354 MINERALOGY zone plane, cutting it at right angles, is the " zone axis " ; it is parallel to all the faces, and intersections of the faces (if they are extended enough to intersect), of the zone. A face may be common to two or more zones; its normals will then coincide Avith the intersections of the several zone planes. In the absence of actual spheres upon which to detail the facts which go to form the "sphere of projection" of each substance, the hemisphere is represented on a plane surface. This has of necessity the disadvantage, except as regards the circumferential zone, of introducing spherical distance- distortion foreshortening of all parts lying near the cir cumference ; but the eye soon gets accustomed to this. Fig. 43 presents the principal zones of the cubic system, and shows the position of the poles of the faces of the cube, the octahedron, and the rhombic dodecahedron. o 1} o 2 , o 3 , &c., are the poles of the octahedral faces; a 1? a 2 , a 3 , &c., those of the faces of the cube ; and d ly d 2 , d z , &c., those of the rhombic dodecahedron. It will be observed that the faces of the cube fall into the zone circles of the octahedron and dodecahedron, while those of the octahedron fall into those O lO ttn 1.10 2. ,0 120 Fia. 44. Principal Poles of Cubic System in Octant of Sphere. of the rhombic dodecahedron. Considering this as a delineation of a globe, these zone circles come to represent latitude and longitude ; and, as almost all the faces in this system fall into some zone circle, it is clear that the latitude and longitude of all normals may be readily laid down, and their relations at once determined by spherical trigonometry Fig. 44 shows the arrangement of tho poles of all the forms belonging to the cubic system noticed above, or referred to in the present article, delineated on an octant of the sphere of projection. It displays the perfect regularity of the system. Hcmihcdral and Tetartohedral Forms. The exception to the Hemi- second law (that of symmetry), which was formulated by Weiss, hedral was to the effect that one-halt or even one-fourth only of the faces forms, which go to form i: holohedral crystal may be present. When but one-half of the faces present themselves, the form is termed hemi- hedral ; when only one-fourth, it is tetartohedral. These restrained developments have now to be considered. In hemihedral forms the development is restrained, but symmetry is not deranged ; half the similar parts are still alike, though unlike the other half. There are two classes of hemihedral forms : I. Those forms in which halt the similar angles or edges are modified independently of the other half ("hemi-holohedral "), producing 1. In the monometric and dimetric systems tetrahedral " and "sphenoidal" forms, by the independent replacement of the alter nate angles; their opposite faces are not parallel, and they are hence called "inclined" hemihedrons; as in chalcopy rite, boracite. 1 The replacement in the dimetric system of two opposite basal edges at one base and the other two at the opposite base is of the same kind; as in edingtonite. 2. In the trimetric system "monoclinic" forms, by the replace ment of half the similar parts of one base and the diagonally opposite of the other, unlike the other half; as in datholite, humite. 3. In the trimetric and hexagonal systems "hemimorphic " iornis, by independent replacements at the opposite extremities of thu crystal ; as in topaz, calarnine, tourmaline. 4. In the rhombohedral system, by the replacements of the alternate basal edges or angles of the rhombohedron, forms usually called "tetartohedral" or quarter forms, on the ground that mathematically the rhombohedron is a hemihedral form derived from the hexagonal prism, which is the type of the hexagonal system. Rock crystal is usually developed according to this law. II. Those forms in which all the similar angles or edges are modified, but by half the full or normal number of planes (" holo- hemihedral"), producing 1. In the monometric system "pyritohedral" forms, by a replace, ment of the edges or angles ; as in pyrites. Such forms have opposite faces parallel, and are often called parallel hemihedrons. 2. In the dimetric system "pyramidal" and " scalenoidal" forms, by a replacement of the eight solid angles of the primary prism, according to two methods. 3. In the hexagonal system "pyramoidal and "gyroidal" forms, by a replacement of the solid angles of the hexagonal prism, or of the six lateral angles of the rhombohedron, according to two methods ; as in quartz and apatite. The above illustrations show that hemihedrism is not only divided into two classes, but is of various kinds, and these have been systematized as follows: " holomorphic," in which the occurring planes pertain equally to the upper and lower (or opposite) ranges of sectants, as in ordinary hemihedrons; and (2) "hemi morphic, " in which each jet of planes pertains to either the upper or the lower range, but not to both. As to the relative position of the sectants which contain the planes, the forms may be vertically direct, as in baryte ; vertically alternate, as in the tetrahedron, the rhombohedron, and the plagihedral faces of quartz ; and vertically oblique, as in many forms of chona roditc. In hemimorphic forms symmetry is deranged ; the crystals are Hemi- bounded at the opposite ends of their main axes by faces belonging morphic to distinct forme or modifications, always, fc / forms, however, of the same system ; hence only the upper or the under half of each crystal can be regarded as complete, as regards the form there seen ; and so for each end it is half formed. Fig. 45 represents a crystal of tourmaline, which is bounded on the upper end by the planes of the rhombohedrons R (P) and - 2R (o), and on the lower end by the basal pinacoid (& ). In fig. 46 of smith- sonite the upper extremity shows the Fig. 45. base k, two brachydomes o and p, and two macrodomes m and I; 1 As the parts of either half are alternate, there still results a symme trical solid. As either one or other half may be the one thus modified, there may result two such symmetric solids, which stand in an inverse position to one another. When the modifications affect the upper right- hand solid angle, the resulting form is called + when the upper left-
hand -aiT lt; it is - .