Page:Encyclopædia Britannica, Ninth Edition, v. 16.djvu/368

Rh 350 MINEKALOGY and there are certain advantages in considering them at least first by the former method. We consider first, as the more essential, the relative lengths of the axes, and, secondly, the angular inclination of these. 1. In the cubic system the axes are all equal, and all intersect at right angles. Here is the most perfect sim plicity, and the most perfect regularity. 2. In the tetragonal system two only of the axes are equal ; but all still intersect at right angles. Here is a departure from simplicity as regards the length of one axis, but no departure as regards the angular inclination. 3. In the right prismatic system none of the axes are equal, but all still intersect at right angles. Here is total loss of regularity in the first particular, but still none in the second. 4. In the oblique prismatic system none of the axes are equal, and only two intersect at right angles. Here there is again a total loss of simplicity in the first particular, and a certain amount of departure from it in the second. 5. In the anorthic system none of the axes are equal, and none of them intersect at right angles, so that here, as expressed by the name, there is a total departure from regularity in both particulars. 6. The hexagonal system is anomalous in relation to this mode of consideration. It is regarded as having four axes, three of which lie in one plane, parallel to the base, and intersect each other at equal angles Unique axis made vertical. Fig. 27. (necessarily angles Fig. 28. Fig. 29. of 60). The fourth axis intersects these at right angles, and may be longer, shorter, or equal to them. This system is generally considered after the tetragonal system, as having one axis which differs in length from the others, and only one which cuts the others at right angles. By some a rhombohedron is con sidered as the primary of this system ; it then comes to have three axes, all equal, but none intersecting at right angles. In considering these sys tems, or in describing the form of a crystal, the vertical or erect axis is named the principal axis of the figure, and that axis is chosen as the vertical which is the only one of its kind. In the cubic system there is no such axis, .so. that any one may be chosen as the vertical. It will be convenient, before proceeding to the considera tion of the laws of crystallography and the combinations Fig. 30. of forms, especially in view of the terminology that must be employed in illustrating those general aspects of the subject, to give an outline of one of the six systems here. For this preliminary description the cubic system, as the simplest and most regular, naturally suggests itself as the most suitable. I. The Cubic System. Here the axes are all equal, and CuLic all intersect at right angles. The &quot;cube&quot; (fig. 26), &quot;octa- system. hedron&quot; (fig. 30), and &quot; rhombic dodecahedron &quot; (fig. 33), which are here included, are alike in their perfect symmetry ; the height, length, and breadth are equal ; and their axes are equal, and are rectangular in their intersections. In the cube (fig. 5) these axes connect the centres of opposite faces ; in the octahedron (fig. 1 5) the apices of opposite solid angles; in the dodecahedron (fig. 18) the apices of opposite acute solid angles. The relation of these Relatio forms to each other, and the correspondence in their axes, ot siin l will be made manifest through a consideration of the transi- l tion between the forms. If a cube be projected with the axes in the above position, or if a model of it in any sectile material be employed, and if the eight angles are sliced off evenly, keeping the planes thus formed equally inclined to the original faces, we first obtain the form in fig. 27, then that in fig. 28 and fig. 29, and finally a regular octahedron (fig. 30) ; and the last disappearing point of each face of the cube is the apex of each solid angle of the octahedron. Hence the axes of the former, being in no way displaced, necessarily connect the apices of the solid angles of the latter. By cutting off as evenly the twelve edges of another cube, the knife being equally inclined to the faces, we have the form in fig. 31, then fig. 32, and finally the rhombic dodecahedron (fig. 33), with the axes of the cube connecting the acute angles of the new form. These forms are thus mutually derivable. Moreover, they are often pre sented by the same mineral species, as is exemplified in galena, pyrites, and the dia mond. The process may be re versed, and the cube made from the octahedron, as will be readily understood from a comparison, in reverse order, of figs. 26 to 30. Or the cube may be similarly derived from the dodecahedron, as seen by inspecting figs. 33, 32, 31, 26. The octahedron also is changed to a rhombic dodeca hedron by removing its twelve edges (figs. 34, 35), and con tinuing the removal till the original faces are obliterated, thus producing the dodeca hedron. It will be observed that throughout all these changes the position of the axes, as determinants of dimensions, need not be altered, that, in fact, one set of axes has served for all the forms. The relationships of the principal forms of this system being thus disclosed, the forms themselves have next to be- considered.