352 MINERALOGY angles. Symme- try. derivation of forms, it is necessary to state briefly the following laws, which have been established in crystallo graphy. It is to be remembered that these laws apply, not merely to the cubic system just described, but to all the systems. Invari- 1. The Law of the Invariability of the Angles of Crystals, abihty of w hich was established by Rome" de 1 Isle, may be thus stated : the angles of inclination of the faces of a crystal are constant, however unequally the faces may be developed. The corresponding angles of different crystalline specimens of the same body do not, however, always absolutely agree. Differences have been found, amounting sometimes even to 10 . 2. The Law of Symmetry, discovered by Haiiy, may be thus expressed : (1) similar parts of crystals faces, edges, angles, and consequently axes are all modified in the same manner, and dissimilar parts are modified separately or differently; (2) the modifications produce the same effect on the faces or edges which form the modified part, when they are equal ; when they are not equal, they produce a different effect. That is, if an edge be truncated or bevelled, every similar edge will be similarly truncated or bevelled ; if an angle be truncated or acuminated, every similar angle will be similarly truncated or acuminated ; and consequently every similar axis will be equally affected by the modifications. Thus the cube has eight similar angles and twelve similar edges. In the physical produc tion of the cube, if one of the angles or edges be modified, all will be similarly modified. This, which is the most important law of crystallography, is, however, subject to an exception which was fully formulated by Weiss. The law was all the similar parts of crystals, faces, edges, angles, and consequently axes, are modified at the same time and in the same manner ; the forms resulting from this law are termed " holohedral." The exception is that half of them or one-fourth of them only may be similarly modified. When only half of the similar parts are modi fied, we get the " hemihedral " forms ; when one-fourth only are modified, which occurs only rarely, we get " tetartohedral " forms. 3. The Law of the Parallelism of the Faces of a Crystal, discovered by Rome" de 1 Isle, may be expressed as follows : every face of a crystal has a similar face parallel to it ; or every figure is bounded by pairs of parallel faces (with the exception of certain hemihedral forms). 4. The Law of Zones, first established by Weiss, may be thus enunciated : the lines in which several faces of a crystal intersect each other (or would do so if they were produced until they met) frequently form a system of parallels. Such a series of faces is termed a "zone." Sometimes the zones are parallel to one of the symmetrical axes. Thus, in every prism, the faces of the prism con stitute a zone which encircles the axis of the prism. Faces may be in a zone although they do not actually intersect on the form. 5. The Law of the Rationality of the Parameters of the faces of crystalline series, first indicated by Malus, is that ^ e P 08 ^ 011 ^ pl ane s may be assigned by numbers bearing some simple ratio to the relative lengths of the axes of the crystal. This law was the outcome of investigations into the relationship of forms glanced at in commencing the consideration of the cubic system, and was arrived at through the study of the mode of derivation of forms. The derivation of forms is that process by which, from one form chosen for the purpose, and considered as the type, the funda mental or primary form, all the other forms of a system may be produced, according to fixed principles or general laws. In order to understand this process or method of derivation, it must be noted that the position of any plane is fixed when the position of any three points in it, not all in one straight line, is known. To deter mine the position, therefore, of thu face of u crystal, it is only Parallel ism of faces. Zones. Ration ality of the para meters. Deriva tion of forms. necessary to know the distance of three points in it from the centre of the crystal, which is the point in which the axes intersect each other. As the planes of all crystals are referred to their axes, the points in which the face (or its supposed extension) meets the three axes of the crystal are chosen, and the portions of the axes between these points and the centre are named parameters of the face ; and the position of the face is sufficiently known when the relative length or proportion of these parameters is ascertained. When the position of one face, of a simple form is thus fixed or described, all the other faces of the form are in like manner fixed in accordance with law 2, since they are all equal and similar, and have equal parameters that is, intersect the axis in the same proportions. Hence the expression which marks or describes one face marks and describes the whole figure, with all its faces. The octahedron is adopted as the primary or fundamental form of the cubic system, and distinguished by the first letter of the name, 0. Its faces cut the half-axes at equal distances from the centre ; so that these semiaxes, the parameters of the faces, have to each other the proportion 1:1:1. In order to derive the other forms from the octahedron, the following construction is employed. Suppose a plane to be laid down perpendicular to one axis, and consequently parallel to the two other axes (or to cut them at an in finite distance, expressed by oo , the sign of infinity) ; then the hexahedron or cube is produced, designated by the crystallographic sign ooOoo, expressing the proportion of the parameters of its faces, or oo: 1 ; oo. If a plane is supposed placed on each edge, parallel to one axis, and cutting the two other axes at equal dis tances, the resulting figure is the rhombic dodecahedron, designated by the sign ooO, the proportion of the parameters of its faces being co : 1 : 1. The triakisoctaheclron arises when, on each edge of the octahedron, planes are placed cutting the axis not belonging to that edge at a distance from the centre m, which is a rational number greater than 1. The proportion of its parameters is therefore //i:l : 1, and its sign mO ; the most common varieties are ^0, 20, and 30, seen in diamond and fluorite. AVhen, on the other hand, from a similar distance m in each two semiaxes prolonged a plane is drawn to the other semiaxis, or to each angle, an icositetrahedron is formed ; the parameters of its faces have consequently the pro portion m : m : 1, and its sign is mOm ; the most common varieties are 202 and 303, the former very frequent in leucite, analcime, and garnet, the latter in gold and amalgam. When, again, planes are drawn from each angle, or the end of one semiaxis of the octa hedron, parallel to a second axis, and cutting the third at a distance n, greater than 1, then the tetrakishexahedron is formed ; the para meter of its faces is co ; n : 1 ; its sign is ooOft; and the most com mon varieties in nature are 0f, oo02, and oo03. Finally, if in each semiaxis of the octahedron two distances m and n be taken, each greater than 1, and m also greater than ft, and planes be drawn from each angle to these points, so that the two planes lying over each edge cut the second semiaxis belonging to that edge at the smaller distance n, and the third axis at the greater distance m, then the hexakisoctahedron is produced ; the parameters are m : 11 : 1, its sign mOn, and the most common varieties 30f, 402, and 50f , seen iu diamond and fluorite. It must be observed that the numbers in the above signs refer to the parameters of the faces, not to the axes of the crystal, which are always equal. One parameter also has always been, in the above, assumed =1, and then, either one only of the two other para meters, marked by the number before 0, or both of them, marked by the numbers before and after 0, have been changed. In the above consideration of the mode of derivation of these forms actually found in nature, which belong to the cubic system, it will be observed (though the illustrations were limited) that the value of m and n in these indicated, by the precision of the propor tions |, 2, or 3, a definite numerical relationship. This at once led up to the extended observations which established the law above stated of proportionality in the modification of crystals, or the rationality of the parameters, which gives a mathematical basis to the science, adding to symmetry of arrangement a numerical rela tion in the position of the planes. To illustrate this in a general form (and not merely with special reference to the mode of notation or expression of Kaumaiin, which is that adopted in the subsequent descriptions), let AOA , BOB , COG (fig. 42) be the three axes of a crystal, drawn in perspective, and cutting one another in the centre 0. The semiaxes OA, OB, 0(J are three parameters. Now in the line OA take Off 2 = |OA, and Oa 3 =^OA, making as many points as may be necessary be tween OA, rational fractions of OA. Subdivide OB and 00 in a similar manner. Further produce OA, OB, OC to Ao, Bo, Co, in each direction to an infinite distance, or to a supposed infinite distance, as expressed by the arrow-head ; and suppose these ex tended axes to be divided in a manner similar to the subdivisions of the parameters, by rational multiples of OA, OB, and OC. All the planes of a crystal will be parallel to one or other of the planes icliich pass through three of the points thus determined. First, in order to apprehend the relationship of faces to these axes, or to the half axes, the parameters of the faces, let us suppose one Para meters. Propor tion of para meters expressec by symbols. Notation of Nau-
mami.