THE FOURTH DIMENSION. QI 



no matter how we try, be made to coincide, that is, be so fitted the 

 one into the other that they shall both stand as one pyramid. But 

 the reflected image of the one could be brought into coincidence 

 with the other. Two spatial structures whose sides and angles are 

 thus equal to each other, and of which each may be viewed as the 

 reflected image of the other, are called symmetrical. For instance, 

 the right and the left hand are symmetrical ; or, a right and a left 

 glove. Now just as in two dimensions it is impossible by simple 

 displacement to bring into congruence triangles which like those 

 above mentioned can only be made to coincide by circumversion, 

 so also in three dimensions it is impossible to bring into congruence 

 two symmetrical pyramids. Careful mathematical reflection, how- 

 ever, declares that this could be effected, if it were possible, while 

 holding one of the surfaces, to move the pyramid out of the space 

 of experience, and to turn it round through a four- dimensioned 

 space until it reached a point at which it would return again into 

 our experiential space. This process would simply be the four- 

 dimensional analogue of the three-dimensional circumversion in 

 the above-mentioned case of the two triangles. Further, the interior 

 surfaces in this process would be converted into exterior surfaces, 

 and vice versa, exactly as in the circumversion of a triangle the an- 

 terior and posterior sides are interchanged. If the structure which 

 is to be converted into its symmetrical counterpart is made of a flex- 

 ible material, the interchange mentioned of the interior and exterior 

 surfaces may be effected by simply turning ihe structure inside out ; 

 for example, a right glove may thus be converted into a left glove. 

 Now from this truth, that every structure can be converted, by 

 means of a four-dimensional space inclusive of the world, into 'a 

 structure symmetrical with it, it has been sought to establish the 

 probability of the real existence of a four- dimensioned space. Yet 

 it will be evident, from the discussion of the preceding section, that 

 the only inference which we can here make is, that the idea of a 

 four- dimensioned space is competent, from a mathematical point of 

 view, to throw some light upon the phenomena of symmetry. To 

 conclude from these facts that a space of this kind really exists, 

 would be as daring as to conclude from the fact that the uniform 



