382 William Hallowes Miller. 
ately disciplined in mathematics. 
ut, however comprehensive and perfect in its details, the 
system of Naumann was cumbrous, and lacked elegance o 
mathematical form. This arose chiefly from the fact that the 
old methods of analytical geometry were unsuited to the prob- 
lems of crystallography; but it resulted also from a habit of 
the German mind to dwell on details and give importance to 
systems of classification. To Naumann the six crystalline sys- 
tems were as much realities of nature as were the forms of the 
integrant molecules to Haiiy, and he failed to grasp the larger 
thought which includes all partial systems in one comprehen- 
sive plan. 
Our late colleague, Professor Miller, on the other hand, had 
that power of mathematical generalization which enabled him 
to properly subordinate the parts to the whole, and to develop 
a system of mathematical crystallography of such simplicity 
and beauty of form that it leaves little to be desired. This 
was the great work of his life and a work worthy of the Uni- 
versity which had produced the “ Principia.” It was published 
in 1839 under the title “A Treatise on Crystallography,” and 
in 1863 the substance of the work was reproduced in a more 
perfect form, still more condensed and generalized, in a thin 
volume of only eighty-six pages, which the author modestly 
ealled “ A Tract on Crystallography.” 
Miller began his study of crystallography with the same 
materials as Naumann; but in addition he adopted the beauti- 
ful method of Franz Ernst Neumann of referring the faces 
