CRYSTALLOGRAPHY. 



445 



Theory, touch by one of their faces, and the ranges themselves 

 ^f*' are simply placed contiguous to each other. The mole- 

 cules parallel to the diagonals touch only by an angle, 

 and the ranges are indented into each other. When 

 secondary crystals are formed by this last kind of de- 

 crement, the new faces are not merely channelled, as 

 happens in the case of decrements on the edges ; they 

 are all bristled with points, which being exceedingly 

 minute, and all in the same plane, escape the eye, so 

 that the faces appear smooth. 



Having thus explained the meaning of the terms, 

 let us illustrate, this kind of decrement by an example; 

 and we cannot get a better than the formation of a re- 

 gular octahedron from a cubic nucleus. This is the 

 consequence of the superposition of plates upon each 

 i'ace of the cube with decrements of a single range of 

 p L ,t Tr: molecules on the angles. LetAEOI (Fig. 17. A) be 



iCXXII. one of the faces of the cubic nucleus subdivided into 

 Pig-. 17. A. yi little squares, which are the bases of so many mole- 

 cules, of which the face is conceived to be composed. 

 Fig. 17. B. Fig. 17- B, represents the first plate of superposition, 

 which ought to be placed above AEOI (Fig. 17. A) 

 in such a manner, that the point e' corresponds with 

 the point e ; the point a' with the point a ; the point o' 

 with the point o ; and the point i' with the point i. It 

 is obvious, from this manner of placing it, that the 

 squaresEe,A«, It, Oo, (Fig. 17. A), remain uncovered; 

 which is the initial effect of the decrement on the an- 

 gles. We see likewise, that the edges QV, PN, LC, FG, 

 (Fig. 17. B), exceed by a range of molecules the edges 

 £A,EO,OI,IA, (Fig.l7-A). This is necessary to pre- 

 vent re-entering angles, and is merely the consequence 

 of the increase of size of the crystal, without any change 

 of form in these quarters. 



The upper face of the second plate of superposition, 

 Fig. 17. C. is represented by BKHD (Fig. 17. C). It must be ap- 

 plied to the first plate in such a manner, that the points 

 e'',a", i", o", coincide with the points e', a', i',o', (Fig. 17. 

 B ) , which leaves bare another row of molecules parallel to 

 the diagonal. This plate also increases by a row of 

 molecules at all its edges B, K, H, D> for the same rea- 

 son as the first plate did. 



The figure of these plates of superposition, which at 

 first was an octagon, has now become a square. It is 

 no longer necessary to continue the addition of rows 

 of molecules at the edges ; so that the succeeding plates 

 retain the square shape, but constantly diminish in 

 size, in consequence of the abstraction of a row of mole- 

 cules from each edge parallel to the diagonal of the 

 face of the cubic nucleus. These different plates 

 Fig. 17. D, are represented by Fig. D, E, F, G, H, and I, in 

 *■* F ' G » H » each of which the small accented letters denote the 

 points of the plate that coincide with the same letters 

 in the preceding plate. Eight plates are necessary, as 

 appears from the Figure, and the last of them consists 

 only of a single molecule. 



If we suppose the same number of plates, of the 

 same form, to be applied successively upon each face 

 of the cubic nucleus, it is obvious that we raise upon 

 each of the six faces of the cube a four-sided pyramid. 

 Hence it would appear, at first sight, that the secondary 

 crystal would have 24 faces. Each of these faces will 

 have four edges, as must appear evident upon a little 

 consideration, and will have the form represented in 

 Kg. 1 &. Fig. 1 8. in which the angle o is conceived to coincide 

 with the angle O of the cubic nucleus, and the diago- 

 nal tx represents the edge HK (Fig. 17. C) of the 

 plate BKHD. The triangle t o x, being composed of 

 those plates of superposition., the edges of which un- 



dergo an increment, will be much shorter than the ton* Theory. 

 angle t s x formed of those plates of superposition whose ■—~,'-— ' 

 edges undergo no increment; because the number of 

 the first is much smaller than that of the second, they 

 being to each other as 2 to b'. 



Thus the surfaces of the secondary crystal is com- 

 posed of 24 quadrilateral faces, arranged, three and 

 three, round each angle of the cubic nucleus. But as 

 - in the decrements, by one range of molecules on the 

 edges, the faces produced on both sides of the same 

 edge are in the same plane, so in decrements by one 

 range of molecules on the angles, the faces formed on 

 the three sides of each angle are in the same plane. 

 This plane is represented in Fig. 19. where the three *' LA J e 

 quadrilaterals surrounding the angle of the cube o, Sr^^ 

 coincide to form the equilateral triangle m n s. Thus °* 

 the faces of the secondary crystal are reduced to eight 

 equilateral triangles, and of course the figure is that 

 of the regular octahedron. 



If these decrements were to stop before they termi- 

 nated in a point, the consequence would be, that faces 

 would remain parallel to the original faces of the cube. 

 The consequence would be, that the crystal would 

 have fourteen faces, eight those of the octahedron, and 

 six those of the cube ; so that it would at once have the 

 form of the cube and of the octahedron. Nothing is 

 more common than to find such crystals both in py- 

 rites and galena. 



If the decrements were more rapid, as, for example, 

 if two or more ranges of molecules were abstracted, 

 then the three trapezoids stox, ml or, nrox, (Fig. 19.) 

 formed round the same solid angle of the nucleus, would 

 not be in the same plain, but would be inclined upon 

 each other, and the secondary crystal would have 24 

 trapezoidal faces. 



As another example of this kind of decrement, let 

 us take the rhomboid, Fig. 20. which differs somewhat pjg., o^ 

 from a cube by having acute angles. Let us suppose 

 that the plates applied upon all the faces of this rhom- 

 boid suffer decrements only at the angles contiguous to 

 the summits A, O', and that these decrements take 

 place by two ranges; then, instead o'f 24 faces, only 

 six would be formed : and if we conceive these pro- 

 longed till they meet each other, they would compose 

 a very obtuse rhomboid, which would be the secondary 

 crystal. Fig. 21. represents such a rhomboid, with its f,>. gl. 

 primitive nucleus enclosed. We see that its summits 

 A, O' coincide Avith the summits of the primitive rhom- 

 boid, from which the decrements commenced, and that 

 each of its faces, as A e o 7, corresponds with one of the 

 faces AEOI of the nucleus, so that the diagonal which 

 passes through the points e, i is parallel to the diago- 

 nal EI of the face of the nucleus, and only somewhat- 

 more elevated. This kind of crystal is found among 

 the secondary crystals of Oligiste iron ore. 



The decrements whicl> take place upon the angle, 

 whether superior or inferior, are susceptible of different 

 variations, respecting which, it may be proper to make 

 some observations before we proceed farther. Let G g 

 (Fig. 1.) beany rhomboid whatever, the summits of Plats 

 which are S, s. Let Sg" s G" (Fig. 2.) be a quadrila- ^^^I"' 

 teral figure formed by cutting through the rhomboid ~£" '" 

 G g in the direction of a plain formed by the two '*' 

 oblique diagonals Sg", *G" (Fig. 1.) and the edges 

 SG", s g" contained betweei* these two diagonals. This 

 quadrilatei'al figure, termed by Hauy the principal 

 section of the rhomboid, is divided in the Figure into 

 a number of similar small quadrilaterals, representing, 

 the principal section of as many molecules. Let SGg"G.' 



