LESSON 17

The Senses of Relative

Chapter 15: 1020b 26-1021b 11

             492. Some things are said to be relative (ad aliquid) directly, as double to half and triple to a third part; and in general what is multiplied to a part of what is multiplied, and what includes to what is included in it. And in another sense as what heats to what can be heated, and what cuts to what can be cut; and in general everything active to everything passive. And in another sense as what is measurable to a measure, and what is knowable to knowledge, and what is sensible to sense.

             493. The first things which are said to be relative numerically are such, either without qualification, or in some definite relation to them, or to unity; as double is related to half as a definite number. And the multiple is related numerically to the unit, but not in a definite numerical relation such as this or that. But what is one and a half times as great as something else is related to it in a definite numerical relation to a number. And the superparticular is related to the subparticular in an indefinite relation, as what is multiple is related to a number. And what includes is related to what is included in it as something altogether indefinite in number, for number is commensurable. For what includes is related to what is included in it according to so much and something more; but this something more is indefinite. For whatever the case may be, it is either equal or not equal to it. Therefore all these relations are said to be numerical and are properties of number.

             494. Further, equal, like and same are said to be relative, but in a different way, because all these terms are referred to unity. For those things are the same whose substance is one; and those are alike whose quality is one; and those are equal whose quantity is one. And unity is the principle and measure of number. Hence all these are said to be relative numerically, yet not in the same way.

             495. Active and passive things are relative in virtue of active and passive potencies and the operations of potencies; for example, what can heat is relative to what can be heated, because it can heat it; and what is heating is relative to what is being heated; and what is cutting to what is being cut, inasmuch as they are doing these things. But of those things which are relative numerically there are no operations, except in the sense stated elsewhere; and operations which imply motion are not found in them. Moreover, of things which are relative potentially, some are said to be relative temporally also, as what makes to what is made, and what will make to what will be made. For in this way a father is said to be the father of his son, because the former has acted, whereas the latter has been acted upon. Again, some things are said to be relative according to the privation of potency; for example, the incapable and other terms used in this way, as the invisible.

             496. Therefore things which are said to be relative numerically and potentially are all relative because the subject of the reference is itself referred to something else, not because something else is referred to it. But what is measurable and knowable and thinkable are said to be relative because in each case something else is referred to them, not because they are referred to something else. For by what is thinkable is meant that of which there may be a thought. However, a thought is not relative to the one whose thought it is, for then the same thing would be expressed twice. And similarly sight is relative to that of which it is the sight and not to the one whose sight it is (although it is true to say this); but it is relative to color or to something of this sort. But then the same thing would be said twice, that sight is of the one whose sight it is. Things which are said to be relative directly, then, are spoken of in this way.

             497. And other things are said to be relative because their genera are such; for example, medicine is relative because its genus, science, seems to be relative. Furthermore, of this type are all things which are said to be relative by reason of their subject; for example, equality is said to be relative because equal is relative; and likeness, because like is relative.

             498. But other things are said to be relative indirectly, as man is relative because he happens to be double, and this is relative; or the white is said to be relative because the same thing happens to be white and double.

COMMENTARY

             1001. Here the Philosopher establishes the meaning of the relative or relation; and in regard to this he does two things. First (492:C 1001), he gives the senses in which things are said to be relative directly; and second (497:C 1030), those in which things are said to be relative indirectly ("And other things").

             In regard to the first he does two things. First, he enumerates the senses in which things are said to be relative directly. Second (493:C 1006), he proceeds to deal with these ("The first things").

             He accordingly gives, first (492), three senses in which things are said to be relative directly. The first of these has to do with number and quantity, as double to half and triple to a third, and "what is multiplied," i.e., the multiple, to a part "of what is multiplied," i.e., the submultiple, "and what includes to what is included in it." But what includes is here taken for what is greater in quantity. For everything which is greater in quantity includes within itself that which it exceeds. For it is this and something more; for example, five includes within itself four, and three cubits include two.

             1002. The second sense is that in which some things are said to be relative according to acting and undergoing, or to active and passive potency; for example, in the realm of natural actions, as what can heat to what can be heated; and in the realm of artificial actions, as what can cut to what can be cut; and in general as everything active to everything passive.

             1003. The third sense of relation is that in which something measurable is said to be relative to a measure. Here measure and measurable are not taken quantitatively (for this pertains to the first sense, in which either one is said to be relative to the other, since double is said to be relative to half and half to double), but according to the measurement of being and truth. For the truth of knowledge is measured by the knowable object. For it is because a thing is so or is not so that a statement is known to be true or false, and not the reverse. The same thing applies in the case of a sensible object and sensation. And for this reason a measure and what is measurable are not said to be related to each other reciprocally, as in the other senses, but only what is measurable is related to its measure. And in a similar fashion too an image is related to that of which it is the image as what is measurable is related to its measure. For the truth of an image is measured by the thing whose image it is.

             1004. These senses are explained as follows: since a real relation consists in the bearing of one thing upon another, there must be as many relations of this kind as there are ways in which one thing can bear upon another. Now one thing bears upon another either in being, inasmuch as the being of one thing depends on another, and then we have the third sense; or according to active or passive power, inasmuch as one thing receives something from another or confers it upon the other, and then we have the second sense; or according as the quantity of one thing can be measured by another, and then we have the first sense.

             1005. But the quality as such of a thing pertains only to the subject in which it exists, and therefore from the viewpoint of quality one thing bears upon another only inasmuch as quality has the character of an active or passive power, which is a principle of action or of being acted upon. Or it is related by reason of quantity or of something pertaining to quantity; as one thing is said to be whiter than another, or as that which has the same quality as another is said to be like it. But the other classes of things are a result of relation rather than a cause of it. For the category when consists in a relation to time; and the category where in a relation to place. And posture implies an arrangement of parts; and having (attire), the relation of the thing having to the things had.

             1006. The first things (493).

             Then he proceeds to deal with the three senses of relation which have been enumerated. First (494:C 1022), he considers the first sense. Second (495:C 1023), he treats the second sense ("Active and passive"). Third (496:C 1026), he attends to the third sense ("Therefore, things").

             In regard to the first he does two things. First (493), he describes the relations which are based simply on number; and second (494:C 1022), he treats those which are based simply on unity ("Further, equal").

             He says, first (493), that the first way in which things are relative, which is numerical, is divided inasmuch as the relation is based on the ratio of one number to another or on that of a number to unity. And in either case it may be taken in two ways, for the number which is referred to another number or to unity in the ratio on which the relation is based is either definite or indefinite. This is his meaning in saying that the first things which are said to be relative numerically are said to be such "without qualification," i.e., in general or indefinitely, "or else definitely." And in both ways "to them," namely, to numbers, "or to unity," i.e., to the unit.

             1007. Now it should be borne in mind that every measure which is found in continuous quantities is derived in some way from number. Hence relations which are based on continuous quantity are also attributed to number.

             1008. It should also be borne in mind that numerical ratios are divided first into two classes, that of equality and that of inequality. And there are two kinds of inequality: the larger and smaller, and more and less. And the larger is divided into five kinds.

             1009. For a number is larger whenever it is multiple with respect to a smaller number, i.e., when it includes it many times, as six includes two three times. And if it includes it twice, it is called double; as two in relation to one, or four to two. And if it includes it three times, it is called triple; and if four times, quadruple; and so on.

             1010. But sometimes a larger number includes a whole smaller number once and some part of it besides; and then it is said to be superparticular. If it includes a whole smaller number and a half of it besides, it is called sesquialteral, as three to two; and if a third part besides, it is called sesquitertian, as four to three; and if a fourth part besides, it is called sesquiquartan, as five to four; and so on.

             1011. Sometimes a larger number includes a whole smaller number once and not merely one part but many parts besides, and then it is called superpartient. And if it includes two parts, it is called superbipartient, as five to three. Again, if it includes three parts, then it is called supertripartient, as seven to four; and if it includes four parts, it is superquadripartient, and then it is related as nine to five; and so on.

             1012. Sometimes a larger number includes a whole smaller number many times and some part of it besides, and then it is called multiple superparticular. If it includes it two and a half times, it is called double sesquialteral, as five to two. If it includes it three and a half times, it is called triple sesquialteral, as seven to two. And if it includes it four and a half times, it is called quadruple sesquialteral, as nine to two. And the species of this kind of ratio can also be considered in the case of the superparticular, inasmuch as we speak of the double sesquitertian ratio when a greater number includes a smaller number two and a third times, as seven to three; or of the double sesquiquartan, as nine to four; and so on.

             1013. Sometimes too a larger number includes a whole smaller number many times and many parts of it besides, and then it is called multiple superpartient. And similarly a ratio can be divided from the viewpoint of the species of multiplicity, and from that of the species of the superpartient, provided that we may speak of a double superbipartient, when a greater number includes a whole smaller number twice and two parts of it, as eight to three; or even of triple superbipartient, as eleven to three; or even of double supertripartient, as eleven to four. For it includes a whole number twice and three parts of it besides.

             1014. And there are just as many species of inequality in the case of a smaller number. For a smaller number is called submultiple, subpartient, submultiple superparticular, submultiple superpartient, and so on.

             1015. But it must be noted that the first species of ratio, namely, multiplicity, consists in the relation of one number to the unit. For any species of it is found first in the relation of some number to the unit. Double, for example, is found first in the relation of two to the unit. And similarly a triple ratio is found in the relation of three to the unit; and so on in other cases. But the first terms in which any ratio is found give species to the ratio itself. Hence in whatever other terms it is subsequently found, it is found in them according to the ratio of the first terms. For example, the double ratio is found first between two and the unit. It is from this, then, that the ratio receives its meaning and name; for a double ratio means the ratio of two to the unit. And it is for this reason [too that we use the term in other cases; for] even though one number is said to be double another, this happens only inasmuch as a smaller number takes on the role of the unit and a larger number the role of two; for six is related to three in a double ratio, inasmuch as six is to three as two is to one. And it is similar in the case of a triple ratio, and in all other species of multiplicity. Hence he says that the relation of double is a result of the fact that a definite number, i.e., two, "is referred to unity," i.e., to the unit.

             1016. But the term multiple implies the relation of a number to the unit, not of any definite number but of number in general. For if a definite number were taken, as two or three, there would be one species of multiplicity, as double or triple. And just as the double is related to two and the triple to three, which are definite numbers, so too the multiple is related to multiplicity, because it signifies an indefinite number.

             1017. Other ratios, however, cannot be reduced to the relation of a number to the unit: either a superparticular ratio, or a superpartient, or a multiple superparticular, or a multiple superpartient. For all of these species of ratios are based on the fact that a larger number includes a smaller number once, or some part of it, and one or several parts of it besides. But the unit cannot have a part, and therefore none of these ratios can be based on the relation of a number to the unit but on the relation of one number to another. Thus the double ratio is either that of a definite number, or that of an indefinite number.

             1018. And if it is that of a definite number, then "it is what is one and a half times as great," i.e., sesquialteral, or "that which it exceeds," i.e., supersesquialteral. For a sesquialteral ratio consists first in these terms: three and two; and in the ratio of these it is found in all other cases. Hence what is called one and a half times as great, or sesquialteral, implies the relation of one definite number to another, namely, of three to two.

             1019. But the relation which is called superparticular is relative to the subparticular, not according to any definite number, as the multiple is relative also to the unit, but according to an indefinite number. For the first species of inequality given above (493:C 1008) are taken according to indefinite numbers, for example, the multiple, superparticular, superpartient, and so on. But the species of these are taken according to definite numbers, as double, triple, sesquialteral, sesquiquartan, and so on.

             1020. Now it happens that some continuous quantities have a ratio to each other which does not involve any number, either definite or indefinite. For there is some ratio between all continuous quantities, although it is not a numerical ratio. For there is one common measure of any two numbers, namely, the unit, which, when taken many times, yields a number. But no common measure of all continuous quantities can be found, since there are certain incommensurable continuous quantities, as the diameter of a square is incommensurable with one of its sides. The reason is that there is no ratio between it and one of its sides like the ratio of one number to another or of a number to the unit.

             1021. Therefore, when it is said in the case of quantities that this quantity is greater than that one, or is related to that one as what includes is related to what is included in it, not only is this ratio not considered according to any definite species of number, but it is not even considered according to number at all, because every number is commensurable with another. For all numbers have one common measure, which is the unit. But what includes and what is included in it are not spoken of according to any numerical measure; for it is what is so much and something more that is said to have the relation of what includes to what is included in it. And this is indefinite, whether it be commensurable or incommensurable; for whatever quantity may be taken, it is either equal or unequal. If it is not equal, then it follows that it is unequal and includes something else, even though it is not commensurable. Hence it is clear that all of the above-mentioned things are said to be relative according to number and to the properties of numbers, which are commensuration, ratio, and the like.

             1022. Further, equal (494).

             He now treats those relative terms which have a reference to unity or oneness and are not based on the relation of one number to another or to the unit. He says that equal, like and same are said to be relative in a different way than the foregoing. For these are called such in reference to unity. For those things are the same whose substance is one; and those are alike whose quality is one; and those are equal whose quantity is one. Now since unity is the principle and measure of number, it is also clear that the former terms are said to be relative "numerically," i.e., in reference to something belonging to the class of number. But these last terms are not said to be relative in the same way as the first. For the first relations seen are those of number to number, or of a number to the unit; but this relation has to do with unity in an absolute sense.

             1023. Active and passive (495).

             Here he proceeds to treat the second type of relations, which pertains to active and passive things. He says that relative beings of this kind are relative in two ways: in one way according to active and passive potency; and in a second way according to the actualizations of these potencies, which are action and passivity; for example, what can heat is said to be relative to what can be heated in virtue of active and passive potency. For it is what is capable of heating that can heat, and it is what is capable of being heated that can become hot. Again, what is heating in relation to what is heated, and what is cutting in relation to what is being cut, are said to be relative according to the operations of the aforesaid potencies.

             1024. Now this type of relation differs from those previously given; for those which are numerical are operations only figuratively, for example, to multiply, to divide, and so forth, as has also been stated elsewhere, namely, in Book II of the Physics, where he shows that the objects of mathematics abstract from motion, and therefore they cannot have operations of the kind that have to do with motion.

             1025. It should also be noted that among relative terms based on active and passive potency we find diversity from the viewpoint of time; for some of these terms are predicated relatively with regard to past time, as what has made something to what has been made; for instance, a father in relation to his son, because the former has begot and the latter has been begotten; and these differ as what has acted and what has been acted upon. And some are used with respect to future time, as when what will make is related to what will be made. And those relations which are based on privation of potency, as the impossible and the invisible, are reduced to this class of relations. For something is said to be impossible for this person or for that one; and the invisible is spoken of in the same way.

             1026. Therefore, things (496).

             Next he proceeds to deal with the third type of relations. He says that this third type differs from the foregoing in this way, that each of the foregoing things is said to be relative because each is referred to something else, not because something else is referred to it. For double is related to half, and vice versa; and in a similar way a father is related to his son, and vice versa. But something is said to be relative in this third way because something is referred to it. It is clear, for example, that the sensible and the knowable or intelligible are said to be relative because other things are related to them; for a thing is said to be knowable because knowledge is had of it. And similarly something is said to be sensible because it can be sensed.

             1027. Hence they are not said to be relative because of something which pertains to them, such as quality, quantity, action, or undergoing, as was the case in the foregoing relations, but only because of the action of other things, although these are not terminated in them. For if seeing were the action of the one seeing as extending to the thing seen, as heating extends to the thing which can be heated, then just as what can be heated is related to the one heating, so would what is visible be related to the one seeing. But to see and to understand and actions of this kind, as is stated in Book IX (746:C 1788) of this work, remain in the things acting and do not pass over into those which are acted upon. Hence what is visible or what is knowable is not acted upon by being known or seen. And on this account these are not referred to other things but others to them. The same is true in all other cases in which something is said to be relative because something else is related to it, as right and left in the case of a pillar. For since right and left designate starting points of motion in living things, they cannot be attributed to a pillar or to any nonliving thing except insofar as living things are related to a pillar in some way. It is in this sense that one speaks of a right-hand pillar because a man stands to the left of it. The same holds true of an image in relation to the original; and of a denarius, by means of which one fixes the price of a sale. And in all these cases the whole basis of relation between two extremes depends on something else. Hence all things of this kind are related in somewhat the same way as what is measurable and its measure. For everything is measured by the thing on which it depends.

             1028. Now it must be borne in mind that, even though verbally knowledge would seem to be relative to the knower and to the object of knowledge (for we speak both of the knowledge of the knower and of the knowledge of the thing known), and thought to the thinker and to what is thought, nevertheless a thought as predicated relatively is not relative to the one whose thought it is as its subject, for it would follow that the same relative term would then be expressed twice. For it is evident that a thought is relative to what is thought about as to its object. Again, if it were relative to the thinker, it would then be called relative twice; and since the very existence of what is relative is to be relative in some way to something else, it would follow that the same thing would have two acts of existence. Similarly in the case of sight it is clear that sight is not relative to the seer but to its object, which is color, "or something of this sort." He says this because of the things which are seen at night but not by means of their proper color, as is stated in The Soul, Book II.

             1029. And although it is correct to say that sight is of him who sees, sight is not related to the seer formally as sight but as an accident or power of the seer. For a relation has to do with something external, but a subject does not, except insofar as it is an accident. It is clear, then, that these are the ways in which some things are said to be relative directly.

             1030. And other things (497).

             He now gives three ways in which some things are said to be relative not directly but indirectly. The first of these is that in which things are said to be relative because their genera are relative, as medicine is said to be relative because science is relative. For medicine is called the science of health and sickness. And science is relative in this way because it is an accident.

             1031. The second way is that in which certain abstract terms are said to be relative because the concrete things to which these abstract terms apply are relative to something else. For example, equality and likeness are said to be relative because the like and the equal are relative. But equality and likeness are not considered relative as words.

             1032. The third way is that in which a subject is said to be relative because of an accident. For example, a man or some white thing is said to be relative because each happens to be double; and in this way a head is said to be relative because it is a part.