Now, whatever one may be inclined to think about the infinity of space, it is clear that this argument is an absurd one.  Let me write it out more at length:  “We are altogether unable to conceive space as bounded—­as finite; that is, as a whole in the space beyond which there is no further space.”  “We find ourselves totally unable to imagine bounds, in the space beyond which there is no further space.”  The words which I have added were already present implicitly.  What can the word “beyond” mean if it does not signify space beyond?  What Sir William and Mr. Spencer have asked us to do is to imagine a limited space with a beyond and yet no beyond.

There is undoubtedly some reason why men are so ready to affirm that space is infinite, even while they admit that they do not know that the world of material things 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]