us 



CRYSTALLOGRAPHY. 



Theory. 



Plate 

 CCXXII. 

 fig. 12. 



Plate 

 CCXXHI, 

 Fig. 7. 



Plate 

 CCXXIII 

 Fig. 7. 

 Plate 

 CCXXII. 

 Fig. 12. 



Plate 

 CCXXII 

 Kg. 1. 



Plate 

 CCXXII 

 Fig. 5. 



cleus. This increase of size was necessary to preserve 

 the size of the nucleus. This will be better understood 

 by the following illustration. 



If we wished to obtain the nucleus from the icosa- 

 liedron of Fig. 7, it is evident that the fractures must 

 be made in directions parallel to the edges r s, t n, p q 

 (Figs. 12. and 7.), so that they should be equally in- 

 clined upon the faces of which they form the junction. 

 These planes would pass at the same time through the 

 equilateral triangles p s t, s n p', &c. and we would ob- 

 tain the nucleus when they all met at the centres of the 

 equilateral triangles. 



It follows from this, that the nucleus, the edges of 

 which OI, OE, &c. (Fig. 12.) were uncovered upon 

 the surface of the dodecahedron, is entirely enveloped 

 in the icosahedron (Fig. ']•), excepting its solid an- 

 gles, which are only points, and which constitute the 

 centres of the equilateral triangles. This being under- 

 stood, in order to form an accurate idea of the struc- 

 ture of the icosahedron, we must conceive that the 

 plates applied to the nucleus for a certain period un- 

 dergo decrements only at the angles, as if the second- 

 ary solid were to be a regular octahedron. Beyond 

 this term (the decrement on the angles continuing al- 

 ways) a new decrement takes place and combines with 

 the preceding ; and this new decrement being relative 

 to the dodecahedron, produces the twelve isosceles tri- 

 angles. In this manner we see how the nucleus is en- 

 tirely inclosed in the dodecahedron, excepting the so- 

 lid angles. The first plates of superposition, which on- 

 ly underwent a decrement on the angles, continued to 

 envelope the nucleus by those portions of their edges 

 which underwent no decrements. It is sometimes ne- 

 cessary to suppose, in this manner, different epochas to 

 the different decrements, which concur to produce a 

 compound secondary form when we wish to give a par- 

 ticular account of the mechanism of the structure. 



From this statement it follows, that the distance be- 

 tween the centres of the equilateral triangles pts, at s' 

 (Fig. 7-), ought to be equal to the corresponding 

 edge OI of the nucleus (Fig. 12.), as it evidently is to 

 the eye, as any one may satisfy himself by inspecting 

 the two Figures. 



The icosahedron just described, occurs among the 

 secondary crystals of pyrites. Naturalists at first were 

 disposed to consider this as the regular geometrical ico- 

 sahedron. But it has been demonstrated by Hauy, 

 that the regular icosahedron does not exist among cry- 

 stals, and cannot be produced by any law of decrement 

 whatever. The same remark applies to the dodecahe- 

 dron of mathematicians, a solid bounded by twelve re- 

 gular and equal pentagons. No such crystal exists, nor 

 can -be produced by any law of decrement whatever. 

 Of the five regular solids of mathematicians, the cube, 

 the tetrahedron, the octahedron, the dodecahedron, and 

 the icosahedron, the first three occur in the mineral 

 kingdomj but not the last two. 



It will be worth while to give another example of a 

 compound secondary form ■ and we shall take for that 

 purpose the regular six-sided prism of calcareous spar 

 (Fig. 1.) From the account formerly given of the 

 manner of dissecting this prism, it is easy to conceive 

 that its rhomboidal nucleus A A' (Fig. 5.) has its solid 

 lateral angles E, O, I, K, G, H situated in the middle 

 of the faces of the prisms ; from which it follows, that 

 these angles are the points from which the decrements 

 set out that produce these faces. 



These decrements act at once upon the three plain 



angles EOI, EOA', IOA' ; but we may satisfy ourselves Theory. 

 with considering the decrement relative to one of these > ""'~Y~"" 

 angles, supposing the face which results from it ex- 

 tends itself upon the two adjacent rhombs belonging to 

 the same angle. Let us agree, therefore, to restrict the 

 whole to the six angles EOI, EHG, IKG, HGK, OIK, 

 HGO, the three first of which are turned towards the 

 summit A, and the three last to the summit A'. If 

 we suppose a decrement by two ranges of rhomboidal 

 molecules on these different angles, six faces will be 

 produced parallel to the axis, as has been already ob- 

 served. 



The plates of superposition, at the same time that 

 they undergo a decrement towards their inferior an- 

 gles, will extend by their superior parts so as to re- 

 main always contiguous to the axis, the length of which 

 will progressively augment. The faces produced by 

 the decrement will gradually increase, and when they 

 touch each other we shall have the solid A A' (Fig. 4.), Platk 

 where each of the faces, as oO o, is marked by the CCXXII. 

 same letter as the angle O (Fig. 5.), to which it be- F 'S S - 4 > s - 

 longs, and which is now situated in the middle of the 

 triangle oOo, because it constitutes the common point 

 from which the three decrements set out. 



In proportion as new plates are applied after this to 

 the preceding ones, the points o, o rise up, while the 

 point O sinks down, so that at a certain period we shall 

 have the solid represented by Fig. 3, where the faces 

 produced by the decrements are become pentagons, 

 such as ooiO e. 



Things being in this state, let us suppose a second 

 decrement to concur with the first, and to take place 

 by a single range upon the superior angle EAI (Fig. 5.), 

 and its opposite angle HA'K, always with this condi- 

 tion, that the fkee produced by it on both ends of the 

 figure is continued upon the two rhombs adjacent to 

 that to which the angles EAI, HA'K belong. The 

 effect of this decrement will be, to produce two faces 

 perpendicular to the axis ; and when it has reached the 

 point at which these faces cut the six faces parallel to 

 the axis, produced by the first decrement, the second- 

 ary solid will be completed, and will be a regular six- 

 sided prism (Fig. 1.) 



We have already said that this result is general, Fig. i 

 whatever be the measure of the angles of the primitive 

 rhomboid. We now see why, in the mechanical divi- 

 sion of the prism, the cut pp oo ( Fig. 2. ) has its Fig. 2. 

 sides pp, oo parallel to each other, and to the hori- 

 zontal diagonal EI (Fig. 5.); since the two decrements pig. 5. 

 taking place, the one upon the angles EOI, the other 

 upon the angle EAI, the plates of superposition ought to 

 have their edges turned towards this same diagonal. 



In the case which we have been considering, and 

 which is the most usual, the axis of the secondary cry- 

 stal is longer than that of the nucleus ; so that this nu- 

 cleus having its lateral angles contiguous to the faces of 

 the prism, its summits are inclosed within the prism, 

 at a certain distance above the centre of the bases. If 

 we were to suppose that the two decrements began at 

 the same time, in that case the axis of the prism would 

 be equal to that of the nucleus, and the lateral angles 

 and summits of the nucleus would be tangents, the one 

 to the faces of the prism, the other to its bases. If the 

 decrements on the superior angles of the nucleus were 

 anterior to the other decrements, which is the opposite 

 of the first case, the summits of the nucleus would then 

 be contiguous to the bases of the prism, while its late- 

 ral angles would be wholly within the prism, between 



Plate 

 CCXXII. 



