90 THE FOURTH DIMENSION. 



mathematicians to the space from which the spirit-friends of the 

 spiritualistic mediums entertain us with rappings, knockings, and 

 bad English. Before taking this step we will first discuss the ques- 

 tion of the real existence of a four-dimensional space, not deciding 

 the question whether this space, if it really does exist, is inhabited 

 by reasonable beings who consciously act upon the world in which 

 we exist. 



Among the reasons which are put forward to prove the exist- 

 ence of a four-dimensional space containing the world, the least rep- 

 rehensible are those which are based on the existence of symmetry. 

 We spoke above of two triangles in the same plane which have aH 



Fig. 38. 



their sides and angles congruent, but which cannot be made to co- 

 incide by simple displacement within the plane; but we saw that 

 this coincidence could be effected by holding fast one side of one 

 triangle and moving it out of its plane until it had been so far turned 

 round that it fell back into its plane. Now something similar to 

 this exists in space. Cut two figures, exactly like that of Fig. 38, out 

 of a piece of paper, and turn the triangle ABF about the side AB, 

 ACE about the side AC, BCD about the side BC, and in one figure 

 above and in the other below; then in both cases the points D, , F 

 will meet at a point, because AE is equal to AF, BF is equal to 

 BD t CD is equal to CE. In this manner we obtain two pyramids 

 which in all lengths and all angles are congruent, yet which cannot, 



