30 NEW YORK STATE MUSEUM 
is by far the one in most general use and is at present given by many of the 
German crystallographers in addition to the Naumann or Bravais-Miller 
symbols. This system of Goldschmidt consists of two indexes for every 
form, derived from the corresponding Bravais-Miller symbol by dividing 
the first and second indexes of the latter by the last index. Thus the 
general symbol h k i 1 of Bravais becomes }-+ of Goldschmidt. 
Rhombohedral system of Miller. The system 
"of symbols developed in 1839 by W. H. Miller '* 
S is, as applied to crystals of the rhombohedral 
division of the hexagonal system, largely used by 
the English and some of the German writers on 
crystallography. It is in fact the predecessor of 
Fig. 17 a the Bravais system which latter was derived from 
it. Muller refers rhombohedral-hexagonal crystals to three axes instead of 
four, the three Miller axes being parallel to the edges of the unit rhombo- 
hedron of the species. Figure 17 indicates the position of the Miller axes 
for the species calcite, the unit rhombohedron of 4 
which is shown with the Miller symbols for its vo 
respective planes. The indexes of the Miller = Beas 
symbol for any plane are derived from its inter- 
cepts on the three axes by a method analogous to 
that discussed under the Bravais system. The 
general type of Miller symbol is (h k 1). These Pin. 18 
indexes are not susceptible of division into types as is the case with 
those of the Bravais system; for instance the Bravais symbol (4483) 
conforming to the general type (h.h.2h.1) indicates a second order 
pyramid, whereas the corresponding Miller symbol (513) can not be 
referred to any such general type. Familiarity with the Miller symbols, 
however, develops certain characteristics by which forms geometrically 
related may be recognized. The principal advantage of the Miller system 
1 Miller, W. H. A Treatise on Crystallography. London 1839. 
