Categories, Section 1, 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.

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

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.

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.

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 arts
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.

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.

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.

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.

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.

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.

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.

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.

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.