Page:Encyclopædia Britannica, Ninth Edition, v. 16.djvu/376

Rh 358 MINEKALOGY Prinii- one of the series must be chosen as the primary form, and for this tive purpose a tetragonal pyramid of the first variety, designated by P pyramid, as its sign, is selected. The angle of one of its edges, especially the middle edge, found by measurement, determines its angular dimensions, whilst the proportion of the principal axis a to the Fig. 83. pyra mids. Prism. Fig. 82. lateral axes, supposed equal to 1, gives its linear dimensions. The parameters, therefore, of each face of the fundamental form are 1 : 1 : a. Now if in be any (rational) number, either less or greater than unity, and if from any distance ma in the principal axis planes be drawn to the middle edge of P, then new tetragonal pyramids of the first order, but more or less acute or obtuse than P, are formed. Derived The general sign of these pyramids is mP, and the most common varieties JP, 2P, and 3P, with the chief axis half, twice, or thrice that of P. If m becomes infinite, then the pyramid passes into a prism, indefinitely extended along the principal axis, and with the sign oo P. If m = 0, which is the case when the lateral axes are supposed infinite, then it becomes a Pinacoid. pinacoid, consisting properly of two basal faces open towards the lateral axes, and designated by the sign OP. Theditetra- gonal pyramids are produced by taking in each lateral axis distances n greater than 1, and drawing two planes to these points from each F - of the intermediate polar edges. The parameters of these planes are therefore m : 1 : n, and the general sign of the form mPn, the most common values of n being f, 2, 3, and oo. When n = oo, a tetragonal pyramid of the second order arises, designated generally by mP&amp;lt;x&amp;gt;, the most common in the mineral kingdom being Poo and 2Poo. The relation of these to pyramids of the first order is shown in fig. 85, where ABBBC is the first and ACCCX the second order of pyramid. In like manner, from the prism ooP, the ditetragonal prisms O&amp;gt;PTI Fig. 88. Fig. 89. are derived, and, finally, when w=oo the tetragonal prism of the second order, whose sign is ooPoo. The combinations of the tetragonal system are either holohedral or hemihedral ; but the latter are rare. Prisms and pinacoids must always be terminated on the open sides by other forms. Thus in fig. 86 a square prism of the first order is terminated by the primary pyramid, and has its lateral angles again replaced by another more acute pyramid of the second order, so that its si&quot;n is oo P, P, 2Poo. In fig. 87 a prism of the second order is first bounded by the fundamental pyramid, and then has its edges of combination replaced by a ditetragonal pyramid ; its sign is ooPoo, P, 3P3. In fig. 88 the polar edges of the pyramid are replaced by another pyramid, its sign being P, Poo. In fig. 89 a hemihedric form, very characteristic of chalcopyrite, is represented, P and P being the two sphenoids, a the basal pinacoid, and b, c two ditetragonal pyramids. III. The Hexagonal System. The essential character of Hexa- this system is that it has four axes, three equal lateral g nal axes intersecting each other in one plane at 60, and one s y stem - principal axis at right angles to these. The plane through the lateral axes, or the base, from its hexagonal form, gives the name to the system. As in the last system, its forms are either closed or open. They are divided into holohedral, hemihedral, and tetartohedral, the last, which are rare, having only a fourth part of the faces developed. Only a few of the more common forms require to be here described. The hexagonal pyramids (figs. 90, 91) are bounded by twelve Pyra- isosceles triangles, and are of three kinds, according as the lateral mids. axes fall in the angles, in the middle of the lateral edges, or in another point of these edges, the last being hemihedral forms. They are also classed as acute or obtuse, but with out any precise limits. The trigonal pyramid is bounded by six triangles, and may be viewed as the hemihedral form*)f the hexagonal. The dihexa- gonal pyramid is bounded by twenty- four scalene triangles, but has never been observed alone, and rarely even in combinations. The more common prisms are the hexagonal of six sides ; in these the vertical axis may be either longer than the lateral, as in fig. 92, or shorter, as in fig. 93. There are also dihexagonal, of twelve sides. A particular pyramid P is chosen as the fundamental form of this system, and its dimensions deter mined either from the proportion of the lateral to the principal axis (1 : a) or from the measurement of its angles. From this form (mP) others are de rived exactly as in the tetragonal system. Thus dihexagonal pyramids are produced with the general sign mPn, the chief peculiarity being that, whereas in the tetragonal system n might have any rational value from 1 to oo, in the hexagonal system it can only vary from 1 to 2, in consequence of the geometric character of the figure. When 7i = 2, the dihexagonal changes into an hexagonal pyramid of the second order, whose sign is mP 2. When m = oo, various prisms arise from similar changes in the value of n ; and when m = the basal pinacoid is formed. Few hexagonal min eral species form per fect holohedral com binations. Though quartz and apatite ap pear as such, yet pro perly the former is a tetartohedral, the lat ter a hemihedral spe cies. In holohedral species the predomi nant faces are usually those of the hexagonal Fig. 93. Fig. 94. prisms ooP (fig. 92) arid ooP2, or of the pinacoid OP (fig. 93); whilst Prisms, the pyramids P and 2P2 are the most common subordinate forms. Fig. 94 represents the prism, bounded on the extremities by two pyramids, one, P, forming the apex, the other, 2P2, the rhombic