THE RECOGNITION OF THE FOURTH DIMENSION. 203 
In four-dimensional space in which all four dimensions 
were commensurable the intensity of the action transmitted 
by the medium would vary inversely as the cube of the 
distance. Now, the action of a current on a magnetic pole 
varies inversely as the square of the distance; hence over 
measurable distances the extension of the ether in the fourth 
dimension cannot be assumed as other than small in com¬ 
parison with those distances. This extension being small, 
the effect of a vortex sheet would be equivalent to a num¬ 
ber of jets on one side and suctions on the other. 
Such an arrangement in the case of a liquid would pro¬ 
duce velocities in the liquid which coincide in direction 
with the tendency of motion of a magnetic pole. But 
analogies of this kind leave out of sight the fact that the 
action is a reciprocal one. Non-magnetic matter shows no 
tendency to move. To arrive at a definite conclusion it will 
be necessary to investigate the resultant pressures which 
accompany the collocation of solid vortices with surface 
ones. 
To recapitulate: 
The movements and mechanics of four-dimensional space 
are definite and intelligible. A vortex with a surface as its 
axis affords a geometric image of a closed circuit, and there 
are rotations which by their polarity afford a possible defini¬ 
tion of statical electricity. 
Note. —This paper, as originally presented , was accompanied by a mathe¬ 
matical exposition of the subject demonstrating the theoretical existence of the 
vortices that are described and showing that Hamilton’s quaternions represent 
the four-dimensional rotations of an A or a B kind. 
29—Bull. Phil. Soc., Wash., Vol. 14. 
