188 
HINTON. 
Thus, if the cube be turned by an x to w turning, both 
the edge A B and the edge A C remain stationary ; hence 
the whole face A B E F in the y z 
plane remains fixed. The turn¬ 
ing has taken place about the face 
A B E F. 
Suppose this turning to con¬ 
tinue till A C runs to the left from 
A. The cube will occupy the po¬ 
sition shown in Fig. 8. This is 
the looking-glass image of the 
cube in Fig. 3. By no rotation in 
three-dimensional space can the 
cube be brought from the position 
in Fig. 3 to that shown in Fig. 8. 
We can think of this turning as 
a turning of the face A B C D about 
A B, and a turning of each section 
parallel to A B C D round the vertical line in which it in¬ 
tersects the face A B E F, the space in which the turning takes 
place being a different one from that in which the cube lies. 
One of the conditions, then, of our inquiry in the direction 
of the infinitely small is that we form the conception of a 
rotation about a plane. The production of a body in a state 
in which it presents the appearance of a looking-glass image 
of its former state is the criterion for a four-dimensional 
rotation. 
There is some evidence for the occurrence of such trans¬ 
formations of bodies in the change of bodies from those which 
produce a right-handed polarization of light to those which 
produce a left-handed polarization ; but this is not a point to 
which any very great importance can be attached. 
Still, in this connection, let me quote a remark from Prof. 
John G. McKendrick’s address on Physiology before the 
British Association at Glasgow. Discussing the possibility 
of the hereditary production of characteristics through the 
material structure of the ovum, he estimates that in it there 
C A , * * 
2- posit/on /-position 
Fig. 8. 
