RECENT PROGRESS IN CRYSTALLOGRAPHY 107 



its centre of gravity, is taken to represent it, and the distances 

 apart of these points along the three principal directions within 

 the crystal are represented relatively by the topic parameters. 

 We thus arrive, on purely experimental igrounds, at a system 

 of points as representing the crystal structure, and are quite 

 independent of the as yet unknown shape and size of the 

 molecules. This brings us in touch with a remarkable develop- 

 ment of the geometrical theory of crystal structure, derived 

 from a study of the possible modes of partitioning space, which 

 demands a brief explanation. 



The fundamental crystalline elements of Haiiy were sup- 

 posed to be arranged in a parallel or net-like manner, and 

 in 1833 Frankenheim investigated the possible varieties of such 

 space lattices, and concluded that there were fifteen possible, 

 corresponding with fifteen types of cleavage and of crystal 

 symmetry. Bravais in 1850 reduced them to fourteen, by 

 demonstrating the identity of two of the fifteen, and also 

 showed that the fourteen fell by pairs into only seven truly 

 different systems of symmetry. These seven types are, however, 

 entirely holohedral, and leave hemihedral and tetartohedral 

 crystals unexplained. Camille Jordan, in a memoir in 1869 on 

 "groups of movements," most materially assisted the solution 

 of the problem, from the purely mathematical standpoint, by 

 defining possible types of regular repetition in space of identical 

 parts. He showed that when such identical repetition of parts 

 is exhibited by a rigid system, a definite series or group of 

 correlated movements may be employed, each term being such 

 a movement as, while shifting the system, leaves the appearance 

 the same as before, every point being moved to a position 

 previously occupied by a homologous point. 



This principle was applied in 1879 to crystal structures by 

 Sohncke. He showed that it was unnecessary to assume with 

 Bravais that the molecules are all arranged parallel to each 

 other in lattices, but merely that the arrangement about every 

 molecule, as represented by its " point," is the same as about 

 every other. He thus arrived at sixty-five " regular point 

 systems," of which the space lattices of Bravais are special 

 cases, and which include nearly, but not quite, all the thirty- 

 two known types of crystal symmetry. The next and final step 

 is a very remarkable one, for no less than three independent 

 workers, Fedorow in Russia, Schonflies in Germany, and 



