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 pass 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 (Lect. 3, n. 773.)), 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. (Lect. 7, n. 712.)

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.

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. (Lect. 3, n. 770.) 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, when he says, 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.