83 



ill point- /' will be obtained by joining them to a point M opposite 

 N., which is called the pole of the projection, and if the inter- 

 tit >ns S of V with the straight -%r 

 les MP are considered. All 

 >ints 5 thus obtained, form 

 together the stereographical pro- 

 jection of the crystal F. *) 

 Now Gadolin determines the 

 irection of the symmetry-axes * 



of the perpendiculars to the 

 i \-tul-faces just in the same way. 

 ily he superposes the two 

 lages which would be obtained 

 >y projection of the upper and 

 ic lower half of the polyhe- 

 jn, if observed from M or 

 rom N respectively, and he distinguishes the faces above and 

 jneath the plane of projection V simply by different signs, e. g. 

 >y X and O. The period of the axes is denoted in the way 

 lescribed further on. 



For the purpose of illustrating the application of this method 

 >r the representation or the eventual determination of the specific 

 symmetry of a body or of its general form, we will apply it in the 

 of the cubic system only, and deduce in this way the most 

 irestricted polyhedral forms in every class of it. It will then be 

 5y in the same way to extend such considerations to every other 

 lass of crystals. 



Moreover it may be mentioned that the method indicated here 

 lay be recommended in all cases where the special symmetry of 

 )me complicated form or object has to be found. Thus the special 

 imetry of many complicatedly built radiolaries, e. g. of Etmo- 

 Maera siphonophora (Haeckel), etc., or the arrangement of their 

 Mcula, or the type of symmetry of a flower or of some animal, may 

 )ften easily be found, if the repeatedly occurring parts of the object 

 be projected in the way considered, upon a spherical surface, and 



!) For the full application of the stereographical projection and its proper- 

 ties, we may refer here to the numerous treatises on crystallography, in which 

 this method is explained in detail. Cf. more particularly: H. E. Boeke, Die 

 Anwendung der stereographischen Projektion bei krystallographischen Unter- 

 suchungen, Berlin, (1911). 



