82 



All polyhedra of both the trigonal and hexagonal system 

 are determined by one single measurement, just as was 



the case in the tetragonal system. 

 VII. The cubic system includes the groups: K H , T H , T D , K, 



and T. 



In this system no measurement is required to characterise 

 any form completely : all forms have special and invariable 

 values of their dihedral angles. 



From this it is obvious that quite independently of the introduction 

 of conceptions such as: hemihedrism, tetartohedrism, holohedrism, 

 etc., into the science of crystallonomy, a grouping such as above 

 explained, presents itself as a very natural one, in so far as such 

 groups which have all certain characteristic properties in common, are 

 gathered into one and the same greater unit. Thus e.g., all groups K H , 

 T H , T D , K, and T, have four ternary axes in common ; the groups : 

 D*$, Z) 6 , C*%, C V 6 and C 6 possess all a single senary axis, etc. It 

 is upon this basis that the arrangement in "crystal-systems" is 

 really founded; and the deduction of the lower symmetrical forms 

 of each system from the higher ones by partial suppression of their 

 faces appears to be artificial and unnecessary. 



5. An easy and clear review of all symmetry-properties, as 

 well as of the most unrestricted forms of each class, may now be 

 obtained in connection with the above stated facts, if a way of re- 

 presenting axes, planes of symmetry, and crystal-faces be made use 

 of, which takes its origin also from Gadolin. *) 



This author uses for that purpose a special form of the so-called 

 "stereographical projection" in which the axes, planes of symmetry, 

 and faces of the polyhedral object are represented in a simple way; 

 and this method may also be made use of 'in cases where the deter- 

 mination of the real symmetry of a given form in nature is required, 

 e. g. in morphological work. Some short remarks upon this method 

 in general, seems therefore to be in place here. 



A stereographical projection of a crystal for instance, is obtained, 

 if from some point in space perpendiculars are drawn upon all 

 faces of a crystal (fig. 87), and if these perpendiculars are continued 

 to their intersection with a spherical surface, described with a radius 

 R round the point as a centre. If now the diametrical plane 

 VV e. g., be chosen as the plane of projection, the projections of 



A. Gadolin, loc. cit.; Ostw. Klass. No. 75, p. 32. (1896). 



