Commentary Aristotle Physics
Lecture 7
The comparison of motions and what is required for things to be comparable
A difficulty may be raised as to whether every motion is commensurable with every other. Now, if they are all commensurable, and if two things must accomplish an equal motion in an equal time in order to have the same velocity, then we may have a circumference equal to a straight line, or of course, the one may be greater or less than the other.
Further, if one thing alters and another accomplishes a locomotion in an equal time, we may have an alteration and a locomotion equal to one another. Thus, a passion will be equal to a length, which is impossible.
But is it not only when an equal motion is accomplished by two things in an equal time that the velocities of the two are equal? Now, a passion cannot be equal to a length.
Therefore there cannot be an alteration equal to or less than a locomotion, and thus it is not the case that every motion is commensurable with every other.
But how will our conclusion work out in the case of the circle and the straight line? It would be absurd to suppose that the motion of one in a circle and of another in a straight line cannot be similar, but that the one must inevitably move more quickly or more slowly than the other, just as if the course of one were downhill and of the other uphill.
Moreover, it does not as a matter of fact make any difference to the argument to say that the one motion must inevitably be quicker or slower than the other, for then the circumference can be greater or less than the straight line, and if so, it is possible for the two to be equal.
For if, in the time A, the quicker B passes over the distance B’ and the distance C’ is passed over by the slower C, then B’ will be greater than C’; for this is what we took “quicker” to mean.
And so quicker motion implies that a thing traverses an equal distance in less time. Consequently, there will be a part of A in which B will pass over a part of the circle equal to C’ and C will occupy the whole of A in passing over C’.
None the less, if the two motions are commensurable, we are confronted with the consequence stated above: there may be a straight line equal to a circle. But these are not commensurable, and so the corresponding motions are not commensurable either.
But may we say that things are always commensurable if the same terms are applied to them without equivocation?
For instance, a pen, a wine, and the highest note in a scale are not commensurable: we cannot say whether any one of them is sharper than any other.
They are incommensurable because the same term “sharp” is applied to them only equivocally; the ultimate note in a scale is commensurable with the leading note because the term “sharp” has the same meaning as applied to both.
Can it be, then, that the term “quick” has not the same meaning as applied to straight motion and to circular motion respectively? If so, far less will it have the same meaning as applied to alteration and to locomotion.
Or shall we in the first place deny that things are always commensurable if the same terms are applied to them without equivocation? For the term “much” has the same meaning whether applied to water or to air, yet water and air are not commensurable in respect of it.
Or, if not, “double” at least would seem to have the same meaning as applied to each (denoting in each case the proportion of two to one), yet water and air are not commensurable in respect of it.
But, here again, may we not take up the same position and say that the term “much” is equivocal? In fact, there are some terms of which even the accounts are equivocal.
For instance, if “much” were defined as “so much and more,” then “so much” would mean something different in different cases. “Equal” is similarly equivocal; and “one” again is perhaps inevitably an equivocal term; and if “one” is equivocal, so is “two.”
Otherwise, why is it that some things are commensurable while others are not, if the nature of the attribute in the two cases is really one and the same?
Can it be that the incommensurability of two things in respect of any attribute is due to a difference in that which is primarily capable of carrying the attribute?
Thus horse and dog are so commensurable that we may say which is the whiter, since that which primarily contains the whiteness is the same in both, namely, the surface; similarly, they are commensurable in respect of size.
But water and speech are not commensurable in respect of clearness, since that which primarily contains the attribute is different in the two cases.
It would seem, however, that we must reject this solution, since clearly we could thus make all equivocal attributes univocal and say merely that that which contains each of them is different in different cases.
Thus, “equality,” “sweetness,” and “whiteness” will severally always be the same, though that which contains them is different in different cases.
Moreover, it is not any casual thing that is capable of carrying any attribute: each single attribute can be carried primarily only by one single thing.
Thus, if things are commensurable in respect of any attribute, not only must the attribute be applicable without equivocation, but also there must be no differences either in the attribute itself or in what has the attribute.
These, I mean, must not be divisible in the way in which color is divided into kinds.
Thus, in this respect one thing will not be commensurable with another; that is, we cannot say that one is more colored than the other where only color in general is meant, and not any particular color; but they are commensurable in respect of whiteness.
928. After the Philosopher has shown that it is necessary to posit a first in mobiles and movers, now, because things that are of one order seem capable of being compared, and because to be “prior” or “subsequent” implies a comparison, he wishes to inquire about comparison of motions.
Concerning this, he does two things:
first, he shows which motions can be mutually compared;
second, how motions are mutually compared, at now, wherever there is a mover (249b27; [956]).
About the first, he does three things:
first, he raises a question;
second, he objects to both parts of the question, at but is it not only (248a15; [929]);
third, he settles the question, at but may we say that (248b7; [933]).
First, he raises a general question (248a10), and asks whether just any motion at random may be compared to any other motion.
Then he raises a particular question about motions in some one genus. Now, if any motion at random may be compared to just any other motion with respect to swiftness and slowness (it having been said in book 6 that the equally swift is what is moved in equal time over an equal space), it will follow that a circular motion will be equal to or greater than or less than a rectilinear one in swiftness, and further, it will follow that a curved line will be equal to a straight line in quantity, or larger or smaller, from the fact that the equally swift is that which traverses an equal distance in equal time.
Then he raises a question about motions in diverse genera. For if all motions may be compared with respect to speed, it will follow that, if, in an equal time, A is altered and B is moved locally, then a local motion is equal in swiftness to an alteration. Further, by virtue of the definition of the equally swift, it will follow that a passion—that is, passible quality (in respect of which there is alteration)—is equal to the length of the distance traversed by the local motion. But this is plainly impossible, because they do not agree in the same account of quantity.
929. Then, at “but is it not only” (248a15), he raises objections against the question proposed:
first, he objects against comparing alteration with local motion;
second, against comparing circular motion with rectilinear, at but how will our (248a18; [930]).
First (248a15), therefore, from the foregoing argument, which leads to an impossibility, he comes to the contrary of what he posited, as though saying that, since it has been said that a passion cannot be equal to a length, but whenever something is moved through an equal space in equal time, it is said to be equally swift, therefore, since no passion is equal to a length, it follows that a local motion is not equal in swiftness to an alteration, or greater or less. From this, it may be further concluded that not all motions can be compared.
930. Then, at but how will our (248a18), he considers the other part of the question, that is, concerning circular and rectilinear motion.
First, he objects against a circular motion’s being as swift as a rectilinear motion;
second, he takes the contrary position, at none the less, if the two motions (248b4; [932]).
About the first, he does two things:
first, he objects against the proposition;
second, he dismisses a quibbling response, at moreover, it does not (248a22; [931]).
He objects first (248a18) in the following manner. Circular motion and rectilinear are differences of local motion, just as upward and downward are. But, as soon as something is moved upward and something else downward, it is at once necessary that one be being moved faster or slower than the other—the same is true if the same thing is moved now upward and later downward. It seems, therefore, that, in like manner, we must say that a rectilinear motion is swifter or slower than a circular one, whether it be the same thing or two different things that are being moved in a straight line and in a circular one.
It should be noted that, in this argument, he makes no mention of the equally swift, but of the swifter and slower, because this argument is based on the likeness of an upward motion (whose principle is lightness) to a motion that is downward (whose principle is heaviness). Some, indeed, have held that heaviness and lightness are the same as swiftness and slowness—an opinion he rejected in book 5.
931. Then, at moreover, it does not (248a22), he rejects a quibble. For someone could concede on account of the foregoing reason that a circular motion is either swifter or slower than a rectilinear one, but not equally swift.
This he rejects, saying that it makes no difference, so far as the present discussion is concerned, once someone grants that it is necessary for what is being moved circularly to be moved more swiftly or more slowly than what is being moved in a straight line. For according to this, the circular motion will be faster or slower than the rectilinear. Hence it follows that it could also be equal.