11 



Another instance of such a figure which is in different ways 

 similar to its mirror-image, is the cube (fig. 6). 



From fig 6 it appears that the cube, reflected in each plane abed, 

 brought through two opposite edges, will coincide with its original 

 position; and evidently there are six of such mirror-planes present. 

 In the same way the cube will coincide with itself if reflected at 

 one of the three possible planes like A BCD. The cube is thus a 

 figure which in nine different ways is equal to its mirror-image. 

 On the other hand, if an irregular tetrahedron A BCD (fig. 7) is 

 taken, it is easily seen to be different from its mirror-image A 'B'C'D' . 



Such a tetrahedron, therefore, is 

 an instance of a figure which is 

 in no way equal to its mirror- 

 image. 



6. We have dwelt somewhat 

 long on this matter, because in 

 the beginning it often appears 

 difficult to obtain a complete and 

 clear insight into these relations, 

 which on the other hand must be 

 considered as of fundamental sig- 

 nificance for the following. 



IB 



Fig. 6. 



Now there are innumerable stereometrical figures which are 

 similar to themselves as mirror-images: the cube, the octahedron, 

 the cone, etc., are all instances of this kind. They all possess a 

 property which fig. 7, and fig. 2 and 5, if considered only in their 

 own plane, evidently lack. However, figures such as fig. 2 or 5, are 

 surely "symmetrical" figures, and a fortiori, or at least for the 

 same reason, the figures mentioned above must also be called "sym- 

 metrical" ones. 



Therefore our previous definition of a "symmetrical figure" needs 

 an extension in such a way, that we say: symmetrical figures are 

 such as are similar to themselves or to their mirror-images in more than 

 one way. 2 ) In more than one way: for every figure is at least equal 

 to itself by mere identity. 



relation: /\?=\, it follows, that /\ itself can be + i or - - /. The case 

 of A = -f- / corresponds to the transformation of the system to a position 

 in which it remains congruent with itself; the case : A = J however, to 

 that in which it is the mirror-image of the initial system. 



2 ) Indeed, there is a dualistic character in this definition, although it is 



