MINERALOGY. 47 
right angles to it and to each other, these lines to be called the secondary 
axes. These three axes measure the three dimensions of a body, and it 
will be found on examining these dimensions that the natural distances 
between the surfaces bounding the crystals are not without a rule, but are 
rather determined by laws referable to variations in the ratio of the length 
and angular relations of these axes. According to various definite ratios 
between the three axes, we may group a large number of, at first, apparently 
different crystalline forms. However different the ratio of three such lines 
may be in respect to their length, estimated from their middle, or point of 
intersection, or however varied the angle of inclination, nevertheless we 
know of but six essentially different sets of axial proportions. The simplest 
figures determined directly by these axes are called primary forms. All 
the primary forms, with the secondary forms derived from them, however 
different they may seem, are referable to one of six systems of crystal- 
lization. 
2. Crystallography. 
That part of mineralogy relating to systems of crystallization, is called 
Crystallography. The systems referred to in the preceding paragraph are 
briefly as follows : 3 
I. The Monometric System, Dana. The Regular System (Das regulaire 
System), Weiss. Tessular, Mohs. Tesseral, Naumann. Isometric, 
Hausmann. 
The character of this system is such that, if a certain point be taken, 
and three axes be drawn through this point, at right angles to each other, 
they will all be bounded at equal distances by a solid angle, a face, or an 
edge. It is therefore a matter of indifference which axis we make the 
vertical or primary, all three being of equal value. The regular octahedron 
is generally taken as the type of this system, the others being derived from 
it. The most general forms are as follows: 
1. The regular octahedron (pl. 32, figs. 11 and 12). This is a solid 
inclosed by eight equilateral triangular faces, intersecting each other, in 
six solid angles, and twelve edges. Connecting each opposite pair of solid 
angles will give us the three axes, intersecting each other at right angles in 
a common point, and of the same length from the point of intersection. 
Whatever two opposite angles be selected as the limits of the vertical axis, 
the others will always have the same relative situation. Natural crystals, 
however, do not always exhibit the perfect symmetry thus indicated. 
Distortions frequently occur, one of which is given hereafter as a deri- 
vative of the octahedron. Thus if we intersect any two parallel faces of 
this solid, by a plane parallel to another face. or if we move one face 
parallel to itself, nearer to the centre, a hexagon (fig. 11, a, a’, a’, a'"), 
will be obtained by the lines of intersection. 
By moving two parallel faces towards the centre, these with the six othet 
abbreviated faces will inclose an octahedron abbreviated to a six-sided 
plate (fig. 18). Octahedrons distorted in this manner are more abundantly 
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