THE RECOGNITION OF THE FOURTH DIMENSION. 187 
This is a three-dimensional space determined by two of 
the axes we have drawn, x and z, and in place of y the 
fourth axis, w. We cannot, keeping x and z, have both y 
and w in our space ; so we will let y go and draw w in its 
place. What will be our view of the cube? 
Evidently we shall have simply the square that is in the 
plane of x z, the square A C D B. The rest of the cube 
stretches in the y direction, and, as we have none of the 
space so determined, we have only the face of the cube. 
This is represented in Fig. 4. 
Now, suppose the whole cube to be turned from the x to 
the w direction. Conformably with our method, we will not 
take the whole of the cube into consider¬ 
ation at once, but will begin with the face 
A B C D. 
Let this face begin to turn. Fig. 5 
represents one of the positions it will 
occupy; the line A B remains on the 
z axis. The rest of the face extends be¬ 
tween the x and the w direction. 
Now, since we can take any three 
axes, let us look at what lies in the space 
of z y w, and examine the turning there. 
We must now let the z axis disap¬ 
pear and let the w axis run in the di¬ 
rection in which z ran. 
Making this representation, what do 
we see of the cube? Obviously we see 
only the lower face. The rest of the 
cube lies in the space of x y z. In th e 
space of x y w we have merely the base 
of the cube lying in the plane of x y, 
as shown in Fig. 6. 
Now let the x to w turning take place. 
The square A C E G will turn about the 
line A E. This edge will remain along 
the y axis and will be stationary, how¬ 
ever far the square turns. 
Fig. 4. 
27—Bull. Phil. Soc., Wash., Vol. 14. 
