THE RECOGNITION OF THE FOURTH DIMENSION. 183 
All attempts to visualize a fourth dimension are futile. It 
must be connected with a time experience in three space. 
The most difficult notion for a plane being to acquire 
would be that of rotation about a line. Consider a plane 
being facing a square. If he were told that rotation about a 
line were possible, he would move his square this way and 
that. A square in a plane can rotate about a point, but to 
rotate about a line would seem to the plane being perfectly 
impossible. How could those parts of his square which 
were on one side of an edge come to the other side without 
the edge moving ? He could understand their reflection in 
the edge. He could form an idea of the looking-glass image 
of his square lying on the opposite side of the line of an edge, 
but by no motion that he knows of can he make the actual 
square assume that position. The result of the rotation would 
be like reflection in the edge, but it would be a physical im¬ 
possibility to produce it in the plane. 
The demonstration of rotation about a line must be to him 
purely formal. If he conceived the notion of a cube stretch¬ 
ing out in an unknown direction away from his plane, then 
he can see the base of it, his square in the plane, rotating 
round a point. He can likewise apprehend that every par¬ 
allel section taken at successive intervals in the unknown 
direction rotates in like manner round a point. Thus he 
would come to conclude that the whole body rotates round 
a line—the line consisting of the succession of points round 
which the plane sections rotate. Thus, given three axes, x, 
y , 3, if x rotates to take the place of y, and y turns so as to 
point to negative x, then the third axis remaining unaffected 
by this turning is the axis about which the rotation takes 
place. This, then, would have to be his criterion of the axis 
of a rotation—-that which remains unchanged when a rota¬ 
tion of ever}" plane section of a body takes place. 
There is another way in which a plane being can think 
about three-dimensional movements; and, as it affords the 
type by which we can most conveniently think about four¬ 
dimensional movements, it will be no loss of time to con¬ 
sider it in detail. 
