This appearance would last, supposing the motion of the wheel to continue until it had traversed the plane by the extent of its thickness, when there would remain in the plane only the small disk which is the section of the axle. There would be no means obvious in the plane at first by which the axle could be reached, except by going through the substance of the wheel. But the possibility of reaching it without destroying the substance of the wheel would be shown by the continued existence of the axle section after that of the wheel had disappeared. In a similar way a four-dimensional wheel moving transverse to our space would appear first as a solid sphere, completely surrounding a smaller solid sphere. The outer sphere would represent the wheel, and would last until the wheel has traversed our space by a distance equal to its thickness. Then the small sphere alone would remain, representing the section of the axle. The large sphere could move round the small one quite freely. Any line in space could be taken as an axis, and round this line the outer sphere could rotate, while the inner sphere remained still. But in all these directions of 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 one-third 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.