REPORT ON MINERALOGY. 
331 
these fundamental radii. Neumann (a different person from 
Naumann,) has also endeavoured to simplify the subject by the 
introduction of normals and “ index planes,” as I learn from 
Mr. Hessel, who has followed him in the use of these terms. 
Mr. Hessel himself (Professor at Marpurg,) in his work entitled 
“ Crystallometry” (Leipzig 1831), has adopted several other 
new denominations and modes of considering forms. Thus, by 
way of example, he states concerning the rhombic dodecahe¬ 
dron, that its twelve faces “ are perpendicular to doubly-two- 
membered normals, (the edges are doubly-one-membered, like- 
sided, unlike-ended,) which are perpendicular to doubly-one- 
membered four-and-three-spaced rays,” &c. 
The principle of this and similar methods of treating this 
subject consists in the permutations and combinations of various 
kinds of symmetry in lines, surfaces, and solids. One kind of 
symmetry, which occurs frequently in crystals, is not easily 
described by any common expression; and Mr. Hessel, who 
justly attaches much importance to the consideration of it, has 
introduced a peculiar term to designate it. The symmetry here 
spoken of is that which is seen in comparing the two ends of an 
oblique prism ; and they are called by him “ gerenstelli g” gore- 
wise-placed, in opposition to “ gleichstellig” alike-placed. One 
or two new phrases in such cases may perhaps be introduced 
■with advantage: but the systems to which I here refer are so far 
laden with new phraseology and new views of the relations of 
space, that they will probably not be found by many a con¬ 
venient mode of mineralogical study. It may be readily allowed, 
that when a person has mastered the fundamental views of 
these methods, the application of them to crystallometry may 
have some advantages of order or of generality; but this is for 
most readers too long and indirect a road to the results ; and if 
crystallography leads to new views on the subject of elementary 
geometry, the prosecution of these will interest the pure ma¬ 
thematician, but the mineralogist will find it necessary to con¬ 
fine himself to investigations more peculiarly professional. 
The consideration of the faces of crystals as distributed into 
zones, points out a mode of transition from one system of cry¬ 
stallization to another. Thus, if a rhombic dodecahedron be 
placed so that an axis is vertical which is terminated by three 
plane angles at each vertex, we may then, by prolonging or con¬ 
tracting the axis, make the form pass from the tessular to the 
hexagonal system. But if the vertical axis be one which is ter¬ 
minated by four planes at each end, its prolongation or con¬ 
traction converts the same form into one belonging to the te¬ 
tragonal system. This mode of deriving one set of forms from 
another has been followed up by Breithaupt, who has thus 
