MINERALOGY 357 with a vertical-faced pentagonal dodecahedron; while fig. 68 shows an increase in the amount of truncation effected by the latter. Fig. 69 shows the combination of the cube with the dy akidodecahedron, Fi the former being dominant. In fig. 70 an octahedron, in dominance, is combined with the vertical-faced pentagonal dodecahedron; in Fig. 70. Fig. 71. fig. 71 the faces of these forms are of nearly equal size, while in fig. 72 the octahedral faces are nearly removed. The solid angles of Fig- 72. Fig . 73. the octahedron are modified in fig. 73 by the faces of the dyakis- dodecahedron. In fig. 68 a vertical-faced pentagonal dodeca- Fig. 74. Fig. 75. hedron is the prevailing form in combination with the cube; while in fig. 74 the faces of the octahedron are superadded. In fig. 75 its octahedral angles are modified by the faces of the icositetrahedron, Fig. 76. Fig. 77. and in fig. 76 by those of the octahedron in addition. In fig. 77 they are modified by the faces of the dyakisdodecahedron. In each of the five systems which follow there is this difference from the cubic system that one axis is always unequal to (longer or shorter than) the others. This is placed erect, and named the chief axis ; its ends are poles, Dimetric and the edges connected with them polar edges. The and other axes are named subordinate or lateral axes, and the trimetric plane that passes through them is the base. A plane f ms through the chief and a lateral axis is a normal chief section. In these systems also occur the three forms of "pyramids," "prisms," and "pinacoids." (1) The pyra mids have their faces triangles. Pyramids in crystallo graphy are each composed of two geometric pyramids placed base to base, and named "closed forms," as the crystals are shut in by definite faces on every side. (2) The prisms are bounded by plane faces parallel to one axis. They are thus of unlimited extent in the direction of that axis, and therefore named "open forms, "but in solid crystals are shut in by faces of other forms. (3) The pinacoids, or tables, have two faces intersecting one axis and parallel to the others, and thus are also open forms, or unlimited in the direction of these axes. Forms (2) and (3), when con joined, mutually shut in each other, or produce closed forms. II. Pyramidal or Tetragonal System. This system has Pyra- three axes at right angles, two of them equal, and the chief mijal axis longer or shorter. The name tetragonal is derived s y stem - from the form of the base, which is usually quadrangular. There are eight tetragonal forms, of which five are closed. (1) Tetragonal pyramids (figs. 78, 79) are enclosed by eight isosceles triangles, with four middle edges all in one plane, and eight polar edges. There are three kinds of this form, Fig. 79. distinguished by the position of the lateral axes. In the first these axes unite the opposite angles ; in the second they intersect the middle edges equally ; and in the third they lie in an intermediate position, or divide these edges unequally, the last being hemihcdral forms. These pyramids are also dis tinguished as obtuse (fig. 78) or acute (fig. 79), according as the vertical angle is greater or less than in the regular octahedron. (2) Ditetragonal pyra mids (fig. 80) are bounded by sixteen scalene triangles, whose base-lines are all in one Fig. 80. Fig. 81. plane. This form rarely occurs except in combinations. (3) Tetra gonal sphenoids (fig. 81), bounded by four isosceles triangles, are the hemihedral forms of the first variety of tetragonal pyramids. (4) The tetragonal scalenohedron (fig. 82), bounded by eight scalene triangles, whose bases rise and fall in a zigzag line, is the hemi hedral form of the ditetragonal pyramid. Nos. (3) and (4) are rare. (5) The tetragonal trapezohedron is not found in minerals as a simple form. The three open forms are (1) tetragonal prisms, bounded by four planes parallel to the principal axis, which may be either longer (fig. 83) or shorter (fig. 84) than the lateral axes; (2) dite tragonal prisms, bounded by eight similar planes; and (3) the basal pinacoid, consisting merely of two parallel faces bounding the prisms at the ends, above and below. The various series of tetragonal crystals are distinguished from
each other only by their relative dimensions. To determine these,