things have quantity. And fourthly there is the quantity in respect to which the increaser increases and in respect to which that which is increased is increased. These four things are also found in alteration, namely, the alterer, that which is altered, the quantity of passion in respect to which alteration occurs (which passion is present according to more and less), and the quantity of time in which the alteration takes place. These four are likewise found in local motion.

             963. Next where he says, '. . . thus in twice as much . . .' (250 b 1), he gives the true comparisons.

             He says that if a power with respect to these motions moves so much in so much time, in twice the time it will move twice as much, and if it moves twice as much, this will occur in twice the time. And similarly the same power will move half in half the time; or if it moves in half the time, half will be moved. Or if the power is double, it will move twice as much in an equal time.

             964. Next where he says, 'On the other hand . . .' (250 b 4), he rejects a false comparison.

             He says that if in a motion of alteration or increase a power moves so much in so much time, it is not necessary that half of the power move half as much in the same time or the same amount in half the time. For perhaps it would happen that it will increase or alter nothing, just as it was said that a halved power cannot move a whole weight, either through a whole space or some part of it. When he says, '. . . in half as much time half of the object will be altered: or again, in the same amount of time it will be altered twice as much' (250 b 3), 'half' and 'double' (which are in the accusative must not be understood to mean 'half or double of the mobile object' but 'half or double of that in which the motion occurs', that is, of quality or quantity, which are found in these two motions in the same way that length of space is found in local motion. Other than this these motions and local motion are not similar. For it was said in regard to local motion that if so much power moves so much of a mobile object, then half of the power will move half of the mobile object. But here it is said that half of the power may move nothing. This must be understood of the whole, complete mobile object, because a halved motive power will not move it either through so much quantity or quality, or through half of it.

Book VIII