118 



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

 bility of which needs no further discussion, only the values n= 3, 

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

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

 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 7). 



6. The restric- 

 tions to be made in 

 the discussion of the 

 possible symmetries 

 of space-lattices, as 

 following from the 

 considerations in J, 

 will of course diminish 

 very appreciably the 



number of possible 

 B r a v a i s' Fourteen Types of Space-lattices. . 



symmetrical space- 



lattices, and at the same time simplify the deduction of them 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. IQJ. Their symmetry corresponds to that of the holohedral 

 classes of the seven well-known crystal-systems; for the triclinic 



1) 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). 



J&xaqonal 



I0 5- 



