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III. On the Geometrical Isomorphism of Crystals. 
By Heney James Beooke, F.R.S., Hon. M.C.P.S. 
Eeceived June 11, — Head June 19, 1856. 
Crystals, as it is well known, are solids bounded by plane surfaces termed faces, which 
vary in shape and in their relative positions. They occur generally in small imperfect 
fragments, and all that we can be said to know about them, is the shapes and positions 
of their faces, and the angles which those faces make with each other. 
All the crystals at present kno-wn have been classed in six groups or systems, each 
group consisting of a series of elementary solids, or as they will be termed primary forms, 
and of other forms, termed secondary, derived from the primary according to laws to be 
afterwards explained. These different groups or systems are distinguished from each 
other by the shapes and positions of some of their faces, and the crystals of the different 
minerals comprised in each group or system, except the cubic, are distinguished by the 
angles which particular faces make with each other. 
The six systems, and the elementary solids proper to each, are as follows : — 
The cubic, consisting of cubes which are bounded by six square faces. 
The pyramidal, consisting of prisms having their terminal faces square, and their 
lateral faces rectangular. 
The rhombohedral, consisting of solids bounded by six equal rhombs. 
The prismatic, consisting of solids contained within three pairs of parallel rectan- 
gular faces at right angles to each other. 
The oblique, consisting of solids, also contained within three pairs of parallel faces, of 
which only two intersect at right angles, the thuxl being an oblique pair. 
The anorthic. It is said that in the few solids comprised in this system, no two faces 
intersect each other at a right angle ; and hence the name of the system. 
The accompanying Tables of the pyi’amidal and rhombohedral systems contain lists of the 
crystallized minerals comprised in each, arranged in a manner new, as the writer believes, to 
crystallography ; presenting that science under a new aspect, and affording some altogether 
unexpected results, which it is the object of this paper to communicate to the Society. 
Notation. — Previously, however, to stating these results, it is necessary to explain briefly 
the ciystallogi’aphic language and notation that will be employed in describing them. 
The term form is used in crystallography in two different senses : in one of these it 
simply means shape ; in the other — a sense peculiar to this science — ^it denotes particular 
sets of faces that may or may not enclose a finite solid. Thus, in some instances, single 
pairs of parallel faces constitute ciystallographic forms. 
MDCCCLVII. F 
