144 PYRAMIDAL SYSTEM. [82. 2. The Right Square Prismatic, or Pyramidal System.-In this system there are three axes, all at right angles to each other, but two only, a a, a a (fig. 60), are equal; the third, c c, being either longer or shorter than the others. The crystals of this class, from this circumstance, are said to be dimetric. Generally there is no simple relation between the length of the axis, c c, and that of the other two. Expansion by heat is equal in two directions. The crystals of this system have only one axis of single refraction, c c (117, 118), as they, as well as those of the four other systems not yet described, exert double refraction on light. Four principal forms belong to this system-viz., two prisms with a square base, and two octohedra. The prisms differ from each other according as the equal axes, a a, a a, terminate in the angles of the base, as seen in fig. 60, I; or in the sides of the base, as at 2. Similar differences exist in the two octohedra. The octohedron is said to be direct, when the axes end in the angles, and inverse, when they end in the edges. 3 represents a right square prism the axes of which terminate in the edges of the crystal; in 4, the axes terminate in the sides of the prism; 5 is the direct octohedron, with its axes in the solid angles; 6, the inverse octohedron, with the axes in the edges. Examples of this system are seen in potassic ferrocyanide, mercuric cyanide, tinstone, and anatase. In consequence of the absence of any fixed relation in length between the principal axis, c c, and the other two axes, in the four different prismatic systems, these prisms may vary in length indefinitely. In some cases, the axis, c c, is so short that the crystal assumes the form of a flattened plate, when it is said to be a tabular crystal; in others it forms a long prism of indefinite length. In the octohedron of the various prismatic systems, the principal axis, cc, does not, even in the same compound, always bear 82.] RHOMBOHEDRAL OR HEXAGONAL SYSTEM. 145 the same proportion in length to the other two axes; though in these various octohedra, the axis, c c, always bears some simple ratio in length to those of the other octohedra of the same body. 3. The Rhombohedral, or Hexagonal System.-In this system there are four axes; three of them, a a, a a, a a, are of equal lengths, are situated in the same plane, and cross each other at angles of 60°; whilst the fourth, c c, is perpendicular to these, and may vary in length. The crystals of this class produce, in a very marked manner, the effects of double refraction on light. They have one axis, c c, of single refraction; and by the application of heat expand equally in two directions. In this system the principal forms (fig. 61) are the bi-pyramidal dodecahedron, 3, (of which there are two varieties, according as the axes terminate in the angles of the base, 1, when it constitutes a direct dodecahedron; or in its sides, 2, when the dodecahedron is said to be inverse); the rhombohedron, 4, and the six-sided prism, 5. Of each of these forms there are likewise two varieties, depending upon the position of the axes. 4 is an inverse rhombohedron. Among crystals which belong to this system are ice, quartz, beryl, Iceland spar, and sodic nitrate. Fig. 62 represents, in one view, the manner in which the 146 OBLIQUE OR MONOCLINIC SYSTEM. [82. principal forms in each of the first three systems can be described about the crystallographic axes. I exhibits the octohedron inscribed in the cube; 2 shows both varieties of the octohedron and of the square prism; 3 the six-sided prism, containing the rhombohedron and bi-pyramidal dodecahedron. The relations of the first three systems are simple, and easily traced; the other three systems are more complicated, owing to the variety introduced by the irregular lengths and obliquities of the axes. 4. The Right Rectangular Prismatic, or Prismatic System.The crystals of this system have three axes, a a, b b, c c (fig. 63), Prismatic, or Right Rectangular Prismatic System. all at right angles to each other; each axis differs from the others in length, and they usually bear no simple proportion to each other. Crystals of this class are hence termed trimetric by some writers. In this and in the two remaining systems, the crystals expand unequally by the application of heat, in the three directions of these axes; and they have two other resultant axes in which there is no double refraction (118). The principal varieties of the prismatic system are the right octohedron with a rhombic base (fig. 63, 4), or right rhombic octohedron; and the right prism with a rhombic base or right rhombic prism, 5. Both these figures have a rhombic base, 1; the axes terminate in the solid angles of the octohedron, and in the edges of the prism. Owing to the inequality in the lengths of the axes, the sections of the octohedron through a b a b, 1, caca, 2, and c b c b, 3, though all rhombic in form, are each different in dimensions. The faces of the octohedron are all equal, but the length of each side of its triangular faces is different. To this class belong nitre, aragonite, topaz, baric sulphate, and sulphur obtained by evaporation from carbonic disulphide. 5. The Oblique, or Monoclinic System.-The three axes of this system may all differ in length; two of them, c c, a a, cross 82.] DOUBLY OBLIQUE SYSTEM. 147 each other obliquely (fig. 64, 2); the third, b b, is perpendicular to both the others; generally there is no simple proportion between the lengths of the different axes. The principal forms are the oblique octohedron with a rhombic base, 4, and the oblique rhombic prism, 5, in both of which the axes are in the angles of the crystal. The base of the figure in each case is a rhombus, I, in which the axes a a, b b, cross each other at right angles. In the octohedron, the section through the two oblique axes, a a, c c, 2, is a rhomboid; the axis, c c, crosses the third axis, bb, perpendicularly, and a section through these axes produces the rhombus shown in 3. The octohedron of this system is not perfectly symmetrical. Each of the three sides forming its triangular faces differs from the others in length, and the faces are of two kinds. The two upper front faces of 4, fig. 64, correspond to the two lower back faces, and the other four faces are alike. Besides the oblique rhombic octohedron, there are three forms of the oblique rhombic prism; the kind of prism being defined by the axis with which the long axis of the prism coincides. Sodic sulphate, hydric disodic phosphate, sulphur crystallized by fusion and slow cooling, borax, and ferrous sulphate offer examples of crystals belonging to this class. 6. The Doubly Oblique, Triclinic, or Anorthic System.-In this system each of the three axes may differ from the others in length, and all cross each other obliquely. The principal varieties 148 ISOMORPHISM. [82. of crystalline form are the doubly oblique octohedron (fig. 65, 2), the base of which is seen at 1, and the doubly oblique prism, 3. The octohedron is not symmetrical in its form: its four upper faces are all alike, but each face corresponds to the lower face which is parallel to it. Cupric sulphate and bismuth nitrate belong to this class, which, however, contains comparatively few substances. Some of the varieties of crystalline forms which it includes are very complicated, and difficult to define. Isomorphism-Dimorphism-Allotropy. (83) Isomorphism.-Owing to the comparatively small number of forms which belong to the regular system, and to the perfect symmetry which characterizes them, it necessarily happens that a variety of bodies, very dissimilar in properties and in chemical composition, assume crystalline forms which are not distinguishable from each other, since they coincide exactly in their angular measurements. For example, the elements-carbon, gold, and copper, and the compounds-plumbic sulphide, iron pyrites, fluor spar, alum, and spinelle, all crystallize in cubes or octohedral which perfectly resemble each other; yet these substances present no similarity to each other either in properties or in chemical composition. Crystals which belong to the other systems, however, do not so frequently present this exact similarity in form; for though they may crystallize in similar prisms or octohedra, yet a measurement of the angles will suffice to show considerable differences in the length of the axes, and, in the case of the two oblique systems, in the inclination of the axes to each other. But in these systems likewise, as well as in the regular system, cases occur in which an exact, or almost exact identity in crystalline form, even in these respects, is found. In the larger number of these instances, as Mitscherlich has proved, the chemical composition of the substances which thus correspond in form is analogous. Bodies which possess this similarity in form are termed isomorphous (from loos, equal, μoppǹ, form). The term isomorphous is, however, restricted to such substances as exhibit not only similarity in form, but at the same time, the analogy in their chemical composition just alluded to. The diamond (C), magnetic oxide of iron (FeO, Fc,O), and alum (K,Al, 4 SO, 24 H2O), all crystallize in octohedra, yet they are not usually cited as instances of isomorphism: but the spinelle-ruby (MgO, Al,O,), magnetic oxide of iron (FeO, FeO3), and chrome-iron ore (FeO, |