85 



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 

 all points P will be obtained by joining them to a point M opposite 

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

 sections 5 of V with the straight 

 lines MP are considered. All 

 points 5 thus obtained, form 

 together the stereographical pro- 

 jection of the crystal F. l ) 



Now Gadolin determines the 

 direction of the symmetry-axes^ 

 and of the perpendiculars to the 

 crystal-faces just in the same way. 

 Only he superposes the two 

 images which would be obtained 

 by projection of the upper and 

 the lower half of the polyhedron, 

 if observed from M or from N 

 respectively, and he distinguishes the faces above and beneath 

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

 x and O. The period of the axes is denoted in the way described 

 further on. (fig. po). 



For the purpose of illustrating the application of this method 

 for the representation or the eventual determination of the specific 

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

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

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

 easy to extend in the same way such considerations to every other 

 class of crystals. 



Moreover, it may be mentioned that the method indicated here 

 may be recommended in all cases where the special symmetry of 

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

 symmetry of many complexly built Radiolaries, e. g. of Ethmo- 



*) 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 Pyojektion bei krystallographischen Unter- 

 suchungen, Berlin, (1911); and the valuable papers of V. Goldschmidt, Zeits. f. 

 Kryst. 28.401,414. (1897) ; 29. 364. (1898); 30. 254. (1899); idem, Ueber Entwicke- 

 lung der Kry stall formen,and: Atlas der Krystallformen, Heidelberg, (1913). 



