108 SCIENCE PROGRESS 



Barlow in this country, in the years 1890 to 1894, showed that, 

 by the introduction of another factor, the principle of mirror- 

 image symmetry (enantiomorphism), which is characteristic of 

 the crystalline forms not explained by Sohncke's sixty-five point 

 systems, all the thirty-two forms of crystalline symmetry are 

 explained. The geometrical methods adopted are different in 

 the three cases, yet all arrive at the same result — namely, that 

 there are 230 types of structure (groups of movements) possible, 

 all of which fall into one or other of the thirty-two classes of 

 crystal symmetry — which completely explains the whole of 

 those classes. Whether it is the molecules or the atoms com- 

 posing them which must be considered as the parts forming 

 the crystal structure, they must be arranged according to one 

 of the 230 types of symmetry, of which both Bravais space 

 lattices and Sohncke point systems are special cases. 



Sohncke has further developed the theory by considering 

 the atoms instead of the centres of gravity of the molecules, and 

 derived a general theory of crystal structure which was stated 

 by Prof. v. Groth, in his memorable address to Section B 

 of the British Association at the Cambridge meeting in 1904, 

 in the following terms: "A crystal — considered as indefinitely 

 extended — consists of n interpenetrating regular point systems, 

 each of which is formed from similar atoms ; each of these point 

 systems is built up from n interpenetrating space lattices, each 

 of the latter being formed from similar atoms occupying parallel 

 positions. All the space lattices of the combined system are 

 geometrically identical, or are characterised by the same 

 elementary parallelopipedon." The combined system conforms 

 to one or other of the 230 possible homogeneous arrangements 

 in space, and is expressed outwardly in the symmetry of one 

 of the thirty-two classes of crystals, while the space lattice 

 determines the crystal system, the angles, and the concordance 

 with the law of rational indices. 



The completion of the geometrical theory renders it un- 

 necessary to assume the existence of any molecular forces to 

 keep the structure together, beyond the interatomic forces. 

 This is entirely in accordance with the conclusion of the writer, 

 from direct experiment, in the 1896 memoir on the structural 

 unit, that there is no chemical union between the molecular 

 constituents of the double salts, and that there is merely 

 aggregation in accordance with such a particular type of 



