LECTURE 7 (248 a 10-249 a 7)

THE COMPARISON OF MOTIONS. HE SHOWS IN GENERAL WHAT IS REQUIRED FOR THINGS TO BE COMPARABLE

             928. The Philosopher has shown that there must be a first in mobile objects and movers. Now since things which are of one order seem to be comparable, and since the state of being prior and posterior implies a comparison, he wishes next to inquire into the comparison of motions.

             Concerning this he makes two points. First he shows what motions are comparable to each other. Secondly, where he says, 'Now since wherever . . .' (249 b 26), he shows how motions are compared to each other.

             Concerning the first part he makes three points. First he raises a difficulty. Secondly, where he says, 'But is it not only . . .' (248 a 16), he objects to the parts of this difficulty. Thirdly, where he says, 'But may we say that . . .' (248 b 7), he answers the difficulty.

             He raises the difficulty first in general, asking whether or not every motion is comparable to every other motion, and secondly, in particular, raising the difficulty in regard to motions of one genus. Now if every motion is comparable to every other motion with respect to its speed or slowness (it was shown in Book VI that a thing is equally fast when it is moved in equal time and through equal space), it follows that circular motion is equal to straight motion and is greater and less in velocity. Further it follows that a circular line is equal to a straight line in quantity, or greater and less. From this it follows that an equal velocity is that which is moved through an equal space in an equal time.

             Next he raises the difficulty in respect to motions of different genera. If all motions are comparable in velocity, it will follow that if in an equal time one thing is altered and another thing is moved with respect to place, then the alteration is equal in velocity to the change of place. And further, from the definition of equal velocity it will follow that a passion, that is, a passive quality, with respect to which there is alteration, is equal to the length of space which is crossed by the local motion. This is obviously impossible because they do not agree in having the same nature [ratio] of quantity.

             929. Next where he says, 'But is it not only . . .' (248 a 16), he objects to the proposed difficulty. He does this first in respect to the comparison of alteration and change of place, and secondly, where he says, 'But how will our . . .' (248 a 19), in respect to the comparison of circular and straight motion. From the preceding argument which leads to an impossibility, he first concludes to the contrary of that which was assumed. It is as if he were to say: it is granted that it is impossible for a passion to be equal to a length. But a thing is equally fast when it is moved in an equal time through an equal space. Therefore, since no passion is equal to a length, it follows that change of place is not equal in velocity to alteration, nor is it greater or less. From this it can be concluded further that not all motions are comparable.

             930. Next where he says, 'But how will our . . .' (248 a 19), he deals with the other part of the difficulty, namely, straight and circular motion.

             First he objects to the point that circular motion is equal in velocity to straight motion. Secondly, where he says, 'None the less, if the two motions . . .' (248 b 5), he objects to the contrary.

             Concerning the first part he makes two points. First he objects to the proposition. Secondly, where he says, 'Moreover it does not . . .' (248 a 22), he rejects a trivial answer.

             He objects first as follows. Circular motion and motion in a straight line are different local motions, just as are upward and downward motions. But it is at once necessary that a thing is moved faster or slower if one thing is moved up and another down, or even if the same thing is sometimes moved up and sometimes down. It seems, therefore, that it must be said likewise that motion in a straight line is faster or slower than circular motion, whether the same thing is moved in a circle and in a straight line, or whether there be two different things.

             It should be noticed that in this argument he makes no mention of equal velocity, but only of faster and slower velocity. For this argument is taken from the comparison of upward motion, whose principle is lightness, and downward motion, whose principle is heaviness. Some have thought that heaviness and lightness are the same as fastness and slowness (which he has rejected in Book V .

             931. Next where he says, 'Moreover it does not . . .' (248 a 22), he rejects a certain trivial explanation. On the basis of the previous argument one could concede that circular motion is faster or slower than straight motion, but not equally fast.

             He rejects this by saying that it makes no difference to the present argument if one were to say that that which is moved in a circle must be moved faster or slower than that which is moved in a straight line. For according to this circular motion will be either greater or less in velocity than straight motion. From this it follows that it may also be equal.

             And that this follows, he shows in this way. Let A be the time in which a faster motion passes through B, which is a circle; and let another slower motion pass in the same time through C, which is a straight line. Since, therefore, in the same time a faster motion crosses a greater distance, it will follow that the circle B will be greater than the straight line C, for thus we defined 'faster' above in Book VI. But in the same place we said that the faster crosses an equal distance in less time. Therefore, in part of the time A the body which is moved in a circle will cross part of the circle B, and in the same time it will cross C. For in the whole time A the slower body will cross the whole of C. It follows, therefore, that that part of the circle will be equal to the whole of C, because the same thing crosses an equal distance in an equal time. And thus a circular line will be equal to a straight line, and the circular motion consequently will be equally as fast as the straight motion.

             932. Next where he says, 'None the less if the two motions . . .' (248 b 5) he objects to the contrary. If circular motion and straight motion are comparable in velocity, there follows what was just said, namely, that a straight line is equal to a circle. For that which is equally fast is moved through an equal space. But a circular line and a straight line are not comparable such that they can be called equal. Therefore, circular and straight motion cannot be said to be equally fast.

             933. Next where he says, 'But may we say that . . .' (248 b 7), he answers the above difficulty.

             First he inquires in general as to what may be comparable to what. Secondly, where he says, 'Similarly in the case of . . .' (249 a 8), he adapts this to the problem.

             Concerning the first part he makes three points. First he treats one thing which is required for comparison. Secondly, where he says, 'Can it be that the . . .' (248 b 21), he treats the second requirement. Thirdly, where he says, 'Must we then say . . .' (249 a 3), he concludes with the third requirement.

             Concerning the first part he makes three points. First he states what is required for comparison. Secondly, where he says, 'Or shall we in the first place . . .' (248 b 12), he objects to the contrary. Thirdly, where he says, 'But here again may we not . . .' (248 b 15), he answers the objection.

             934. He says, therefore, first that whatever things are not equivocal seem to be comparable. Or in other words, the subjects of things which are not predicated equivocally may be comparable to each other. For example, 'sharp' is used equivocally: in one sense it is applied to magnitudes, according to which an angle is said to be sharp and a pen is said to be sharp; in another sense it is applied to flavours, according to which wine is said to be sharp; in a third sense it is applied to tones, according to which the ultimate tone, that is, the highest, in melodies, or the string of a harp, is said to be sharp.

             Therefore, a comparison cannot be made so as to designate which may be sharper; the pen, the wine, or the highest tone. For 'sharp' is predicated equivocally of them. But the highest tone may be compared in regard to sharpness to the one next to it in the order of melody. For 'sharp' is predicated of both not equivocally, but according to the same nature [ratio].

             Therefore, in regard to the proposed question it can be said that a straight motion and a circular motion are not comparable with respect to velocity. For velocity is used equivocally in these two cases. And much less is there the same definition [ratio] of velocity in alteration and change of place. Hence these are much less comparable.

             935. Next where he says, 'Or shall we in the first place . . .' (248 b 12), he objects to what has been said. He says that at first sight it does not seem to be true that if things are not equivocal, then they are comparable. For there are some non-equivocal things which are not comparable. For example, the word 'much' is applied in the same sense [ratio] to air and water, but nevertheless air and water are not comparable with respect to multitude.

             Moreover, if someone is unwilling to concede that 'much' signifies the same thing because of its commonness, at least he will agree that 'double', which is a species of the multiple, signifies the same thing in air and water. In each case it signifies a proportion of two to one. Nevertheless air and water are not comparable with respect to double and half, so that it may be said that water is the double of air, or vice versa.

             936. Next where he says, 'But here again may we not . . .' (248 b 15), he answers this objection.

             Concerning this he makes two points. First he gives his answer. Secondly, where he says, 'Otherwise why is it that . . .' (248 b 20), he strengthens his answer by raising a question.

             He says, therefore, first that 'much' and 'double' are not comparable when predicated of air and water for the same reason mentioned above in respect to 'sharp' which is predicated of a pen, wine, and a tone. For the term 'much' is also equivocal.

             And since one might object that 'much' has the same meaning [ratio] when it is predicated of both, to reject this he adds that even the definitions of certain things are equivocal. For example, if one were to say that the definition of 'much' is 'so much and more', then 'just so much' and 'equal', which are the same, are equivocal. For the 'equal' is that which has one quantity. However 'one quantity' does not have the same meaning [ratio] in all things. Moreover, it was granted that the meaning [ratio] of 'much' implies a comparison, such that it is opposed to 'few', and it is not taken absolutely, such that it is opposed to 'one'.

             What he had said about 'much' he says also about 'double'. For although the meaning [ratio] of 'double' is a proportion of two to one, nevertheless this meaning [ratio] also contains an equivocation. For perhaps it can be said that 'one' is equivocal. And if one is predicated equivocally, the same is true of two, because two is nothing other than twice one.

             Moreover, it must be realized that, although many things are not equivocal according to the abstract consideration of logic or mathematics, nevertheless they are sometimes predicated equivocally according to the consideration of one who applies the concrete nature [ratio] of natural things to matter. For they are not received in every matter according to the same nature [ratio]. For example, quantity and unity, which is the principle of number, are not found according to the same nature [ratio] in celestial bodies and fire and air and water.

             937. Next where he says, 'Otherwise why is it that . . .' (248 b 20), he confirms what has been said by raising a question. For if it is said that there is one nature of 'much' and 'double' and other such things which are not comparable, just as there is for things which are predicated univocally, then there remains the question of why some things which have one nature are comparable, and others are not. For it seems that the same judgment should be made for similar things.

             Next where he says, 'Can it be that the . . .' (248 b 21), he answers this question by establishing the second requirement for comparison.

             Concerning this he makes two points. First he establishes the second requirement for comparison. Secondly, where he says, 'It would seem, however . . .' (248 b 25), he shows that this is not yet sufficient.

             He says, therefore, first that this may be the reason why some things which are of one nature are comparable and others are not: if one nature is received in different things according to one primary subject, then these things will be mutually comparable. For example, horse and dog can be compared in respect to whiteness, such that one of them may be called whiter. For not only is the same nature of whiteness in each, but there is also one primary subject in which whiteness is received, namely, the surfaces. And similarly, magnitude is comparable in each, so that one may be called larger. For there is the same subject of magnitude in each, namely, the substance of the mixed body. But water and tone are not comparable with respect to magnitude, so that it may be said that a tone is greater than water, or vice versa. For although magnitude with respect to itself is the same, the thing receiving it is not the same. Insofar as it is predicated of water, its subject is a substance, but insofar as it is predicated of a tone, its subject is sound, which is a quality.

             938. Next where he says, 'It would seem, however . . .' (248 b 25), he shows with two arguments that this is not sufficient.

             The first is as follows. If things are comparable only because their subject is not different, it would follow that all things have one nature. For it can be said of any diverse things that they did differ only because they are in different primary subjects. And accordingly it would follow that what is equal and what is sweet and what is white have one and the same nature, and they differ only because they are in different receivers. And it seems impossible for all things to have one nature.

             Moreover it must be realized that the Platonic opinion attributes the diversity of things only to the diversity of the receiver. This opinion attributes unity to form and duality to matter such that the whole nature [ratio] of diversity proceeds from the material principle. And so he held that both unity and being are predicated univocally and signify one nature. And the species of things are diversified according to the diversity of receivers.

             The second argument, which he gives where he says, 'Moreover, it is not . . .' (249 a 2), is that not everything is a receiver for everything. Rather one thing is primarily the receiver for one thing. Thus form and the receiver are predicated of each other. If, therefore, there are many primary receivers, there must be many received natures. Or if there is one received nature, there must be one primary receiver.

             939. Next where he says, 'Must we then say . . .' (249 a 3), he concludes to the third thing which is required for things to be comparable.

             He says that things which are comparable not only must not be equivocal, which was the first requirement, but also they must have no difference, either in respect to the primary subject in which a thing is received, which was the second requirement, or in respect to that which is received, which is form or nature, and this is the third requirement.

             He gives an example of this third point. Since colour is divided into different species of colour, it is not comparable in respect to what is predicated of them. Nevertheless, it is not predicated equivocally, and it has one primary subject, namely, surface, which is the primary subject of the genus, but not of any species of colour. For we cannot say that a thing which is black or white is more coloured. For this comparison is not according to some determined species of colour but according to common colour itself. In regard to white, however, which is not divided into different species, a comparison can be made of all white things such that it may be said which is whiter.