121 



From this it is obvious that besides the value n = 2, the possi- 

 bility of which needs no further discussion, only the values n = j, 

 ^, and 6 fulfil the conditions mentioned above. No value greater 

 than 6 is allowable, and n = 5 is excluded because of the irrational 

 value of the corresponding parameter. It appears, therefore, that 

 in this respect also the space-lattice may be considered as a geome- 

 trical interpretation 

 of a fact previously 

 stated, and intrinsi- 

 cally identical with 

 the significance of 

 Hauy's law, namely: 

 the restriction of the 

 possible crystallogra- 

 phical axes (Chap- 

 ter V)'. 



6. 



The restrictions to 

 be made in the dis- 

 cussion of the possible 

 symmetries of space- 

 lattices, as following 

 from the considera- 

 tions in 5, will, of 

 course, diminish very 

 appreciably the num- 



C. 



Fig. 107. 



Bravais' Fourteen Types of Space-lattices, 

 ber of possible sym- 

 metrical space-lattices, and at the same time simplify their deduction 

 to a considerable degree. 



It was Bravais, 1 ) who in 1848 solved the problem: what are 

 all possible types of symmetrical space-lattices?, -- not only com- 

 pletely, but at the same time applied the theory successfully to 

 various problems concerning the internal structure of crystals. 



Bravais demonstrated, that there are only fourteen possible types 

 of symmetrical space-lattices, the unit-cells of which are represented 

 in fig. 107. Their symmetry corresponds to that of the holohedral 

 classes of the seven wellknown crystal-systems; for the triclinic 



*) A. Bravais, Journ. de 1'Ecole polyt., 19, 1, (1850); 20, 201, (1851); 

 Etudes crystallographiques, Paris, (1866), p. 1128; 101287; L. Sohncke, 

 Pogg. Ann. d. Phys., 132, 75, (1867). 



