The Complete Aristotle (eng.)

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