104 ROGERS— THE VALIDITY OF [March i, 



general, irrational constitute the axial ratios which are constants for 

 each crystallized substance. For convenience in calculation the re- 

 ciprocal ratios of the intercepts are used. These reciprocals are 

 called indices or Miller indices, as Miller, an English crystallogra- 

 pher, was the first to make extensive use of this method. The indices 

 are usually simple numbers such as (no), (210), (130), (211), 

 (321), (441), etc., the unit face being (in). 



If we examine the statements concerning the rationality of the 

 indices of crystal faces in text-books and treatises we find a differ- 

 ence of opinion as to the exact definition of the \d.\v. Some authors 

 insist that the indices are small whole numbers, while others simply 

 state the fact that the indices are whole numbers, usually, but not 

 necessarily small. One crystallographer, Viola/ goes so far as to 

 doubt the validity of the law of rational indices. Another investi- 

 gator, G. H. F. Smith,- believes that the law of simple rational indices 

 is valid except in one particular instance, that of calaverite from 

 Cripple Creek, Colorado. But, as he shows, by assuming several 

 interpenetrant space-lattices it may be valid even in this case. 



Thus there are three possibilities to consider: (i) The indices 

 are always small rational numbers. (2) The indices are rational 

 numbers, but not necessarily small. (3) The indices are not always 

 rational and the law has no meaning. This subject is such a funda- 

 mental one in both theoretical and practical crystallography that it 

 seems advisable to enquire into the history and status of the law. 

 Such is the object of this paper. 



The credit of the discovery of the rationality of the indices is due 

 to Haiiy,^ professor of the humanities in the University of Paris, 

 who developed it from his theory of crystal structure based upon 

 cleavage observations. Haiiy believed that crystals are composed of 

 minute cleavage fragments which he called molecules intcgrantes. 

 Primary faces, according to his view, are due to the association of 

 the molecules in parallel position, while secondary faces are due to 

 the omission of molecules on the exterior of the crystal in step-like 



^ Zeifschrift fur Krystallocjrapliic unci Miiieralogie, Vol. 34, pp. 353-388 

 (1901). 



- Mineralogical Magazine, Vol. 13, p. 122 (1902). 



' " Essai d'une Theorie sur la Structure des Crystaux." Paris, 1784. 



