116 ROGERS— THE VALIDITY OF [March i, 



just as the law of rational indices was deduced by Haiiy from his 

 theory of crystal structure. If chemical compounds are made up 

 of atoms they must necessarily unite in definite proportions. This 

 it will be recalled is precisely analogous to the argument used for 

 proof of the rationality of the indices. If crystals are made up of 

 particles or molecules, the crystal faces necessarily have rational 

 indices. 



Two or more given elements do not unite in all possible pro- 

 portions but in a comparatively few, usually simple, proportions 

 which we explain by the term valence. There are but two oxids of 

 mercury Hg.O, and HgO which we explain by saying that the 

 valence of mercury is one and two. This is analogous to the limita- 

 tion imposed by the law of complication of Goldschmidt or the law 

 of maximum reticulate density of Bravais. 



To complete the analogy between the laws and theories of crys- 

 tallography and chemistry let us consider the periodic law and its 

 analogue. Alendeleef, the Russian chemist, predicted the existence 

 of several chemical elements, scandium and gallium, which he called 

 ekaboron and eka-aluminum, before they were discovered. Not less 

 remarkable was the deduction by Hessel, a German mathematician, 

 ^ of the thirty-two possible types of symmetry in crystals, assum- 

 ing 2-, 3-, 4-, and 6-fold symmetry-axes, in 1830, at a time when 

 only about half of them were known. Of the thirty-two possible 

 types of symmetry, only one remains to be found. 



Summary. 



Judging from various text-books and articles a difference of 

 opinion exists as to the exact meaning of the law of rational indices. 

 Some authors limit the indices to simple numbers while others 

 admit that occasionally the indices are large numbers. Unfortu- 

 nately this question can not be decided by direct measurement of the 

 angles on account of errors in measurement. As crystals possess 

 axes of only 2-, 3-, 4-, and 6-fold symmetry they must consist of 

 regularly arranged molecules, or particles of some sort, whatever 

 their nature may be. Crystal faces, then, necessarily have rational 

 indices. The indices are usually small numbers but may also be 



