Olf THE STRUCTURE OK CRYSTALS. 305 



The general problem of the symmetrical space arrangements available 

 for crystals was at first supposed to be a comparatively simple one. 

 Sohncke remarks ' that all the various extensions of Haiiy's theory put 

 forward by the writers above referred to led to the same conclusion, viz., 

 that the arrangement of the middle points of the crystal elements is that 

 of a parallellepipedal network or 'space-lattice' (Raumgitter),- such as 

 that shown in fig. 4. 



In this simple guise the problem was dealt with exhaustively by 

 M. L. Frankenheim, who investigated the difierent kinds of parallelepi- 

 pedal networks of points (Raumgitter) possible in order to ascertain 

 whether these correspond to the various types of symmetry presented by 

 crystals."' He did not, however, at first furnish any rigid proof, and his 

 classification of the various kinds of symmetry presented is not perfectly 

 satisfactory : he described fifteen forms as distinct from each other, whereas 

 in fact there are but fourteen, as was afterwards shown by Bravais. He 

 states explicitly that the inquiry is founded solely on the symmetrical 

 arrangement in space of the ultimate particles, and is not based on con- 

 siderations of the magnitude or the shape of these particle.s, be they 



Fig. 4. 



plane-faced like small crystals or rounded ; solid spheres or hollow com- 

 pressible shells ; or, indeed, mere centres of force. For the purpose of 

 comparison with the fifteen geometrical systems of points which he has 

 discriminated he classifies crystals into fifteen systems by taking note of 

 differences in cleavage direction as well as of diflferences of crystal form. 



The obvious objection to Frankenheim's treatment of the subject is 

 that unless some appropriate configuration be attributed to the particles — 

 and this he expressly disclaims — no hemihedral or hemimorphous forms 

 are accounted for ; and yet, as pointed out by Delafosse, there is no more 

 justification for regarding these forms as subsidiary than for so regarding 

 the holohedral forms. 



But none the less the solution of the problem of the possible varieties 

 of space lattices, and the establishment of the fact that in their symmetry 

 they correspond to the systems of crystals, marks a very important advance 

 in the theory of crystal structure. 



' Sohncke, Bntivielwluvg einer Theoric der KrystalhtruJduf, p. 17. 

 - Sec above, p. 304. 



' Die Lelire run- dcr Colil'iRion, Breslau, 18.'!5 ; also ' System der Crystalle ' ia 

 Kora Acta Acad. Cas. Leopoldino- CaroUncc Nat, Ciir., 1842, xlx, (2), pp. 471-6G0. 

 1901 X 



