136 Prof. Breithaupt on Thirteen Systems of 



the general primary form, and the II as the especial primary 

 form. 



The crystallographer must in future take a hexagonal axis as 

 the principal one in boracite, as well as in the above-mentioned 

 pyrites. The value of the form R is easily calculated ; from the 

 angles it contains, it appears that the inclination of the planes 

 to the principal axis is 70° 59'. If, then, we use the formula 



*p = 1 to express the principal axis of such a rhombohedron, we 



obtain, from \\ of the planes and the angle they form with the 

 chief axis, the angle 70° 58' 10" ; the other angle we found 

 = 144° 1 7', which, according to calculation, = 144° 1 7' 2". If we 

 deduce R from the rhombohedron of the rhombic dodecahedron, 

 we get the coefficient f#, and from the cube -&$. 



Further, the inclination of the planes to the polar edges of R 

 is found to be 147° 12' 46" ; 



if calculated after ^, = 146° 26' 33" ; 



46' 13", which is an important de- 

 viation, and could easily be found with the hand-goniometer on 

 a crystal the size of the tip of one's finger. 



It is on account of this R that boracite has one optical and 

 one crystallographical axis ; and thus, owing to prejudiced opi- 

 nions, boracite has been considered isometrical for forty-one 

 years. 



If we wish to carry our comparisons between the tesseral sy- 

 stem and the symmetrical tetragonal and hexagonal systems 

 further, we find some of the minor divisions wanting. The 

 tetragonal garnets represent the holohedral division of the tetra- 

 gonal systems. 



May not a mineral be found also of a tetragonal character 

 which has hitherto been called semitesseral ? It seems very pro- 

 bable to me that among those minerals which I have called 

 Clinoedrites, viz. Kupferblende, Tennantite, Fahlerz, Schwarzerz, 

 and crystallized Weisgiiltigerz (from Freiberg), some species may 

 be found having other symmetrical forms than those which they 

 are generally supposed to possess. Pharmakosiderite, helvin and 

 eulytine must have their two-edged forms investigated; and then, 

 I fancy, the missing links will be found. Further, were it pos- 

 sible to construct two hexagonal pyramids out of that tetrakis- 

 hexahedron ^ J (which is the holohedral form of the pentagonal 

 dodecahedron), when placed upright on a hexagonal axis (and 

 this might be occasioned by the necessary difference in the 

 angles), then an analogy would be established between the hex- 

 agonalized tesseral system and the holohedral division of the 

 symmetrical hexagonal system. Perowskite, and perhaps fluor- 



