508 



Transactions. — Chemistry and Physics. 



The method of treatment does not admit of much origi- 

 nality. A straight line bounded by two points will, if moved 

 in a direction perpendicular to itself, trace out a square, 

 bounded by four lines and four points. By moving this 

 square in an independent direction at right angles to the 

 two original directions we shall obtain a cube, bounded 

 by six squares, twelve lines, and eight points. If the cube 

 be now moved in an independent direction compounded of 

 none of the three original directions, but at right angles 

 to them all, it will trace out a four - dimensional figure 

 (called by Mr. Hinton a " tessaract ") which will be bounded 

 by eight cubes, twenty-four squares, thirty-two lines, and 

 sixteen points. 



A very small amount of consideration will show how these 

 latter figures are arrived at. The bounding cubes consist of 

 the cube in its original position, the cube in its final position, 

 and the six cubes traced out by the motion of the six squares 

 which bounded the cube. Of the squares we had six in the 

 initial and six in the final position, while each of the twelve 

 lines of the cube traced a square, making twenty-four in all. 

 So too with the lines : twelve in the initial and twelve in the 

 final position, with eight traced by the eight points, bring up 

 the total to thirty-two. We may tabulate these results as 

 follows : — 



We might, of course, carry on the enumeration for figures 

 in five, six, or " n " dimensions. 



Mr. Hinton remarks that, if we take two equal cubes and 

 place them with their sides parallel and connect the corre- 

 sponding corners by lines, we shall form the figure of a tessar- 

 act. But it seems to the present writer that this suggestion 

 ignores the limitation of our three-dimensional space. It is 

 just these limitations which prevent our placing the cubes in 

 a satisfactory position. The suggestion contains the assump- 

 tion that, as we may project a cube on to a plane, so we may 

 project a tessaract on to a three-dimensional system. In 

 point of fact, such a projection might be made by a being in 

 four dimensions, but we three-dimensional beings must be 

 content with projecting our tessaract upon a plane. 



