MINERALOGY 347 accomplished in two ways (l)by finding the weak joints in that arrangement, through splitting the crystal, and (2) by measuring the angular inclination of the outside surfaces which bound the form and, from these measure ments, by simple mathematical laws, arriving at what has been termed its " primitive " or simplest form. As regards the mere recognition of a substance, such measurement in itself suffices, the angular inclination, if the same surfaces be measured, being unvarying in each species. It can, moreover, be shown that the possible range of external variety of form is governed by fixed mathematical laws, which determine precisely what crys talline forms are or may be produced for each species. Comparatively few of these actually occur in nature ; but crystallographic laws can point out the range of those which can possibly occur, can delineate them even before they are found, and can in all cases show the relationship which subsists between them and the simple or fundamental form from which or out of which they all originate. It must be observed that in crystalline bodies the internal structure that is, the arrangement of the molecules is as regular in an outwardly shapeless mass as in the modelled crystal which presents itself as a perfect whole. Definitions of Crystals, and their Members or Parts. A crystal is a symmetrical solid, either opaque or transparent, contained within surfaces which theoretically are flat, and of a perfect polish, but which are actually frequently curved, striated, or pitted. These surfaces are called "planes," or "faces." The external planes of a crystal are called its " natural planes " ; the flat surfaces obtained by splitting a crystal are called its " cleavage planes." The intersections of the bounding planes are called " edges," and planes are said to be similar when their corresponding edges are proportional and their corresponding angles equal. Crystals bounded by equal and similar faces are termed " simple forms." The cube, bounded by six equal squares, the octahedron, bounded by eight equilateral triangles, and the rhombohedron, bounded by six equal rhombs, are thus simple forms. Crystals of which the faces are not all equal and similar are termed compound forms, or " combinations," being regarded as produced by the union or combination of two or more simple forms. Edges are termed rectangular, obtuse, or acute, according as the angle at which the faces which form the edge meet is equal to, or greater or less than, a right angle. Edges are similar when the planes by the intersection of which they are formed are respectively equal and equally inclined to one another ; otherwise they are unlike or dissimilar. When a figure is bounded by only one set of planes, it is said to be "developed." When an edge is cutoff by a new plane, it is said to be "replaced"; when cut off by a plane which forms an equal angle with each of the original faces which formed the edge, it is said to be " truncated." When an edge is cut off by two new faces equally inclined to the two original faces respectively, it is said to be "bevelled." When a solid angle is cut off by a new face which forms equal angles with all the faces which went to form the solid angle, it is said to be truncated. In classifying crystals and studying their properties, it is found convenient to introduce certain imaginary lines called "axes." Axes are imaginary lines connecting points in the crystal which are diametrically opposite, such as the centres of opposite faces, the apices of opposite solid angles, the centres of opposite edges. Different sets of axes may thus be drawn through the same crystal; but there is always one set, usually of three, but in one special class of crystals of four, axes, by reference to which the geometrical and physical properties of a crystal can be most simply explained. These axes intersect one another, either at right angles, producing " orthometric " forms, or at oblique angles, producing " clinometric " forms. The axes may be all equal, or only two equal, or all unequal. There is a definite conventional position in which for p purposes of description a crystal is always supposed to be tioning held. With reference to this position one of the axes, of crys- that which is erect or most erect,, is termed the "verti- tajs - cal," and the others the "lateral." The planes in which any two of the axes lie are called the "axial" or " diametral planes," sometimes " sections." By these the space around the centre is divided into " sectants." If there are, as is generally the case, only two lateral axes, the space is divided into eight sectants, or octants ; but, if there are three lateral axes, it is divided into twelve sectants. Primitive Forms of Crystals. If we attempt to arrive, p . . through a study of the internal structure of crystals, as tive evidenced by directions of weakness of cohesion, at the forms, total number of primitive or parent forms which can exist, we find that there are thirteen such forms and no more. Nine of these may be regarded as prisms standing upon a base, three as octahedia standing upon a solid angle ; and there is one twelve-sided figure, or dodecahedron. Prisms. Of the prisms eight have a four-sided base Prisms If the base is square and the prism stands erect that is, if its sides or lateral planes, as they are called, are perpendicular to the base the form is termed a "right square prism" (fig. 6). In this the four lateral planes are rectangular and equal ; they may be either oblong or square ; in the latter case the form is the " cube " (fig. 5). When the base is a rectangle instead of a square, the form is a "right rectangular prism" (fig. 7). In each of the Fig. f.. Fig. 6. Fig. 7. above three forms the edges are twelve in number. In the cube all the edges are equal. In the square prism the lateral edges are all equal, but are different from the four equal edges of the base. In the rectangular prism, two at each base differ in length from the other two, while both differ from the lateral ; hence there are here three sets of edges, four in each. In each of the three forms, however, the solid angles are eight in number, all equal, and each enclosed by three right angles. When the base is a rhombus, and the prism stands erect, the form is a "right rhombic prism " (fig. 8). Two of the angles in the base being here acute and two obtuse, two of the solid angles corresponding each with each must differ from the others. So also must two of the lateral angles be acute and two obtuse. The four lateral faces are equal. When the base is a rhomboid, and the prism stands erect, it is only the opposite lateral faces that can be equal. The form is called a "right rhomboidal prism " (fig. 9). When the base is a rhombus, but the prism stands obliquely on its base, the form is called an " oblique rhombic prism " (fig. 10). Here the basal edges of the lateral planes are all equal in length, but on account of the inclination of the prism the angles which these edges form with the lateral edges of the lateral planes are two acute and two obtuse. Fig 10. Fig. 11. Fig. 12. If all the edges of an oblique rhombic prism are equal in lengtk to the breadth of the base, and if the lateral planes are rhombi equal in all respects to the basal, the form is called a" rhombo
hedron " (fig. 11). This is included within six equal planes, like