Physics   

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