196 
HINTON. 
Thus in Fig. 13 the point p moves in a circle parallel 
to the x y plane; in Fig. 14 it moves in a circle parallel to the 
z w plane, indicated by the arrow. 
Now, suppose both of these independent rotations com¬ 
pounded, the point p will move in a circle, but this circle 
will coincide with neither of the circles in which either one 
of the rotations will take it. The circle the point p will 
move in will depend on its position on the surface of the four 
sphere. 
In this double rotation, possible in four-dimensional space, 
there is a kind of movement totally unlike any with which 
we are familiar in three-dimensional space. It is a requisite 
preliminary to the discussion of the behavior of the small 
particles of matter, with a view to determining whether they 
show the characteristics of four-dimensional movements, to 
become familiar with the main characteristics of this double 
rotation. And here I must rely on a formal and logical assent 
rather than on the intuitive apprehension which can only 
be obtained by a more detailed study. 
In the first place this double rotation consists in two varie¬ 
ties or kinds, which we will call the A and B kinds. Con¬ 
sider four axes, x, y, z, w. The rotation of x to y can be ac¬ 
companied with the rotation of z to iv. Call this the A kind. 
But also the rotation of x to y can be accompanied by the 
rotation of not z to w, but w to z. Call this the B kind. 
They differ in only one of the component rotations. One 
is not the negative of the other. It is the semi-negative. 
The opposite of an x to y, z to w rotation would be y to x, 
w to z. The semi-negative is x to y and w to z. 
If four dimensions exist and we cannot perceive them 
because the extension of matter is so small in the fourth 
dimension that all movements are withheld from direct 
observation except those which are three dimensional, we 
should not observe these double rotations, but only the 
effects of them in three-dimensional movements of the type 
with which we are familiar. 
