It is reasonable that there should not be held to be an infinite
in respect of addition such as to surpass every magnitude, but that
there should be thought to be such an infinite in the direction of
division. For the matter and the infinite are contained inside what
contains them, while it is the form which contains. It is natural
too to suppose that in number there is a limit in the direction of the
minimum, and that in the other direction every assigned number is
surpassed. In magnitude, on the contrary, every assigned magnitude
is surpassed in the direction of smallness, while in the other
direction there is no infinite magnitude. The reason is that what is
one is indivisible whatever it may be, e.g. a man is one man, not
many. Number on the other hand is a plurality of "ones" and a
certain quantity of them. Hence number must stop at the indivisible:
for "two" and "three" are merely derivative terms, and so with each of
the other numbers. But in the direction of largeness it is always
possible to think of a larger number: for the number of times a
magnitude can be bisected is infinite. Hence this infinite is
potential, never actual: the number of parts that can be taken
always surpasses any assigned number. But this number is not separable
from the process of bisection, and its infinity is not a permanent
actuality but consists in a process of coming to be, like time and the
number of time.

    With magnitudes the contrary holds. What is continuous is divided ad
infinitum, but there is no infinite in the direction of increase.
For the size which it can potentially be, it can also actually be.
Hence since no sensible magnitude is infinite, it is impossible to
exceed every assigned magnitude; for if it were possible there would
be something bigger than the heavens.

    The infinite is not the same in magnitude and movement and time,
in the sense of a single nature, but its secondary sense depends on
its primary sense, i.e. movement is called infinite in virtue of the
magnitude covered by the movement (or alteration or growth), and
time because of the movement. (I use these terms for the moment. Later
I shall explain what each of them means, and also why every
magnitude is divisible into magnitudes.)

    Our account does not rob the mathematicians of their science, by
disproving the actual existence of the infinite in the direction of
increase, in the sense of the untraversable. In point of fact they
do not need the infinite and do not use it. They postulate only that
the finite straight line may be produced as far as they wish. It is
possible to have divided in the same ratio as the largest quantity
another magnitude of any size you like. Hence, for the purposes of
proof, it will make no difference to them to have such an infinite
instead, while its existence will be in the sphere of real magnitudes.

    In the fourfold scheme of causes, it is plain that the infinite is a
cause in the sense of matter, and that its essence is privation, the
subject as such being what is continuous and sensible. All the other
thinkers, too, evidently treat the infinite as matter-that is why it
is inconsistent in them to make it what contains, and not what is
contained.