444 



CRYSTALLOGRAPHY. 



Throrv. 



Plate 

 CCXXII. 



Fig. 12. 



Fig. 13. 



Fi£. 6. 



m*. ?. 



the edges EO and AI, 01 and O'F, OO' and EE'. We 

 see that these decrements take place upon the different 

 faces of tile cube in three directions, which cross each 

 other at right angles. The decrement, by two ranges 

 in breadth, tending to produce a face more inclined 

 than that which results from a decrement by two 

 ranges in height, the consequence must be, that the 

 structure of plates does not terminate in a point, as in 

 the first example, but in a wedge. The lines p q, tn, 

 (Fig. J 2.), represent the summits of two of these 

 wedges. If we compare these summits p q, tn, with 

 the summit r s of the wedge which covers the face 

 EOO'E' of the cubic nucleus, it will be easy to per- 

 ceive that these three lines are perpendicular to each 

 other respectively. Fig. 13. represents the cubic nu- 

 cleus with wedges raised upon two of its contiguous 

 faces by means of plates pursuing decrements accord- 

 ing to the law above described. The same letters are 

 applied to the same parts of the crystal in Figs. 12. 

 and 1 3. At s' is seen the extremity of the summit of 

 a third wedge raised upon a third face of the cube. 

 Each trapezium, such as Op q I (Figs. 12 and 13.), be- 

 ing in the same plain with the triangle O t I belonging 

 to the adjacent wedge, both together conspire to form 

 the pentagon p O I 1 q, so that the secondary crystal 

 formed by these decrements, instead of 24 faces, has 

 only 12 pentagonal faces, and is therefore a dodecahe. 

 dron as well as the first example, but a dodecahedron 

 of a different kind. 



We shall give a third example of these kind of de- 

 crements, because it contains something peculiar in it, 

 but which often takes- place in the formation of secon- 

 dary crystals ; and it is requisite that the reader should 

 be aware of it. The dodecahedron represented in 

 Fig. 6. is a secondary crystal of calcareous spar. In it 

 the edges EO, OI, IK, &c. where the two opposite py- 

 ramids join, coincide with the edges of the primitive 

 nucleus, as may be perceived by inspecting Fig. 7. 

 The decrements set out from these edges, and do not 

 take place at all upon the other six edges of the primi- 

 tive nucleus EA, AI, AG, OA', &c. Now, it is easy 

 to conceive, that the edges of the plates laid upon the 

 primitive nucleus, form, by their sum, as many trian- 

 gles E s O, I s' O, E s' 0, he. resting upon the edges 

 from which they set out ; and as these lines are six in 

 number, there will be 1 2 triangles, six above, and as 

 many below ,• and all these triangles will be scalene, in 

 consequence of the obliquity of the edges from which 

 the decrements set out. 



With respect to the other edges of the plates of su- 

 perposition, they will be so far from experiencing any 

 decrement, that they will, on the contrary, augment, 

 because they must always remain contiguous to the 

 axis of the crystal, just as happens when the primi- 

 tive crystal increases in size by the superposition of 

 new plates, without undergoing any change of form. 

 It is the province of mathematics, combined with ob- 

 servation, to determine the law of decrement upon 

 which this dodecahedral form depends. If we sup- 

 pose a decrement of one range, it may be demonstrated 

 that the two faces produced on each side of the edge 

 from which the decrement set out, will be in the same 

 plain, and parallel to the axis of the primitive crystal, 

 circumstances which do not suit the present case. If 

 we suppose a decrement of two ranges in breadth, it 

 may be demonstrated that the result will be a dodeca- 

 hedron similar to the one which we are considering. 

 Hauy has pitched upon this law in the present case, 

 •••ifiuenced by several very plausible geometrical consi- 



derations, which, however, we are afraid will not be Theory. 

 found to hold so accurately as he supposed, seduced by ^T"""^ 

 an inaccurate measurement of the angles of the primi- 

 tive crystal of calcareous spar. Fig. 1 4. represents one Plate 

 of the pyramids of this dodecahedron formed by the CCXXfl. 

 superposition of plates following the law of decrements Fl S- ii -. 

 by two ranges of particles. The line E * represents 

 an edge of this pyramid such as it appears to the eye, 

 E s such as it really exists ; but the distance s s' is not 

 sensible, in consequence of the extreme minuteness of 

 the size of the intermolecules, by the abstraction of 

 which the pyramids are formed. The same reason 

 prevents the channels or steps of stairs upon the pyra- 

 mids from being sensible. Though in some cases, 

 when secondary crystals are formed with great rapi- 

 dity, these channels may be perceived by the naked 

 eye. 



We conceive the preceding illustrations are sufficient 

 to explain what is meant by the decrements on the 

 edges of crystals. Let us now proceed to the second 

 kind of decrement. 



2. Decrements on the Angles. 



Decrements on the edges, which have been just de- 

 scribed, are not sufficient to account for all the diver- 

 sity of forms which secondary crystals assume. To 

 give an example ; mineral species, the primitive form 

 of whose crystals is the cube, are found crystallized in 

 secondary forms, some of which are rhomboidal dode- 

 cahedrons, and others regular octahedrons. The for- 

 mation of the rhomboidal dodecahedron has been ex- 

 plained above, by means of decrements on the edges. 

 At first sight, it would appear that the octahedron 

 might also be derived from the cube by decrements on 

 the edges. We have only to take two opposite faces of 

 the cube, and to suppose a four-sided pyramid raised 

 upon each by means of decrements on the edges of 

 the plates successively applied. While this is going on 

 upon these two faces, we may suppose that the other 

 four faces of the cube remain unaltered. Each of these 

 two pyramids may be supposed to prolong itself down- 

 wards till they meet. The consequence would be an 

 octahedron enveloping the cubic nucleus ; but it may 

 be demonstrated, that no law of decrement whatever 

 could in this case form an octahedron with equilateral 

 triangular faces, which is the case with the octahedron 

 derived from the cube. Besides, if we have recourse 

 to mechanical division, in order to obtain the cubic nu- 

 cleus from this kind of octahedron, we shall find that 

 the solid angles of the cube coincide with the central 

 points of the eight faces of the octahedron, which could 

 not be the case if the octahedron had been formed in 

 the way we have been supposing. But if we -suppose 

 the decrements to take place parallel to the diagonal 

 of the faces of the cube, all difficulty vanishes ; we 

 obtain the regular octahedron without difficulty. Such 

 decrements are called decrements on the angles. 



Let OI I'O' (Fig. 15.) be one of the faces of the cu- 

 bic nucleus, divided into a number of little squares, 

 which are the bases of as many molecules. We may 

 conceive these molecules arranged in two different 

 ways ; they may be parallel to the edges, as is the case 

 with the molecules a, n, q, r, s', t', v', z', s' ; or they may 

 be arranged in the direction of the diagonals, as is the 

 case with the molecules a, b, c, d, e,f, g, h, i, and like- 

 wise with the molecules ?i, t, I, m, p, o, r, s, and like- 

 wise with the molecules q, v, k, u, x, y, z. One of these 

 rows of molecules is represented saparately in Fig- 16. 



The molecules parallel to the edges of the square 



Deere- 



meuts on 

 the angles 



tig. IB. 



Fig. If. 



