394 
HINTON. 
Consider a four-dimensional body, with four independent 
axes, x, y , 2 , w. A point in it can move in only one direction 
at a given moment. If the body has a velocity of rotation 
by which the x axis changes into the y axis and all parallel 
sections move in a similar manner, then the point will de¬ 
scribe a circle. If, now, in addition to the rotation by which 
the x axis changes into the y axis the body has a rotation by 
which the 2 axis turns into the iv axis, the point in question 
will have a double motion in consequence of the two turn¬ 
ings. The motions will compound, and the point will de¬ 
scribe a circle, but not the same circle which it would 
describe in virtue of either rotation separately. 
We know that if a body in three-dimensional space is 
given two movements of rotation they will combine into a 
single movement of rotation round a definite axis. It is in 
no different condition from that in which it is subjected 
to one movement of rotation. The direction of the axis 
changes; that is all. The same is not true about a four¬ 
dimensional body. The two rotations x to y and 2 to w are 
independent. A body subject to the two is in a totally 
different condition to that which it is in when subject to 
one only. When subject to a rotation such as that of x to 
2 /, a whole plane in the body, as we have seen, is stationary. 
When subject to the double rotation no part of the body 
is stationary except the point common to the two planes of 
rotation. 
If the two rotations are equal in velocity, every point in 
the body describes a circle. All points equally distant from 
the stationary point describe circles of equal size. 
We can represent a four-dimensional sphere by means of 
two diagrams, in one of which we take the three axes x, y , 
and z; in the other the axes x, w, and 2 . In Fig. 13 we have 
the view of a four-dimensional sphere in the space of x y z. 
Fig. 13 shows all that we can see of the four sphere in the 
space of x y z, for it represents all the points in that space, 
which are at an equal distance from the center. 
Let us now take the x z section, and let the axis of w take 
