Commentary Aristotle Physics
That this follows he now proves. Let A be the time in which the swifter traverses B, which is a circle, and let something slower traverse the straight line C in the same time. Now, since the swifter traverses more in the same time, it will follow that circle B is larger than the straight line—that is the way the swifter was defined in book 6. But we also said in that place that the swifter traverses an equal distance in less time. Therefore, we can take a part of time A during which the circularly moving body will traverse a part of this circle B and during which it will traverse C, while the slower body is traversing C in the entire time A. It will follow, therefore, that this part of the circle is equal to the entire C, because one and the same object traverses an equal distance in equal time. And in this way, a circular line will be equal to a straight line and a circular motion will, consequently, be as fast as a rectilinear.
932. Then, at “none the less, if the two motions” (248b4), he takes the contrary position. For if circular and rectilinear motions may be compared with respect to swiftness, it follows, as was just said, that a straight line will be equal to a circular one, for the equally swift is what traverses an equal distance. But a circular line and a straight line cannot be compared so as to be called equal. Therefore, neither can a circular motion be said to be as swift as a rectilinear.
933. Then, at but may we say that (248b7), he settles the difficulty he raised.
First, he asks in general what may be compared to what;
second, he adapts this to his proposition, at “similarly, in the case of” (249a8; [940]).
About the first, he does three things:
first, he states one thing that is required for comparison;
second, he states a second thing, at “can it be that the” (248b21; [937]);
third, he concludes a third requirement, at “thus, if things are commensurable” (249a3; [939]).
About the first, he does three things:
first, he mentions what is required for comparisons;
second, he takes the contrary position, at or shall we in the first place (248b12; [935]);
third, he settles the matter, at but, here again, may we not (248b15; [936]).
934. He thus says first (248b7) that things seem to be capable of being compared so long as they are not equivocal; that is, in the line of things not predicated equivocally, the subjects of predication may be compared. For example, sharp is an equivocal term. In one sense, it is applied to magnitudes, as when an angle is said to be sharp and when a pen point is said to be sharp. In another sense, it is applied to flavors, as when wine is said to be sharp or dry. In a third sense, it is applied to notes, as when the ultimate—that is, the highest—note in a melody or a chord of a lyre is said to be sharp.
Now, the reason that no answer can be made to the question which of these is “sharpest,” the point, the wine, or the voice, is that “sharp” is predicated of them in equivocal senses. But the highest note can be compared with respect to sharpness to another which is next to it in the scale, for in this case, “sharp” is not taken equivocally, but is predicated of both in the same sense.
Therefore, according to this, it could be replied to the proposed difficulty that the reason that a straight motion cannot be compared to a circular one is that the word “swift” is being used equivocally. And much less is the meaning of “swift” the same in respect to alteration and local motion. Consequently, these two are even less capable of being compared.
935. Then, at or shall we in the first place (248b12), he objects against what was just said. And he says that, at first sight, it does not seem to be true that things may be compared so long as they are not equivocal. For there are some non-equivocal things that cannot be compared; for example, much is used in the same sense when applied to water and to air, yet water and air cannot be compared with respect to muchness.
Now, if someone refuses to admit that much signifies the same thing on account of its general nature, he will at least grant that double, which is a species of muchness, signifies the same thing in regard to air and to water: in both cases, it signifies the ratio of two to one. Nevertheless, air and water cannot be compared in terms of double and half such as to be able to say that the amount of water is double that of the air or vice versa.
936. Then, at “but, here again, may we not” (248b15), he answers this objection.
About this, he does two things:
first, he gives the solution;
second, he confirms it by raising another question, at otherwise, why is it that (248b20; [937]).
He says first that it could be said that, in much and double, we discern the same reason for their inability to be compared when they are applied to water and to air as was discerned in sharp when it was applied to pen and wine and note; for much itself is equivocal.
Now, because someone could object against this on the ground that the same account of much is referred to when it is applied to both, then, in order to reject this, he states that even the accounts, or definitions, of certain things are equivocal. For example, if someone should say that the definition of much is that it is “so much and more,” “to be this amount and to be equal,” which is the same thing, is equivocal, for to be equal is to have one quantity, but the account of one quantity is not the same in all things. (The account of much as used here implies a comparison in the sense that it is the opposite of a little—that is, it is not taken in its absolute sense of being the opposite of one.)
And what he said of much he says consequently of double. For although the account of double is that there is a proportion of two to one, even that account contains an equivocation, for it could be said that one is equivocal; and if one is equivocal, it follows that two is, because two is nothing more than one taken twice.
Now, it should be observed that there are many things that, when considered in an abstract way in logic or mathematics, are not equivocal but that are, in a certain sense, equivocal when they are taken in a concrete way—as the philosopher of nature takes them when he applies them to matter, for they are not taken according to the same account in all matter. For example, quantity and unity (which is the principle of number) are not found according to the same account in the heavenly bodies and in fire, air, and water.
937. Then, at otherwise, why is it that (248b20), he confirms what has been said by raising a certain other question. For if it be held that there is one nature of much and of double and of other like things that cannot be compared, as there is also one nature of things that are predicated univocally, the question still remains why it is that, among things having one nature, some can be compared and some not. For it seems that, when things are similar, there should be the same judgment about them.
Then, at “can it be that” (248b21), he answers this question by positing the second requirement for things to be compared.
About this, he does two things:
first, he mentions the second requirement;
second, he shows that even that one is not enough, at it would seem, however (248b25; [938]).
He says first (248b21) that the reason that some things possessing one nature can be compared, while other things having one nature cannot, could be that, when some one nature is received according to one first subject in diverse things, they will be comparable. Thus, horse and dog can be compared with respect to whiteness—one being able to be called whiter than the other—because not only is the same nature of whiteness in both, but there is one first subject in which whiteness is received, namely, the surface. In like manner, the magnitude of each may be compared, such that one can be called larger than the other, because there is one subject of magnitude in each, namely, the substance of a mixed body. But water and a note cannot be compared with respect to magnitude such as to say that the note is greater than the water or vice versa, for although magnitude in itself is the same, the receiver of it is not the same. For when magnitude is said of water, its subject is a substance, but when it is said of a note, its subject is sound, which is a quality.
938. Then, at it would seem, however (248b25), he shows that this second requirement does not complete the list of requirements for two reasons.
The first of these is this. If things were comparable just because there is a non-differing subject, it would follow that all things have one nature, for it could be said of all things whatever that they do not differ except insofar as they exist in some one or another first subject. And, according to this, it would follow that to be equal and to be sweet and to be white are one and the same nature, differing only by reason of being received in one or another receiver. And this seems to be unacceptable, namely, that all things have one nature.
But it should be noted that positing a diversity of things on the sole ground of diversity of subject is a Platonist opinion, which attributed unity to form and duality to matter, such that the entire reason for diversity came from the material principle. That was why he stated that one and being are predicated univocally, and that they signify one nature, but that the species of things are diversified according to a diversity of receivers.
The second reason that he gives, at “moreover, it is not” (249a2), is that not just anything is capable of receiving anything else at random, but one is primarily the receiver of one; consequently, the form and what receives it are in proportion. Therefore, if there are many first receivers, there must necessarily be many natures capable of being received; or if one nature has been received, then necessarily there is one first receiver.
939. Then, at “thus, if things are commensurable” (249a3), he concludes that there is a third requirement for things to be comparable. And he says that things that can be compared not only must be non-equivocal (which is the first requirement) but must also not possess any difference either on the side of the first subject in which something is received (which is the second requirement) or on the side of what is received, which is a form or a nature (and this is the third).
And he gives examples of this third requirement. Color is divided into various species of color; hence, something cannot be compared solely on the ground that it is predicated of these colors, even though it be not predicated equivocally, and even though it have one first subject, which is a surface and is a first subject of the genus color, but not of any species of color. For we cannot say which is more colored, white or black, because this comparison would not be in terms of some definite species of color, but in terms of color in general. But, in terms of whiteness, which is not divided into various species, all white things may be compared, such that it can be said which one is whiter.