The fourth dimension

THE HIGHER WORLD 73

revolution there would be in reality one line which remained unaltered, that is the line which stretches away in the fourth direction, forming the axis of the axle. The four-dimensional wheel can rotate in any number of planes, but all these planes are such that there is a line at right angles to them all unaffected by rotation in them.

An objection is sometimes experienced as to this mode of reasoning from a plane world to a higher dimensionality. How artificial, it is argued, this conception of a plane world is, If any real existence confined to a superficies could be shown to exist, there would be an argument for one relative to which our three-dimensional existence is superficial. But, both on the one side and the other of the space we are familiar with, spaces either with less or more than three dimensions are merely arbitrary conceptions.

In reply to this I would remark that a plane being having one less dimension than our three would have onethird of our possibilities of motion, while we have only one-fourth less than those of the higher space. It may very well be that there may be a certain amount of freedom of motion which is demanded as a condition of an organised existence, and that no material existence is possible with a more limited dimensionality than ours. This is well seen if we try to construct the mechanics of a two-dimensional world. No tube could exist, for unless joined together completely at one end two parallel lines would be completely separate. The possibility of an organic structure, subject to conditions such as this, is highly problematical; yet, possibly in the convolutions of the brain there may be a mode of existence to be described as two-dimensional.

We have but to suppose the increase in surface and the diminution in mass carried on to a certain extent to find a region which, though without mobility of the