CRYSTALLOGRAPHY. 



443 



Theory, different minerals in which these various integrant 

 V *T~"™'' molecules occur. The tetrahedron, varying, of course, 

 in its respective angles and dimensions, is the integrant 

 molecule of quartz, nitrate of potash, topaz, chiasto- 

 lite, calamine, carbonate of lead, sulphate of lead, phos- 

 phate of lead. The triangular prism, equally various 

 in its dimensions, is the integrant molecule of the eme- 

 rald, aUgite, axinite, granatite, pinite, sommite, vesu- 

 vian, mesotype, sulphate of magnesia, sulphate of ba- 

 rytes, sulphate of strontian, appatite, cinnabar, sul- 

 phuret of copper, titanite, chromate of lead. The pa- 

 rallelopiped is so common, that numerous examples are 

 unnecessary. We may mention common salt, pyrites, 

 calcareous spar, as familiar instances. 



When these integrant molecules happen to be regu- 

 lar mathematical figures, it is not uncommon to find 

 them belonging to more than one species. Thus the 

 cube is the integrant molecule of common salt and of 

 pyrites ; but when this regularity does not exist, we 

 find every species have an integrant molecule of its 

 own, distinct in shape from that of every other species. 

 After having determined the primitive forms of crys- 

 tals^ the next point is to determine the laws which the 

 integrant molecules observe in arranging themselves, 

 so as to produce the great variety of secondary crys- 

 tals, which belong to every mineral species. The Abbe 

 Hauy has shewn, that these secondary forms may be 

 accounted for, and the structure subjected even to cal- 

 culation, by supposing that layers of integrant molecules 

 arranged so as to form plates, are applied successively 

 to all the faces of the primitive crystal, while each suc- 

 cessive plate diminishes in size by the abstraction of a 

 determinate number of integrant molecules (or paral- 

 lelopipeds), either parallel to the edges, or the diagonals 

 of the faces, or in some other direction. We shall en- 

 deavour to make this structure, which constitutes the 

 basis of the theory, intelligible to our readers by some 

 simple examples. The decrements may be either pa- 

 rallel to the edges, to the diagonals, or in an interme- 

 diate direction between the two. It will be proper to 

 give examples of each of these decrements. 



1. Decrements on the Edges. 



Deere- I^* us suppose that the primitive form of a mineral 



ments-on species is the cube ; but that secondary crystals of the 

 the edges, same species likewise occur, having the form of the 

 rhomboidal dodecahedron. How is this dodecahedron 

 derived from the cube ? Let us suppose, as may be 

 done in every case, that the integrant molecule of this 

 species is a cube ; it follows that the primitive cubic 

 crystal is formed by the congeries of a number of 

 cubes. Suppose these cubes of such a size that an edge 

 of the primitive crystal is composed of seventeen of 

 these small cubes applied side by side. Of course 

 every face of the primitive crystal will be composed of 

 289 squares, consisting of the bases of so many inte- 

 grant molecules. According to this supposition, the 

 primitive crystal will be a congeries of 4913 little cubes. 

 Let us now suppose, that a square, consisting of the 

 thickness of one integrant molecule, be applied to every 

 face of the primitive crystal ; but that, instead of be- 

 ing of the size of the face of that crystal, it be less 

 than it by a single row of integrant molecules all round, 

 so that its side, instead of 17 little cubes, contains only 

 15 ; and of course it contains only 225 little cubes in- 

 stead of the 289 that go to the formation of the face 

 of the primitive crystal. Upon each of these first plates 

 applied all round to every face, let another plate be ap- 

 plied similar to the first, but less than it by a row of 



integrant molecules, so that the side contains only IS Theory. 

 squares, and the whole plate only 169 squares. Let *~" "Y~**"' 

 six other plates be applied in succession to each of the 

 faces, diminishing by a row of little cubes all round, 

 so that the sides of each consist of ] 1, 9, 7, 5, 3, 1, 

 squares, respectively. It is obvious, that, by this pro- 

 cess, we have raised upon each of the six faces of the 

 cube a four-sided pyramid, the faces of which, instead 

 of being smooth, will, by their constant diminution in 

 bulk, represent the steps of stairs. Each of these py- 

 ramids having four faces, constitute small 24 triangu- 

 lar faces ; so that, by this process, we have converted 

 the cube into a new crystal. It would seem, at first, 

 that this new crystal ought to have 24 triangular faces ; 

 but a little consideration will satisfy us, that the two 

 adjacent triangular faces, in each pyramid, are in the 

 same plane, and form together a rhomb ; so that, in 

 fact, the cube has been converted into a rhomboidal 

 dodecahedron. Fig. 10. represents the cubic nucleus, Plate 

 with the pyramids raised upon three of its faces ; and CCXXlf. 

 Fig. 11. represents the rhomboidal dodecahedron form- *}%' l0 * 

 ed in this manner. This is an example of a secondary tl &' 

 crystal formed by decrements on the edges of the plates. 

 Suppose us in possession of such a crystal, it is easy to 

 see how, by mechanical division, the cubic nucleus 

 might be extracted. We would have only to cut off 

 all the solid angles formed by four plain angles, by 

 slices parallel to the shorter diagonals EO, OI of the 

 rhombs. ' 



In the preceding example, each plate was only of 

 the thickness of one integrant molecule, and the de- 

 crement was only one row of integrant molecules all 

 round ; but we might have supposed the thickness of 

 the plates to have equalled two or more integrant mole- 

 cules, and the decrements might have been equal to 

 two rows of integrant molecules, or more, at once. In 

 that case, the form of the secondary crystal obtained 

 would have been different from the rhomboidal dode- 

 cahedron. 



It will be necessary here to explain the meaning of Decrement 

 two terms, which we will have occasion to employ fre- '" breadth, 

 quently hereafter. Decrement in breadth is used when 

 the thickness or height of the plate is only equal to one 

 integrant molecule ; but one, two, three, &c. rows of 

 molecules all round, we conceive to be abstracted from 

 the breadth of each succeeding plate. Decrement in Decremem 

 height is used when the plates only diminish by one in heighr. 

 row of integrant molecules in breadth, but their 

 height may be equal to two, three, &c. molecules. In 

 such cases, the decrement is expressed by saying, that 

 it takes place by two, three, &c. rows in height. 



It will be worth while to give another example of a 

 secondary crystal formed by decrements on the edges 

 of the faces. The primitive form of pyrites is a cube ; 

 but, among a great variety of secondary crystals, there 

 is one which occurs in the form of a rhomboid with 

 pentagonal faces. This crystal is represented in Fig. 12. Fig. 12. 

 where the cubic nucleus may likewise be seen. From 

 the inspection of that Figure, it will be obvious, that, 

 instead of a four-sided pyramid, as in the former case, 

 a kind of wedge is formed upon each face of the cubic 

 nucleus, which may be conceived to be the pyramid 

 elongated in one direction. This wedge upon one of 

 the faces of the cube, is represented by OO' tnl V. 

 In this case, the decrements may be conceived to take 

 place by two ranges in breadth between the edges Ol 

 and AE, 1 1' and OO', EO and E'O' ; and in the same 

 manner upon the opposite faces ; while, at the same 

 time, they take place by two ranges in height between 



