MINERALOGY 353 Fig. 42. plane of a crystal to be so situated as to cut the three parameters OA, OB, OC at their extremities A, B, C, which it must be remem bered are points equi distant from the centre ; or let it be supposed that a glass plate rests upon three intersect ing wires at such points. It is evident that such a plane or plate will have a de finite inclination or / slope. Suppose fur- *^ ther a second plane or plate to exist, which cuts the three semiaxes in the points a. 2 , b 2 , c 2 , which have been measured off (along with a lt b lt c x ) as equidistant from 0. It will be evident that such a plane, though smaller, will be parallel to the first, seeing that, like it, it cuts the three parameters at equal distances from 0. A little consideration will show that, whatever the absolute dis tances from the centre may be, so long as the supporting subdivi sions are equal, no new slope of the glass plates or planes is possible ; planes so situated must be parallel and similar. Any sign which may be adopted to express the slope of one of such planes must be applicable to all. A plane, however, cutting the points a v Jj, c, will have quite a different slope. Let us now suppose a plane to cut a different set of the semiaxes, namely, OA , OB , OC , in the points -a ly - b 1} -c lt Such a plane would be parallel to one cutting the points - a 2 , - b 2 , - c 2 , and also to the set of planes first described, but on the opposite side of the centre of the crystal. If again, however, we had a plane cutting the semi- axes OA and OB in - a lt - b lt but the semiaxis OC in the point -c 2 , it is clear that the slope of this plane would be quite different from that of the planes just described, but it would be parallel to the plane cutting the points a lt b lt c 2 . This slope, like the other, evidently depends, not on the absolute lengths of the portions of OA , OB , OC cut off, but upon their proportions or ratios ; and such is the case with all the planes which are referred to the same axes. As there are three axes, and each or all of them may be cut at any points and at any ratios, it is evident that the number of planes which is possible is infinite ; and it must be also evident that the inclinations of all are fixed or determinate if we know the ratios. While, however, the possible number of planes is infinite, the actual number occurring among minerals is either small or moderate, in virtue of the fact that the ratios of subdivision of the axes are always simple, and not numerous. er Naumann s symbols for the notation or individualizing of planes les of have been glanced at. A simpler method is that of employing as i tation. indices the denominators (if simple fractions) of the fractional parts of the axis cut. Thus 111 is used for any plane parallel to that cutting the axes in o^, J lf q ; 122 for those parallel to a 1} b 2 , c 2 ; 313 for a s , b lt c 3 ; and so on. When any of the points referred to have negative signs, the cor responding indices have negative signs placed over_them. Thus 122 is the index for a plane parallel to a ^c*. 103 is the index of the plane a, b<x> , c 3 . <x> here indicates infinity ; that is, the plane never would cut the axis B however far it were extended ; in other words, it is parallel to it. The necessity for elongating the axes is brought about by the occurrence of highly acuminating planes, which in many cases would not meet the axes at all unless these were prolonged. If the axes are unequal, as in the trimetric forms, then the ratio is of the same character, except that the relative lengths of the axes come into consideration ; but here, as in the regular system, irrational values cannot occur, and in even the most complex crystals they seldom exceed seven, either as aliquot parts or multiples. It will thus be seen that in crystals there is no haphazard scatter ing of faces, but a complete subserviency to law, a law which may be said to be the linear equivalent to the law of multiple propor tions by weight, and Gay Lussac s law of multiple proportions in combination by volume. In abbreviation of all the systematic modes of notation, letters of the Latin and Greek alphabets are frequently employed in a more or less arbitrary manner, and with advantage in the case of highly complex forms. 6. The Laio of Symmetry of Crystalline Combination pbina- * s ^ e consequence of the law of symmetry and the law of the rationality of the parameters, and has been partially mme- of stated in enunciating these laws. It is thus expressed . - (1) a substance can only crystallize in forms, whether simple or compound, which have the same relative symmetry, that is, belong to the same crystalline system, and the parameters of the faces of which bear a simple relation to each other, that is, belong to the same axis ; (2) a form cannot be modified by faces belonging to a different system, or a different series. Certain exceptions to the first part of this law occur. Apparent The element carbon occurs as the diamond, which is cubic, excep- and as graphite, which is hexagonal. Sulphur occurs near tlons - volcanoes in needle crystals belonging to the oblique prismatic system, and also in caves (deposited apparently from solution) in crystals belonging to the right prismatic system. Titanic acid is tetragonal in rutile, and right prismatic in brookite. Carbonate of lime is hexagonal in calcite, and prismatic in aragonite. These are probably only apparent exceptions. The elementary substances which go to form them occur in different allotropic states, with different amounts of specific heat; and it is probable that in these different states they go to form the above modifications, which are therefore, in every respect, except in their chemical composition, different mineral bodies. The physical differences between diamond and graphito may suffice as an illustration. The diamond is trans parent, colourless, brittle, and extremely hard ; graphite is opaque, black, tough, and so soft as to be utilized as a lubricant. Spheres of Projection. The foregoing scheme for the development of the relation which subsists between faces of crystals and their axes affords but slight aid in display ing the position of the faces, or their mutual relationships. The delineation even of a considerable series of crystal forms does not indeed go far in effecting this, on account, first, of very unequal development in the size of the faces of crystals, and, secondly, on account of the habit of development of these faces not only differing largely, but being special to certain localities, as in the entire absence of some faces, and in the preponderance of others. Maps of the whole domain occupied by the forms of each Spheres mineral have been happily projected for such display. 9 f P r - The projection is Laid down as on n globe, in accordance J ect with stereographic projection, and admitting of calculation according to the laws of spherical trigonometry. These globe maps are called " spheres of projection." The centre O is the common centre of the crystal and of the sphere in which the axes intersect. The three axes will of course meet the circumference of the sphere in six points, called the "poles of the axes." From the centre radii are supposed to be drawn, meeting each plane perpendicularly. It is evident that such radii will have fixed inclinations to each other. They are called "normals" to the planes, and the points in which when produced they meet the circum ference of the sphere of projection are called the " poles " of the corresponding faces. A face and its pole thus call for only one symbol. The angle included by any two normals is the supplement of that included by the two corresponding faces. It is thus easy to determine the angles of any two normals when that of the corresponding faces is known, or vice versa. Thus, if the angle between two faces is 125, that of the normals will be 55. The spheres of projection are specially adapted to enable us to avail ourselves of the aid to calculation afforded by the forenoted fact that sets of faces lie parallel to each other, forming zones ; for, when Zones, projected on such a sphere, the normals of the parallel faces will all lie in one plane ; and the poles, all cutting its surface in the direction of one line, may be connected, and so form a great circle on the sphere. This is called the " zone circle." A line drawn through the centre of the
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