CRYSTALLOGRAPHY 537 lateral the centres of the opposite lateral faces (fig. 12) or edges (fig. 13); another form is a double 6-sided pyramid (fig. 14), and another a double 12-sided pyramid. Examples : beryl FIG. 12. FIG. 13. FIG. 14. or emerald, apatite. Besides the hexagonal prism, this system includes the rhombohedron and its derivative forms, inasmuch as the sym- metry of these forms is hexagonal. The rhom- bohedron (fig. 15) is a solid, bounded like the cube by six equal faces equally inclined to one another, but those faces are rhombic, and the inclinations are oblique. The relations of the rhombohedron may be explained by compari- son with a cube. If the cube be placed on one solid angle, with the diagonal from that angle to the opposite solid angle vertical, it will have three edges and three faces meeting at the top angle, and as many edges and faces, alternate in position, meeting at the opposite angle below ; while the remaining six edges will form a zigzag around the vertical diago- nal ; these- six edges in zigzag might be called the lateral edges, and the others the terminal. The cube, in this position, is in fact a rhombo- hedron of 90. If the cube were elastic, so that the angles could be varied, a little pres- sure would make it a rhombohedron of an angle greater than 90, that is, an obtuse rhom- bohedron (fig. 15) ; or by drawing it out, it would become a rhombohedron of an angle less than 90, or an acute rhombohedron (fig. 16). The diagonal here taken as the vertical axis is the true vertical axis of the rhombohe- dron ; and as there are six lateral edges situa- ted symmetrically around it, there are three lateral axes crossing at angles of 60, as in the regular hexagonal prism. Fig. 17 shows that a hexagonal prism may be made from a rhom- bohedron by cutting off the edges by a plane FIG. 15. FIG. 16. FIG. 17. parallel to the vertical axis ; another may be made by truncating the lateral angles parallel to the same axis. Examples : calcite, sapphire, quartz. Fig. 17 represents a common form of quartz ; the same with the lateral edges trun- cated so as to make a six-sided prism is more common. IV. The relative values of the axes in any species are constant, and these values may be ascertained from the angles of inclina- tion of the planes on one another. In the iso- metric system the axes are equal (see figs. 1 to 3), and the axial ratio is therefore that of unity. Calling the three axes #, 5, e, it is in all isometric species a:J:c = l:l:l. In the dimetric system the vertical axis (a) is un- equal to the lateral (5, e), and the lateral are equal. Calling the lateral l,a:ft:e = a:l:l, a being of any length greater or less than 1, and whatever the value, it is constant for the species. The axes of the fundamental octahe- dron (fig. 5) of any species being thus a : 1 : 1, the axes of all other octahedrons of the same species may be expressed by the ratio ma : 1 : 1, in which m is any simple number or frac- tion ; and the value of ma being known, the angles of the octahedron may be calculated, and conversely. Which octahedron in a series occurring among the crystals of a species shall be taken as the fundamental octahedron, is generally decided on mathematical grounds, that being so regarded which is of most com- mon occurrence, or is most convenient for exhibiting the mathematical relations of the planes. In zircon (fig. 25) the octahedron as- sumed to be the unit or fundamental one is that having for the value of the vertical axis 0-6407, that of the lateral being a unit ; but it would be as correct mathematically, though less con- venient, to make the octahedron 2a.r 1 : 1 the fundamental one, in which case a would equal 1-2814. In calomel, the assumed fundamental octahedron has the value 1 "232 ; and it is be- yond question that crystallogenically this oc- tahedron in calomel corresponds to 2a of zir- con. In the orthorhombic or trimetric system the three axes are unequal, but the ratio is constant for each species, as in the dimetric. Taking the shorter lateral axis (5) as unity, the ratio for sulphur is a : I : c = 2-344 : 1 : 1-23 ; for heavy spar, 1*6107 : 1 : 1*2276. In obtain- ing these numbers there is the same kind of assumption that is explained above with regard to which octahedron shall be taken as the fun- damental one ; and so under the other systems of crystallization. In the monoclinic system the obliquity of the prism is a constant, as well as the relative values of the axes. In Glauber salt this inclination is 72 15', and the ratio of the axes is a : ft : c = 1-1089 : 1 : 0-8962. In the hexagonal system, as in the dimetric, the vertical () is the varying axis; but its value is constant for each species. In quartz, a : I : c : d = 1-0999 : 1 : 1 : 1 ; in calcite, 0-8543 : 1 : 1 : 1. In other words, taking the lateral axes at unity, the vertical (a) in calcite is 0*8543. Crystallography owes its mathe- matical basis to this law. Constancy of angle for each species is involved. But this constan- cy is not absolute, as explained below. V. Each species, while having a constant axial
