William Hallowes Miller. 881 
and completeness in a work remarkable for the vivacity of its 
style and the felicity of iis illustration. Moreover, a simple 
mathematical expression was given to the system, and the 
notation which Haiiy invented to express the relation of the 
secondary. to the primary forms, as modified and improved by 
Lévy, is still used by the French mineralogists. 
e system of Haiiy, however, was highly artificial and 
only prepared the way for a simpler and more general expres- 
sion of the facts. 'The German crystallozgrapher Weiss seems 
to have been the first to have recognized the truth that the 
decrements of Haiiy were merely a mechanical mode of repre- 
senting the fact that all the secondary faces of a crystal make 
intercepts on the edges of the primitive form which are simple 
multiples of each other; and this general conception once gained 
It was soon seen that these ratios could be as simply measured 
on the axes of symmetry of the crystal as on the edges of the 
fundamental forms; and, moreover, that when crystal forms 
are viewed in their relation to these axes a more general law 
becomes evident, and the artificial distinction between primary 
Freiberg,” belongs the merit of first developing a complete 
system of theoretical crystallography based on the laws of sym- 
