WI 1,1,1 AM IIALI.OWRS MILLER. 468 



law, was n great advance beyond tlio decrements of Ilaily, an ad- 

 vance not unlike that of astronomy from tlio wystem of vortices to the 

 law of giavitation. Yet, as the mcclianism of vortices was a natural 

 prelude to the law of Newton, so the decrements of Iluiiy prepared 

 the way for the wider views of the German crystaliographers. 



Whether Weiss or Mohs contributed most to advance crystallog- 

 raphy to its more philosophical stage, it is not important here to 

 inquire. Each of these eminent scholars did an important work in 

 developing and diffusing the larger ideas, and in showing by their 

 investigations that the facts of nature correHpon(l(!d to the new con- 

 ceptions. But to Carl Friedrich Naumann, Professor at the time in 

 the " IJergakademie zu Freiberg," belongs the merit of first develop- 

 ing a complete system of theoretical crystallography based on the 

 laws of symmetry and axial ratios. Ilis " Lehrbuch der reinen und 

 angewandten Krystallographie," published in two volumes at Leipzig 

 in 1830, was a remarkable production, and seemed to grasp the whole 

 theory of the external forms of crystals. Naumann used the obvious 

 and direct methods of analytical geometry to express the quantitative 

 relations between the parts of a crystal ; and although his methods are 

 often unnecessarily prolix and his notation awkward, his formulae 

 are well adapted to calculation, and easily intelligible to persons 

 moderately disciplined in mathematics. 



But, however comprehensive and perfect in its details, the system 

 of Naumann was cun)brous, and lacked elegance of mathematical 

 form. This arose chiefly from the fact that the old methods of ana- 

 lytical geometry were unsuited to the problems 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 systems were as much realities of nature as were the 

 forms of the integrant molecules to Ilaijy, and he failed to grasp the 

 larger thought which includes all partial systems in one comi»rehensivo 

 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 mathe- 

 matical 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 University which had produced the " Principia." 

 It was published in 1839, under the title "A Treatise on Crystallog- 

 raphy ;" and in 1863, the substance of the work was reproduced in a 

 more perfect form, still more condensed and generalized, in a thin 



