But on the other hand to suppose that the infinite does not exist in
any way leads obviously to many impossible consequences: there will be
a beginning and an end of time, a magnitude will not be divisible into
magnitudes, number will not be infinite. If, then, in view of the
above considerations, neither alternative seems possible, an arbiter
must be called in; and clearly there is a sense in which the
infinite exists and another in which it does not.

    We must keep in mind that the word "is" means either what
potentially is or what fully is. Further, a thing is infinite either
by addition or by division.

    Now, as we have seen, magnitude is not actually infinite. But by
division it is infinite. (There is no difficulty in refuting the
theory of indivisible lines.) The alternative then remains that the
infinite has a potential existence.

    But the phrase "potential existence" is ambiguous. When we speak
of the potential existence of a statue we mean that there will be an
actual statue. It is not so with the infinite. There will not be an
actual infinite. The word "is" has many senses, and we say that the
infinite "is" in the sense in which we say "it is day" or "it is the
games", because one thing after another is always coming into
existence. For of these things too the distinction between potential
and actual existence holds. We say that there are Olympic games,
both in the sense that they may occur and that they are actually
occurring.

    The infinite exhibits itself in different ways-in time, in the
generations of man, and in the division of magnitudes. For generally
the infinite has this mode of existence: one thing is always being
taken after another, and each thing that is taken is always finite,
but always different. Again, "being" has more than one sense, so
that we must not regard the infinite as a "this", such as a man or a
horse, but must suppose it to exist in the sense in which we speak
of the day or the games as existing things whose being has not come to
them like that of a substance, but consists in a process of coming
to be or passing away; definite if you like at each stage, yet
always different.

    But when this takes place in spatial magnitudes, what is taken
perists, while in the succession of time and of men it takes place
by the passing away of these in such a way that the source of supply
never gives out.

    In a way the infinite by addition is the same thing as the
infinite by division. In a finite magnitude, the infinite by
addition comes about in a way inverse to that of the other. For in
proportion as we see division going on, in the same proportion we
see addition being made to what is already marked off. For if we
take a determinate part of a finite magnitude and add another part
determined by the same ratio (not taking in the same amount of the
original whole), and so on, we shall not traverse the given magnitude.
But if we increase the ratio of the part, so as always to take in
the same amount, we shall traverse the magnitude, for every finite
magnitude is exhausted by means of any determinate quantity however
small.

    The infinite, then, exists in no other way, but in this way it
does exist, potentially and by reduction. It exists fully in the sense
in which we say "it is day" or "it is the games"; and potentially as
matter exists, not independently as what is finite does.

    By addition then, also, there is potentially an infinite, namely,
what we have described as being in a sense the same as the infinite in
respect of division. For it will always be possible to take
something ah extra. Yet the sum of the parts taken will not exceed
every determinate magnitude, just as in the direction of division
every determinate magnitude is surpassed in smallness and there will
be a smaller part.

    But in respect of addition there cannot be an infinite which even
potentially exceeds every assignable magnitude, unless it has the
attribute of being actually infinite, as the physicists hold to be
true of the body which is outside the world, whose essential nature is
air or something of the kind. But if there cannot be in this way a
sensible body which is infinite in the full sense, evidently there can
no more be a body which is potentially infinite in respect of
addition, except as the inverse of the infinite by division, as we
have said. It is for this reason that Plato also made the infinites
two in number, because it is supposed to be possible to exceed all
limits and to proceed ad infinitum in the direction both of increase
and of reduction. Yet though he makes the infinites two, he does not
use them. For in the numbers the infinite in the direction of
reduction is not present, as the monad is the smallest; nor is the
infinite in the direction of increase, for the parts number only up to
the decad.

    The infinite turns out to be the contrary of what it is said to
be. It is not what has nothing outside it that is infinite, but what
always has something outside it. This is indicated by the fact that
rings also that have no bezel are described as "endless", because it
is always possible to take a part which is outside a given part. The
description depends on a certain similarity, but it is not true in the
full sense of the word. This condition alone is not sufficient: it
is necessary also that the next part which is taken should never be
the same. In the circle, the latter condition is not satisfied: it
is only the adjacent part from which the new part is different.

    Our definition then is as follows:

    A quantity is infinite if it is such that we can always take a
part outside what has been already taken. On the other hand, what
has nothing outside it is complete and whole. For thus we define the
whole-that from which nothing is wanting, as a whole man or a whole
box. What is true of each particular is true of the whole as
such-the whole is that of which nothing is outside. On the other
hand that from which something is absent and outside, however small
that may be, is not "all". "Whole" and "complete" are either quite
identical or closely akin. Nothing is complete (teleion) which has
no end (telos); and the end is a limit.

    Hence Parmenides must be thought to have spoken better than
Melissus. The latter says that the whole is infinite, but the former
describes it as limited, "equally balanced from the middle". For to
connect the infinite with the all and the whole is not like joining
two pieces of string; for it is from this they get the dignity they
ascribe to the infinite-its containing all things and holding the
all in itself-from its having a certain similarity to the whole. It is
in fact the matter of the completeness which belongs to size, and what
is potentially a whole, though not in the full sense. It is
divisible both in the direction of reduction and of the inverse
addition. It is a whole and limited; not, however, in virtue of its
own nature, but in virtue of what is other than it. It does not
contain, but, in so far as it is infinite, is contained. Consequently,
also, it is unknowable, qua infinite; for the matter has no form.
(Hence it is plain that the infinite stands in the relation of part
rather than of whole. For the matter is part of the whole, as the
bronze is of the bronze statue.) If it contains in the case of
sensible things, in the case of intelligible things the great and
the small ought to contain them. But it is absurd and impossible to
suppose that the unknowable and indeterminate should contain and
determine.