510 



Transactions. — Chemistry and Physics. 



rated might have been considered the generating-cube, which 

 in turn gives us the option of starting from any one of the 

 twenty-four squares or the thirty-two lines. 



In two dimensions revolution takes place about a point, 

 while figures are bounded by lines. In three-dimensional 

 space revolution takes place about a line, and figures are 

 bounded by surfaces. In four dimensions revolution takes 

 place about a plane, and figures are bounded by solids. 



To a two-dimensional being the figures ABC, DEF are 

 essentially different, no amount of revolution about a point 

 effecting coincidence. To us, however, it is obvious that 

 one can be made to coincide with the other by per- 

 forming half a revolution about a line. Similarly, to us the 

 cubes figured below are essentially different, and no revolution 



8 



I 



about a line can make them coincide. To effect this one must 

 be taken into four-dimensional space and revolved about a 

 plane. Another aspect of the same fact is that, as we appre- 

 ciate the identity of the tw T o triangles by concentrating our 

 attention upon one face or the other of either triangle, so a 

 four-dimensional being appreciates the identity of the two 

 cubes by virtue of the fact that either side is equally acces- 

 sible to him. As the faces of 

 8 the cube are no greater hind- 

 rance to him than are the edges 

 of a triangle or square to us, 

 he can apprehend the cube with 

 the angle (6) forward or with 

 £, the angle (3) forward at will ; 

 the latter being the cube B, 

 the former being the cube A, 



[6 



!_ :7 



1 



above. 



