Page:Encyclopædia Britannica, Ninth Edition, v. 16.djvu/366

Rh 348 Octa- hedra. the cube, but these planes have oblique angles. The rhombo- hedron thus bears the same relation to the oblique rhombic prism which the cube does to the right square prism. Of the eight solid angles of a rhombohcdron only two are contained by three equal plane angles, and these two &quot;apices,&quot; as they may be called, are opposite one another. According as the apices are acute or obtuse, we have an acute or obtuse rhombohedron. &quot;When the base of an oblique prism is a rhomboid, the prism becomes an &quot; oblique rhomboidal prism &quot; (fig. 12). In this form, only diagonally opposite edges are similar, as regards equality of length and the value of the included angle. Only opposite solid angles are equal, as are also the opposite and parallel faces. A right prism may have an equilateral six-sided base ; it is then called an &quot;hexagonal prism.&quot; This form may be developed in two positions relatively to each other, one in which the transverse axes pass from the centres of opposite faces (iig. 13), the other in which they pass from the centres of opposite edges of the planes (tig. 14). The faces of the one set mutually truncate the edges of the other. If a rhombo hedron be positioned so as to rest upon one of its apices, the faces of one hexagonal prism would truncate the lateral edges of the rhombohedron, 14. while the faces of the other hexagonal prism would truncate its lateral solid angles. Hexagonal prisms may be longer or shorter than the width of their bases. The interf acial lateral angles are 120. The angle between the lateral and terminal faces is 90. Octahcdra. The planes of these eight-faced solids are triangular, and they may be regarded as made up of two four-sided pyramids applied to each other, base to base. They are always positioned so that they stand upon a solid angle with the &quot; basal plane &quot; that is, the plane which is the common base of the two pyramids hori zontal. In the primitive forms now under consideration the ver tices of the two pyramids will in this position be vertically above and below the centre of the base. The upper and lower solid angles are then termed the &quot; vertical solid angles,&quot; and the four lateral solid angles are called the basal solid angles. There are three octahedrons. In the &quot;regular&quot; octahedron (fig. 15) the base is a square, and the eight faces are equilateral triangles of equal size. There are twelve edges, which are all equal. The faces incline to each other at an angle of 109 28 16&quot;, and have the plane angles all 60. There are six equal solid angles. When the base of the octahedron is square, but the other edges, although Dodeca hedron. Fig. 18. Fig. 15. Fig. 16. Fig. 17. equal to one another, are either longer or shorter than the edges of the base, the form is a &quot; right square octahedron &quot; (fig. 16). In this the faces are isosceles triangles, the equal angles being at the basal edge of the planes. These basal edges are equal and similar, but differ in length and in angles from the eight equal pyramidal edges. When the base of an octahedron is a rhombus, it is called a &quot; right rhombic octahedron&quot; (fig. 17). Dodecahedron. This (fig. 18) has each of its twelve faces a rhombus. It is, like the cube and the octahedron, a solid which is symmetrical. The interfacial angles are all 120, the plane angles are 109 28 16&quot; and 70 U 31 44&quot;. The edges are twenty-four, and similar. There are fourteen solid angles, of which six are formed each by the meeting of four acute plane angles, and eight by the meeting of three obtuse plane angles. Deter- It has been said that the above simple forms were arrived mination at through a study of the internal structure of crystals, of parent chiefly as disclosed by cleavage. Inasmuch, however, as there are some minerals which cleave in only one direction, and many which cannot be cleaved in any direction, this method of investigation fails. Its employment, moreover, frequently led to conflicting or embarrassing results. A conflicting result is when a substance has more than one set of cleavages, that is, splits up in directions which would result in the production of more than one of the above primary or simple forms. Thus the mineral fluorite occurs with much the greatest frequency in the form of the cube, and it might very consistently be held that its frequent occurrence in this form was a clear natural indication that the cube was the primary or simplest form of fluorite ; but it splits up into an octahedron. Galena crystallizes frequently in the form of the octahedron ; yet to cleavage galena yields a cubic primary form. It might be conceived that there had been, in each case, some special tendency to assume the cubic form and the octa hedral form ; but one and the same piece of rock may bear on its surface cubic crystals of fluor and octahedral crystals of galena, each of the minerals having here assumed the primitive cleavage form of the other in pre ference to its own. The mineral blende crystallizes not unfrequently in octahedra, which yield the dodecahedron on cleavage. Fluor crystallizes in dodecahedra, yet yields the octahedron to cleavage. Argyrite crystallizes in cubes and in octahedra, but yields the dodecahedron on cleavage. Pyrite crystallizes in cubes, octahedra, and dodecahedra, and yields both the cube and the octahedron on cleavage. These are most embarrassing results, but they clearly indicate so intimate a relationship to subsist between three of the above simple forms that it is obvious that one alone would serve as a type form for representing the others. The selection of that one should be based upon grounds of most eminent simplicity, and this again is to be arrived at by a consideration of the smallness of number of parts, i.e., of faces, edges, and solid angles. In such a considera tion we find that the dodecahedron, with its higher number of each of these, at once gives place. The cube has six faces, the octahedron eight ; simplicity here is in favour of the cube. The cube has twelve edges, the octahedron has twelve ; in this respect they are equal. The cube has eight solid angles, the octahedron six ; here the greater simplicity is on the side of the octahedron. So that this method of adjudicating by simplicity fails, and we are thrown back upon the relationships which may be unfolded through a consideration of the other elements of crystals, their axes. Systems of Crystals and Laws of Crystallization. This consideration led, first, to the remarkable discovery Relatioi that several of the above primary forms are mere modifica- of faces tions of each other, and ultimately showed that all crystals to axes - found in nature may be referred to six systems, based on certain relations of their axes, and that every face which could occur upon a crystal bears a definite and simple relation, in position and in angular inclination, to these axes. As regards mere geometric measurement, there are several direc- Axes di tions in which axes may with nearly equal advantage be projected, rections For example, in the cube (fig. 19) they may be drawn from the centres of opposite faces, as lettered ; or from opposite solid angles, as lettered C ; or from the centres of opposite edges, as lettered D. There is abundance of evidence that each of these directions must be regarded as lines of dominant accretion of molecules. But the accretion may be not only dominant but overwhelmingly so in one only of these directions in certain cases, or existent FlG. I9._p os itionof three sets of axes, along one set of axes alone in certain others. In a specimen of native silver from Alva in Scotland (fig. 20), along this is so much the case that the con creting molecules have done little more than delineate the form of an octahedron, and this they have only been able to do by of domi nant ac cretion.