492 SCIENCE PROGRESS. 



the necessity of taking into account the nature of the 

 particles ; in other words, the chemical structure of the 

 substance. 



By this time, however, we seem to have lost sight of 

 the simple form of a crystal, in order to account for its 

 symmetry ; the composite interpenetrating lattices and 

 spirals of Sohncke may show every conceivable variety of 

 crystalline symmetry, but they do not lead to simple plane 

 faces which obey the law referred to above as the law of 

 rational indices. In order to limit the faces to such as obey 

 this law, we have to suppose that the only planes possible 

 as crystal faces are those which belong to any one of the 

 lattices contained in the structure. 



It appears, then, that in Sohncke's theory a cluster of 

 particles corresponds exactly to a molecule in the Bravais 

 theory ; the crystal faces and the planes of cleavage are any 

 planes containing a set of these molecules ; and the sym- 

 metry of the Bravais molecule is that of the Sohncke cluster, 

 and upon this in reality the symmetry of the crystal (and of 

 the lattice) depends. 



At this stage we have arrived at the following conclu- 

 sions : If the orderly arrangement of particles is to be taken 

 to represent the structure of a crystal, then, in order to 

 account for the crystal faces we require a lattice structure ; 

 in order to account for the symmetry we require a homo- 

 geneous assemblage ; the latter turns out to consist of inter- 

 penetrating lattices ; but we have further to suppose the 

 lattices to be different in their nature. 



THE THIRTY-TWO TYPES. 



Meanwhile the problem had been approached in a 

 different manner. Including all the merohedral forms, it 

 will be found that the total number of varieties of symmetry 

 possible in crystals amounts to thirty-two ; these, of course, 

 naturally fall into seven groups identical with the seven 

 systems of crystallography. 



It has been only recently pointed out (11) these 

 thirty-two groups were established without any structural 



