THE RECOGNITION OF THE FOURTH DIMENSION. 195 
the place of the y axis. Here, in Fig. 14, we have the space 
of x z w. In this space we have to take all the points which 
are at the same distance from the center, consequently we have 
another sphere. If we had a three-dimensional sphere, as 
has been shown before, we should have merely a circle in 
the x z w space, the x z circle seen in the space of x z w. 
But now, taking the view in the space of x z w, we have a 
sphere in that space also. In a similar manner, whichever 
set of three axes we take, we obtain a sphere. 
£ 
2 
xwz 
In Fig. 13, let us imagine the rotation in the direction x y 
to be taking place. The point x will turn to y , and p to p'. 
The axis zz' remains stationary, and this axis is all of the 
plane z w which we can see in the space section exhibited in 
the figure. 
In Fig. 14, imagine the rotation from 2 to iv to be taking 
place. The w axis now occupies the position previously oc¬ 
cupied by the y axis. This does not mean that the w axis can 
coincide with the y axis. It indicates that we are looking 
at the four-dimensional sphere from a different point of 
view. Any three-space view will show us three axes, and in 
Fig. 14 we are looking at x z w. 
The only part that is identical in the two diagrams is the 
circle of the x and 2 axes, which axes are contained in both 
diagrams. Thus the plane z, x, z' is the same in both, and 
the point p represents the same point in both diagrams. 
Now, in Fig. 14 let the 2 w rotation take place, the 2 axis will 
turn toward the point w of the w axis, and the point p will 
move in a circle about the point x. 
28—Bull. Phil. Soc., Wash., Vol. 14. 
