THE RECOGNITION OF THE FOURTH DIMENSION. 185 
Now, in the same way, the plane being can take another 
point, A', and another line, A' B', in his square. He can 
make the drawing of the two directions at A', one along 
A' B', the other perpendicular to his plane. He will obtain 
a figure precisely similar to Fig. 2, and will see that, as A B 
can turn around A, so A' B' around A'. 
In this turning A B and A' B' would not interfere with 
each other, as they would if they moved in the plane around 
the separate points A and A'. 
Hence the plane being would conclude that a rotation 
round a line was possible. He could see his square as it 
began to make this turning. He could see it half way round 
when it came to lie on the opposite side of the line A C. 
But in intermediate portions he could not see it, for it runs 
out of the plane. 
Coming now to the question of a four-dimensional body, 
let us conceive of it as a series of cubic sections, the first in 
our space, the rest at intervals, stretching away from our 
space in the unknown direction. 
We must not think of a four-dimensional body as formed 
by moving a three-dimensional body in any direction which 
we can see. 
Refer for a moment to Fig. 3. The point A, moving to 
the right, traces out the line A C. The line A C, moving 
away in a new direction, traces out the square A C E G at 
the base of the cube. The square A E G C, moving in a new 
direction, will trace out the cube A ft E G B D H F. The 
vertical direction of this last motion is not identical with 
any motion possible in the plane of the base of the cube. It 
is an entirely new direction, at right angles to every line 
that can be drawn in the base. To trace out a tesseract the 
cube must move in a new direction—a direction at right 
angles to any and every line that can be drawn in the space 
of the cube. 
The cubic sections of the tesseract are related to the cube 
we see as the square sections of the cube are related to the 
square of its base which a plane being sees. 
