I9I2.] THE LAW OF RATIONAL INDICES. Ill 



The intimate connection between rational indices, molecular 

 structure, and symmetry-axes with periods of 2, 3, 4, and 6 can not 

 be denied. If one is true, it is pretty certain that the others are 

 true. There is direct proof of only one of these facts, viz., sym- 

 metry-axes of the kinds mentioned. This is the empirical basis 

 upon which my argument rests. It is absolutely true that only axes 

 of 2-, 3-, 4-, and 6-fold symmetry have ever been found and it is 

 very probable that these are the only ones that ever will be found. 

 Suppose crystals with an axis of 5-fold symmetry should be in- 

 cluded as possible. If five-fold axes are possible, axes of 7-, 8-, 9-, 

 lo-fold, etc., would also be possible, for the minimum possible dis- 

 tance between two particles excludes axes with periods greater than 

 6 for the same reason that it excludes those with a period of 5. 

 Then instead of 32 crystal classes with one gap to be filled, we 

 should have an indefinite number of crystal classes but with only 

 31 of them yet found in nature. 



Even if we grant that the indices are rational numbers, crystal- 

 lography would still be very complicated for the number of possible 

 rational ratios is very large. In the orthorhombic system, for ex- 

 ample, there are 1,037 possible forms with indices not over 10. Yet 

 for the mineral topaz, which leads all orthorhombic minerals in 

 the number of forms there are only about 125 known forms. For 

 all orthorhombic minerals taken together there are only about 386 

 known forms with indices not over 10. Of all known substances 

 calcite has the greatest number of crystal forms, about 325 well- 

 established ones with about 140 uncertain ones. Only about a half 

 (162) of the forms have indices greater than 10,^^ yet the possible 

 number of forms in the calcite class with indices not greater than 10 

 is 876. 



We need an explanation that will reconcile the observed fact that 

 the indices are usually simple with the fact that they are occasion- 

 ally complex, the complexity, in general, increasing with the rarity. 

 Such an explanation is furnished by the structure-theory of 

 Bravais.^° Bravais assumes that the centers of molecules occupy 

 the points of a space-lattice. Fourteen kinds of space-lattices, con- 



^^ That is, h, k, and / in the symbol hk'il are not greater than 10. 

 ^®" Etudes Cristallographiques," Paris (1866). 



