ngs is infinite. To this we shall come back again
later. But if one wishes to affirm it, it is better to do so without
giving a reason than it is to present such arguments as the above.

25. SPACE AS INFINITELY DIVISIBLE.--For more than two thousand years
men have been aware that certain very grave difficulties seem to attach
to the idea of motion, when we once admit that space is infinitely
divisible. To maintain that we can divide any portion of space up into
ultimate elements which are not themselves spaces, and which have no
extension, seems repugnant to the idea we all have of space. And if we
refuse to admit this possibility there seems to be nothing left to us
but to hold that every space, however small, may theoretically be
divided up into smaller spaces, and that there is no limit whatever to
the possible subdivision of spaces. Nevertheless, if we take this most
natural position, we appear to find ourselves plunged into the most
hopeless of labyrinths, every turn of which brings us face to face with
a flat self-contradiction.

To bring the difficulties referred to clearly before our minds, let us
suppose a point to move uniformly over a line an inch long, and to
accomplish its journey in a second. At first glance, there appears to
be nothing abnormal about this proceeding. But if we admit that this
line is infinitely divisible, and reflect upon this property of the
line, the ground seems to sink from beneath our feet at once.

For it is possible to argue that, under the conditions given, the point
must move over one half of the line in half a second; over one half of
the remainder, or one fourth of the line, in one fourth of a second;
over one eighth of the line, in one eighth of a second, etc. Thus the
portions of line moved over successively by the point may be
represented by the descending series:

1/2, 1/4, 1/8, 1/16, . . . [Greek omicron symbol]

Now, it is quite true that the motion of the point can be described in
a number of different ways; but the important thing to remark here is
that, if the motion really is uniform, and if the line really is
infinitely divisible, this series must, as satisfactorily as any other,
describe the motion of the point. And it would be absurd to maintain
that _a part_ of the series can describe the whole motion. We cannot
say, for example, that, when the point has moved over one half, one
fourth, and one eighth of the line, it has completed its motion. If
ev