MINERALOGY 351 arts of The cube (fig. 26) is bounded by six equal squares, has be cube, twelve edges formed by faces meeting at 90, and eight solid trigonal angles. The axes are taken as joining the centres of each two opposite faces. Examples are hallite, galena, and fluor. Fig. 34. Fig. 35. Dodeca hedron. etrakis- exa- edron. Fig. 36. Priakis- pcta- Tlie octahedron (fig. 30), bounded by eight equilateral triangles, has twelve equal edges with planes meeting at 109 28 16", and six tetragonal angles. The principal axes join the opposite solid angles. Examples : magnetite, gold, cuprite. The rhombic dodecahedron (fig. 33) is bounded by twelve equal and similar rhombi, has twenty-four equal edges of 120, and has six tetragonal and eight trigonal angles. Each of the principal axes joins two opposite tetragonal angles. Examples: garnet, cuprite, blende. The tetrakishexahedrons (figs. 36, 37, 38, varieties of icositetrahedron) are bounded by twenty-four isosceles triangles, placed so as to form four-sided pyramids on the faces of the cube, arranged in six groups of four each. They have twelve longer edges, which correspond to those of the pri mitive or inscribed cube, and twenty-four shorter edges placed over each of its faces. The angles are eight hexagonal and six tetragonal, the latter joined two and two by the principal axes. Examples : fluorite, gold. This form varies much in general aspect. The four-sided pyra mid which rests on the edges of each face of the cube may be so low as almost to fall into it (fig. 36) ; or it may rise so high that each side forms a level surface with that which is ad jacent to it upon the nearest cubic face (fig. 38). In the latter case the form has become the rhombic dodecahedron ; so that the more or less acute varieties of the form are but stages of a passage of the cube into the latter figure, through an increasing accretion of matter in the lines of the axes of the cube. This is termed a " tran sition by increment." The triakisoctahedrons, fig. 39 (variety of icositetrahedron, fig. 40), are bounded by twenty- four isosceles triangles, in eight groups of three, arranged as pyramids on the edges of the faces of the octahedron. Like the previous form they vary in general aspect, the variation here being from the octahedron on one side to the rhombic dodecahedron on the other ; while the increased accretion here is in the direction of lines joining the centres of the faces of the octahedron or the solid angles of the cube. The passage of the forms is similar to that illustrated in the last-con- Fig. 39. sidered form. The edges are twelve longer, corresponding with those of the inscribed octahedron, and twenty-four shorter, three and three over each of the faces. The angles are eight trigonal and six ditetragonal (formed by eight faces), the latter angles joined two and two by the principal axes. Examples : galena, diamond. The icositetrahedrons (fig. 40) are bounded by twenty- icosi- four deltoids. This form varies from the octahedron to tetra- the cube, sometimes approaching the former and sometimes the latter in general aspect. A four-sided pyramid rests on the angles of the faces of the cube. When increased accretion takes place along the cubic axes, an octahedron results. When it is along lines joining the solid angles of the cube, that form itself results. The edges are twenty- four longer and twenty-four shorter. The solid angles are six tetragonal joined by the principal axes, eight trigonal, and twelve rhombic or tetragonal with unequal angles. Examples : analcime, garnet. The hexakisoctahedrons (fig. 41), bounded by forty-eight Hexakis- scalene triangles, vary much in general aspect, approaching octa- more or less to all the preceding forms, into all of which hedron - they may pass ; but most frequently they have the faces arranged either in six groups of eight on the faces of the cube, or eight of six on the faces of the octahedron, or twelve of four on the faces of the dodecahedron. There are twenty-four long edges, often corresponding to those of the rhombic dodecahedron or bi secting the long diagonal of the trapezohedron, twenty-four intermediate edges lying in pairs over each edge of the Fig. 41. inscribed octahedron, and twenty-four short edges in pairs over the edges of the inscribed cube. There are six dite tragonal angles joined by the principal axes, eight hexa gonal, and twelve rhombic angles. Examples : diamond, fluorite. General Laws of Crystallography. The seven forms of Laws of crystals now described are related to each other in the crystnilo- most intimate manner. This will appear more distinctly 8 ra P h y- from the account which is to follow of the mode of deriva tion of the forms, with which is conjoined an explana tion of the crystallographic signs or symbols by which they are designated. These symbols were introduced by Naumann, in the belief that they not only mark the forms in a greatly abbreviated manner, but also exhibit the relations of the forms and combinations in a way which
words could hardly accomplish. In order to follow out this