538 CRYSTALLOGRAPHY ratio, may still crystallize in a variety of forms. Thus the diamond, which is isometric, occurs in octahedrons, in dodecahedrons, and in solids that are like octahedrons in general form, but have low pyramids of three or six faces in place of each octahedral face (called tris-octa- hedrons and hex-octahedrons, the number of faces being either 3 x 8 = 24, or 6 x 8 == 48), and in various combinations of these forms. So, dimetric species, as idocrase, may occur in simple square prisms, or in square prisms with the lateral edges truncated or bevelled, or with different planes on the basal edges or angles, or in eight-sided prisms, or in square octahe- drons, &c. In the species calcite, the number of derivative forms amounts to several hun- dreds. This simple fact shows that while cohe- sive attraction in calcite, for example, some- times produces the fundamental rhombohedron, it may undergo changes of condition so as to produce other forms, and as many such changes as are necessary to give rise to all the various occurring forms of the species, with only this limitation, that they are all bused on the fun- damental axial ratio, 0-8543 : 1. VI. In all cases of derivative or secondary forms, either (1) all similar parts (parts similarly placed with reference to the axes) are modified alike, or (2) only half, alternate in position, are modified alike. This law may be explained by reference to a square prism. In this prism there are two sets of edges, the basal and lateral ; the two sets are unlike, that is, are unequal, and included by different planes. One set may therefore be modified by planes when the other is not; moreover, when one basal edge has a plane on it, all the others will have the same plane, that is, a plane inclined at the same angle to the base; or if one has a dozen different planes, all the others will have the same dozen. Again, if a lateral edge is replaced by one plane, that plane will be equally inclined to the lateral planes, because those planes (or, what is equiv- alent, the lateral axes) are equal ; and in addi- tion, all the lateral edges will have the same plane. In a cube, the 12 edges are all equal and similar ; and hence, if one of them has a plane on it, as in fig. 18, there will be a similar plane on each of the 12. Hence, we may dis- tinguish a cube, modified on the edges, however much it may be distorted, by finding the same FIG. 18. planes on all the 12 edges of the solid. The eight angles of a cube are similar, and hence they will all have similar modifications, either one plane, as in fig. 19, or three planes, or six as in fig. 20. Again, the eight angles of a square prism are similar and therefore are modified alike. The square prism and cube differ in this, that in the cube, when there is one plane on each angle, that plane will incline equally to each of the three faces adjoining, because these faces are equal ; while in the square prism, the plane will incline equally to the two lateral planes and at a different angle to the base. This general law, " similar parts similarly modi- fied," is in accordance with what complete symmetry would require. The exception men- tioned, of half the parts modified without the other half, is exemplified in boracite (fig. 21), in which half of the eight solid angles of the cube have planes unlike those of the other FIG. 21. FIG. 22. half a mode of modification that gives rise to the tetrahedron (fig. 22) and related forms; in tourmaline, in which the planes at one end of the crystal differ from those at the other ; and in pyrite, in which on each edge there is only one plane out of a pair of bevelling planes. Fig. 23 represents a cube with all the edges bevelled, that is, replaced by two similar planes a holohedral form; while fig. 24 is that of a hemihedral form, only one of the two bevelling planes being present on each edge, a common form of pyrite. All such forms are said to be FIG. 28. FIG. 24. hemihedral (Gr. ymav, half, and edpa, face), while the former are said to be holohedral (SAof, all, and k 6pa). Many hemihedral crystals, when undergoing a change of temperature, have opposite electrical poles developed in the parts dissimilarly modified. VII. The derivative forms, under any species, are related to one another by simple multiples of the axial ratios. In calcite the fundamental rhom- bohedron has the axial ratio just mentioned, 0-8543 : 1, that is, a = 0*8543. There are a number of derivative rhombohedrons among the crystalline forms of this species ; one has the vertical axis |; another a ; others j?rt, f#, 2, 3, 4a, and so^ on, by simple multi- ples of the vertical axis of the fundamental form. So in zircon, of the dimetric system, as implied above, while a (vertical axis) =0-6407", the lateral being unity, there is one derivative octahedron (1, fig. 25) with the axes a : 1 : 1 ; another, 2a : 1 : 1 ; another, 3a : 1 : 1 ; also a diagonal pyramid, a : 8 : 1 (1 i in fig.) ; and
