CHAPTER V. 



The Limits of the axial periods in Crystalline Matter. Hauy's 

 Law. Crystallographically occurring Symmetry -Axes Groups 

 and Sub-groups; their relation to Holohedral, Hemihedral and 

 Tetartohedral Crystal-Classes. -- Crystal- systems. Gadolin's 

 Projection of Symmetry -Elements. - - The Symmetry -Classes of the 

 Cubic System. -- General and special simple Forms. Symmetry- 

 groups with Axes of Isotropy. The Symmetry of a Physical 

 Phenomemon, of a Physical State, and of a Physical Medium, 

 The "image" of a Physical Phenomenon. - - The Symmetry of 

 Cause and Effect, and their mutual Relation. - - The Symmetry 

 of the Electrostatic and that of the Magnetic Field. The 

 Symmetry of centrically -symmetrical Phenomena in Crystals. - 

 The Superposition of Different Causes. - - Symmetry and Dis- 

 symmetry. - - General Remarks on the symmetrical Arrangement 

 of experimentally determined Numbers. Problems and Investi- 

 gations of the Future. 



1. In the preceding chapters we extended our researches to 

 include all kinds of symmetrical systems. It need hardly be remarked 

 that, as evidently no special circumstances prohibit the occurrence 

 of every kind of symmetry-axes in the objects of living nature, 

 such a general way of treating the problem was the indicated and 

 only effectual one for the application of the doctrine of symmetry 

 in the whole province of natural science. However, in the case of 

 other, non-living natural objects, experience teaches us that such 

 an unlimited variety in the periods of the symmetry-axes by no means 

 manifests itself; and more particularly in the domain of crystalline 

 matter, there must be some reason why certain limits are apparently 

 set to the possible values of the numbers n, and to the characteristic 

 periods of the symmetry-axes, as these are determined by n. 



Indeed, in no field of physical research does the significance of 



