SYSTEMS OF CRYSTALLIZATION. 331 



made by Levy, Naumann, Grassmann, Kupffer, Hessel, and by Profes- 

 sor Miller among ourselves. I may notice that one great improvement 

 was, the method introduced by Monteiro and Levy, of determining the 

 laws of derivation of forces by means of the parallelisms of edges ; 

 which was afterwards extended so that faces were considered as belong- 

 ing to zones. Nor need I attempt to enumerate (what indeed it would 

 be difficult to describe in words) the various methods of notation by 

 which it has been proposed to represent the faces of crystals, and tc 

 facilitate the calculations which have reference to them. 



[2nd Ed.] [My Memoir of 1825 depended on the views of Hatiy in 

 so far as that I started from his " primitive forms ;" but being a general 

 method of expressing all forms by co-ordinates, it was very little 

 governed by these views. The mode of representing crystalline forms 

 which I proposed seemed to contain its own evidence of being more 

 true to nature than Haiiy's theory of decrements, inasmuch as my 

 method expressed the faces at much lower numbers. I determine a 

 face by means of the dimensions of the primary form divided by certain 

 numbers ; Haiiy had expressed the face virtually by the same dimen- 

 sions multiplied by numbers. In cases where my notation gives such 

 numbers as (3, 4, 1), (1, 3, 7), (5, 1, 19), his method involves the 

 higher numbers (4, 3, 12), (21, 7, 3), (19, 95, 5). My method how- 

 ever has, I believe, little value as a method of " calculating the angles 

 of crystals." 



M. Neumann, of Konigsberg, introduced a very convenient and 

 elegant mode of representing the position of faces of crystals by corre- 

 sponding points on the surface of a circumscribing sphere. He gave 

 (in 1823) the laws of the derivation of crystalline faces, expressed geo 

 metrically by the intersection of zones, (Beitrdge zur Jfrystallonomie.) 

 The same method of indicating the position of faces of crystals Avas 

 afterwards, together with the notation, re-invented by M. Grassmann, 

 (Zur Krystallonomie und Geometrischen Combinationslehre, 1829.) 

 Aiding himself by the suggestions of these writers, and partly adopting 

 my method, Prof. Miller has produced a work on Crystallography re- 

 markable for mathematical elegance and symmetry ; and has given 

 expressions really useful for calculating the angles of crystalline faces, 

 (A Treatise on Crystallography. Cambridge, 1839.)] 



Confirmation of the Distinction of Systems by the Optical Properties 

 of Minerals. Brewster. I must not omit to notice the striking con- 

 firmation which the distinction of systems of crystallization received 

 from optical discoveries, especially those of Sir D. Brewster. Of tht 



