MOLECULES IN A CRYSTAL. 493 



theory whatever, as early as the year 1830, by Hessel, who 

 investigated in the most general manner all the possible 

 modes in which a geometrical figure can be symmetrical ; 

 he then introduced a condition corresponding to the law of 

 rational indices (i.e., limited his conclusions to crystals), and 

 deduced thirty-two types of symmetry. 



Hessel's investigation was entirely overlooked for sixty 

 years, and the problem was taken up anew by subsequent 

 authors. 



Gadolin, in 1867 (12), came to precisely the same con- 

 clusions ; he began by introducing the law of rational 

 indices, and found that all figures which obey this law, i.e., 

 all crystals, if grouped by their symmetry, fall into thirty- 

 two classes. 



The essential difference between their methods and that 

 of the lattice consists in the fact that Hessel and Gadolin 

 distinguish between different directions along the same line, 

 whereas in the lattice no distinction is made between the 

 two directions along the same thread of particles ; to in- 

 troduce this distinction something must be said about 

 the shape of the particles ; they must have a polar char- 

 acter. 



Again, in the simple lattice parallel planes are absolutely 

 identical, and a plane has the same aspect viewed from one 

 side as from the other since the shape of the particles is not 

 taken into account ; hence such figures as the tetrahedron, 

 which is not bounded by parallel planes, cannot be explained 

 by the lattice. 



Conclusions identical with those of Hessel and Gadolin 

 have been independently obtained with different methods 

 by subsequent investigators,. All Hessel's varieties of sym- 

 metry belonging to geometrical figures in general have been 

 recently found by Fedorow and Curie ; while Gadolin's 

 thirty-two crystallographic types have been established by 

 Minnigerode and Schonrlies. 



The possible existence of these thirty-two types may 

 now be regarded as a rigidly demonstrated geometrical 

 fact, which must lie at the base of any views on the struc- 

 ture of crystals. Crystals belonging to all the types have 



