Part 6

Quantity is either discrete or continuous. Moreover, some quantities are such that each part of the whole has a relative position to the other parts: others have within them no such relation of part to part.

Quantity is either discrete or continuous. Also, in some quantities, each part has a position relative to the other parts. In other quantities, the parts have no such relation to one another.

Instances of discrete quantities are number and speech; of continuous, lines, surfaces, solids, and, besides these, time and place.

Examples of discrete quantities are number and speech. Examples of continuous quantities are lines, surfaces, solids, time, and place.

In the case of the parts of a number, there is no common boundary at which they join. For example: two fives make ten, but the two fives have no common boundary, but are separate; the parts three and seven also do not join at any boundary. Nor, to generalize, would it ever be possible in the case of number that there should be a common boundary among the parts; they are always separate. Number, therefore, is a discrete quantity.

The parts of a number have no shared boundary where they meet. For example, two fives make ten, but the two fives have no shared boundary. They are separate. The parts three and seven also do not meet at any boundary. More generally, the parts of a number can never have a shared boundary. The parts are always separate. Number, therefore, is a discrete quantity.

The same is true of speech. That speech is a quantity is evident: for it is measured in long and short syllables. I mean here that speech which is vocal. Moreover, it is a discrete quantity for its parts have no common boundary. There is no common boundary at which the syllables join, but each is separate and distinct from the rest.

The same is true of speech. Speech is clearly a quantity because it is measured in long and short syllables. I mean spoken speech here. Speech is also a discrete quantity because its parts have no shared boundary. There is no shared boundary where syllables meet. Instead, each syllable is separate and distinct from the others.

A line, on the other hand, is a continuous quantity, for it is possible to find a common boundary at which its parts join. In the case of the line, this common boundary is the point; in the case of the plane, it is the line: for the parts of the plane have also a common boundary. Similarly you can find a common boundary in the case of the parts of a solid, namely either a line or a plane.

A line, on the other hand, is a continuous quantity because its parts can meet at a shared boundary. For a line, the shared boundary is a point. For a plane, the shared boundary is a line, since the parts of a plane also share a boundary. Similarly, the parts of a solid can share a boundary, either a line or a plane.

Space and time also belong to this class of quantities. Time, past, present, and future, forms a continuous whole. Space, likewise, is a continuous quantity; for the parts of a solid occupy a certain space, and these have a common boundary; it follows that the parts of space also, which are occupied by the parts of the solid, have the same common boundary as the parts of the solid. Thus, not only time, but space also, is a continuous quantity, for its parts have a common boundary.

Space and time also belong to this class of quantities. Time, including past, present, and future, forms a continuous whole. Space is also a continuous quantity because the parts of a solid occupy space and share a boundary. So the regions of space occupied by those parts share the same boundary as the parts themselves. Thus, both time and space are continuous quantities because their parts share a boundary.

Quantities consist either of parts which bear a relative position each to each, or of parts which do not. The parts of a line bear a relative position to each other, for each lies somewhere, and it would be possible to distinguish each, and to state the position of each on the plane and to explain to what sort of part among the rest each was contiguous. Similarly the parts of a plane have position, for it could similarly be stated what was the position of each and what sort of parts were contiguous. The same is true with regard to the solid and to space. But it would be impossible to show that the parts of a number had a relative position each to each, or a particular position, or to state what parts were contiguous. Nor could this be done in the case of time, for none of the parts of time has an abiding existence, and that which does not abide can hardly have position. It would be better to say that such parts had a relative order, in virtue of one being prior to another. Similarly with number: in counting, 'one' is prior to 'two', and 'two' to 'three', and thus the parts of number may be said to possess a relative order, though it would be impossible to discover any distinct position for each. This holds good also in the case of speech. None of its parts has an abiding existence: when once a syllable is pronounced, it is not possible to retain it, so that, naturally, as the parts do not abide, they cannot have position. Thus, some quantities consist of parts which have position, and some of those which have not.

Quantities consist of parts that either have positions relative to one another or do not. The parts of a line have positions relative to one another because each part lies somewhere. We could identify each part and state its position on the plane. We could also explain what kind of part lies next to each one. Similarly, the parts of a plane have positions. We could state each part’s position and what kinds of parts lie next to it. The same is true of solids and space. But we could not show that the parts of a number have positions relative to one another, or particular positions. Nor could we state which parts lie next to one another. We could not do this with time either, because none of the parts of time continues to exist. And what does not continue to exist can hardly have a position. It would be better to say that the parts of time have a relative order, because one comes before another. The same applies to number: in counting, 'one' comes before 'two', and 'two' before 'three'. So the parts of a number may be said to have a relative order. But we could not identify a distinct position for each part. The same is true of speech. None of its parts continues to exist. Once a syllable has been spoken, we cannot hold on to it. So, naturally, because the parts of speech do not continue to exist, they cannot have positions. Thus, some quantities consist of parts that have positions, and some consist of parts that do not.

Strictly speaking, only the things which I have mentioned belong to the category of quantity: everything else that is called quantitative is a quantity in a secondary sense. It is because we have in mind some one of these quantities, properly so called, that we apply quantitative terms to other things. We speak of what is white as large, because the surface over which the white extends is large; we speak of an action or a process as lengthy, because the time covered is long; these things cannot in their own right claim the quantitative epithet. For instance, should any one explain how long an action was, his statement would be made in terms of the time taken, to the effect that it lasted a year, or something of that sort. In the same way, he would explain the size of a white object in terms of surface, for he would state the area which it covered. Thus the things already mentioned, and these alone, are in their intrinsic nature quantities; nothing else can claim the name in its own right, but, if at all, only in a secondary sense.

Strictly speaking, only the things I have mentioned belong to the category of quantity. Everything else called quantitative is a quantity only in a secondary sense. We apply quantitative terms to other things because we have one of these quantities in mind. We call something white large because the surface covered by the white is large. We call an action or a process lengthy because the time it takes is long. These things are not quantitative in their own right. For example, anyone explaining how long an action lasted would describe the time it took. They would say the action lasted a year, or something similar. Likewise, anyone explaining the size of a white object would describe its surface by stating the area it covered. So the things already mentioned, and only those things, are quantities by their very nature. Nothing else is a quantity in its own right, but only, if at all, in a secondary sense.

Quantities have no contraries. In the case of definite quantities this is obvious; thus, there is nothing that is the contrary of 'two cubits long' or of 'three cubits long', or of a surface, or of any such quantities. A man might, indeed, argue that 'much' was the contrary of 'little', and 'great' of 'small'. But these are not quantitative, but relative; things are not great or small absolutely, they are so called rather as the result of an act of comparison. For instance, a mountain is called small, a grain large, in virtue of the fact that the latter is greater than others of its kind, the former less. Thus there is a reference here to an external standard, for if the terms 'great' and 'small' were used absolutely, a mountain would never be called small or a grain large. Again, we say that there are many people in a village, and few in Athens, although those in the city are many times as numerous as those in the village: or we say that a house has many in it, and a theatre few, though those in the theatre far outnumber those in the house. The terms 'two cubits long', 'three cubits long', and so on indicate quantity, the terms 'great' and 'small' indicate relation, for they have reference to an external standard. It is, therefore, plain that these are to be classed as relative.

Quantities have no opposites. For definite quantities, this is obvious. Nothing is the opposite of 'two cubits long' or 'three cubits long', or of a surface or similar quantities. Someone might argue that 'much' is the opposite of 'little', and 'great' of 'small'. But these terms describe relations, not quantities. Things are not large or small in themselves. They are called large or small only by comparison. For example, a mountain is called small and a grain large. The grain is larger than others of its kind, while the mountain is smaller than others of its kind. So these descriptions refer to an outside standard. For if 'great' and 'small' described things in themselves, a mountain would never be called small or a grain large. Again, we say there are many people in a village and few in Athens. Yet the city has many times as many people as the village. Or we say a house has many people in it and a theatre few. Yet the people in the theatre far outnumber those in the house. The terms 'two cubits long', 'three cubits long', and so on describe quantity. The terms 'great' and 'small' describe a relation, because they refer to an outside standard. Clearly, then, 'great' and 'small' must be classified as relative terms.

Again, whether we define them as quantitative or not, they have no contraries: for how can there be a contrary of an attribute which is not to be apprehended in or by itself, but only by reference to something external? Again, if 'great' and 'small' are contraries, it will come about that the same subject can admit contrary qualities at one and the same time, and that things will themselves be contrary to themselves. For it happens at times that the same thing is both small and great. For the same thing may be small in comparison with one thing, and great in comparison with another, so that the same thing comes to be both small and great at one and the same time, and is of such a nature as to admit contrary qualities at one and the same moment. Yet it was agreed, when substance was being discussed, that nothing admits contrary qualities at one and the same moment. For though substance is capable of admitting contrary qualities, yet no one is at the same time both sick and healthy, nothing is at the same time both white and black. Nor is there anything which is qualified in contrary ways at one and the same time.

Again, whether we classify them as quantities or not, they have no opposites. For how can an attribute have an opposite if it cannot be understood on its own, but only through comparison with something else? Again, if 'great' and 'small' are opposites, the same subject will be able to have opposite qualities at the same time. Things will also be opposites of themselves. For sometimes the same thing is both small and great. The same thing may be small compared with one thing and great compared with another. So the same thing comes to be both small and great at the same time. It is therefore capable of having opposite qualities at the same moment. Yet we agreed, when discussing substance, that nothing has opposite qualities at the same moment. For although a substance can have opposite qualities, no one is both sick and healthy at the same time. Nothing is both white and black at the same time. Nor does anything have opposite qualities at the same time.

Moreover, if these were contraries, they would themselves be contrary to themselves. For if 'great' is the contrary of 'small', and the same thing is both great and small at the same time, then 'small' or 'great' is the contrary of itself. But this is impossible. The term 'great', therefore, is not the contrary of the term 'small', nor 'much' of 'little'. And even though a man should call these terms not relative but quantitative, they would not have contraries.

Moreover, if these terms were opposites, they would be opposites of themselves. For suppose 'great' is the opposite of 'small', and the same thing is both great and small at once. Then 'small' or 'great' would be the opposite of itself. But that is impossible. Therefore, 'great' is not the opposite of 'small', nor is 'much' the opposite of 'little'. And even if someone classified these terms as quantities rather than relations, they would still have no opposites.

It is in the case of space that quantity most plausibly appears to admit of a contrary. For men define the term 'above' as the contrary of 'below', when it is the region at the centre they mean by 'below'; and this is so, because nothing is farther from the extremities of the universe than the region at the centre. Indeed, it seems that in defining contraries of every kind men have recourse to a spatial metaphor, for they say that those things are contraries which, within the same class, are separated by the greatest possible distance.

Space is the case where quantity most plausibly seems to have an opposite. For people define 'above' as the opposite of 'below' when 'below' means the region at the centre. The reason is that nothing is farther from the universe’s outer limits than the region at the centre. Indeed, people seem to use a spatial metaphor when defining opposites of every kind. For they say that opposites are things within the same class separated by the greatest possible distance.

Quantity does not, it appears, admit of variation of degree. One thing cannot be two cubits long in a greater degree than another. Similarly with regard to number: what is 'three' is not more truly three than what is 'five' is five; nor is one set of three more truly three than another set. Again, one period of time is not said to be more truly time than another. Nor is there any other kind of quantity, of all that have been mentioned, with regard to which variation of degree can be predicated. The category of quantity, therefore, does not admit of variation of degree.

Quantity does not seem to vary in degree. One thing cannot be two cubits long to a greater degree than another. The same applies to numbers. What is 'three' is not more truly three than what is 'five' is five. Nor is one set of three more truly three than another set. Similarly, one period of time is not said to be more truly time than another. Nor can any other kind of quantity mentioned vary in degree. So the category of quantity does not vary in degree.

The most distinctive mark of quantity is that equality and inequality are predicated of it. Each of the aforesaid quantities is said to be equal or unequal. For instance, one solid is said to be equal or unequal to another; number, too, and time can have these terms applied to them, indeed can all those kinds of quantity that have been mentioned.

The most distinctive feature of quantity is that it can be described as equal or unequal. Each kind of quantity mentioned above can be described in these terms. For example, one solid can be equal or unequal to another. These terms also apply to number and time, and indeed to all the kinds of quantity already mentioned.

That which is not a quantity can by no means, it would seem, be termed equal or unequal to anything else. One particular disposition or one particular quality, such as whiteness, is by no means compared with another in terms of equality and inequality but rather in terms of similarity. Thus it is the distinctive mark of quantity that it can be called equal and unequal.

It seems that nothing other than a quantity can ever be called equal or unequal to anything else. A particular disposition or quality, such as whiteness, is never compared with another in terms of equality or inequality. Instead, such dispositions or qualities are compared in terms of similarity. So the distinctive feature of quantity is that it can be called equal or unequal.