CRYSTALLOGRAPHY. 



453 



Theory. 



Plate 

 CCXXUI. 

 Fig. 22. 



Tigs. 23, 

 24, 25. 



Figs. 26, 



to one of these faces ; as, for example, the one which 

 in the Figure is marked P, because all the others are 

 mere repetitions of this. These observations suggest 

 the following rules : 1. The decrements which set 

 out from the superior angle A, or the superior edge 

 13, will have the figure indicating the number of 

 ranges placed below A and B. 2. Those which set 

 out from the lateral angles E will have their figures situ- 

 ated at the side and towards the top of the same letter. 

 3. With respect to those which set out from the inferi- 

 or angle e, or from the inferior edge D, the figure will 

 be placed above the letter e or D. 



Suppose, for example, that Fig. 22. represents the va- 

 riety of calcareous spar, called analogic by Hauy, its 



> i 

 symbol will be e D B. 

 i 

 erg 



What has been said of the rhomboid is easily applied 

 to the other primitive forms. But probably some illus- 

 trations will be considered as necessary to make the 

 symbols applied to them the more readily understood. 

 On that account we shall take a short review of each 

 of them. 



Fig. 23. represents the octahedron with scalene tri- 

 angles, Fig. 21. the octahedron with isosceles triangles, 

 and Fig. 25. the regular octahedron. 



In placing the figures which accompany the letters 

 in the symbols, the same rule is followed that was de- 

 scribed with respect to the rhomboid. Thus, in Fig. 

 24. the figure is placed below the letter to represent 

 decrements setting out from the angle A or the edge B ; 

 it is placed above for those which set out from the edge 



D, and at the side for those which set out from the angle 



E. If we want to denote the result of a decrement by 



one range upon all the angles of the regular octahe- 



i 

 dron (Fig. 25.) we have only to write A 'A 1 . To in- 

 dicate the result of a decrement by one range on all the 



i 

 edges, we write B B. The first of these decrements 

 i 



produces a cube, the second a rhomboidal dodecahe- 

 dron. 



In some species, as in the nitrate of potash, the pri- 

 mitive octahedron, the surface of which is composed 

 of eight isosceles triangles, similar 4 and 4 to each 

 other, ought to have the position represented in Fig. 26. 

 that the secondary crystals may have the most natural 

 attitude, that is to say, that the edges which join the 

 two pyramids which compose the octahedron, ought to 

 be two of them in a vertical direction, as F, and two 

 in a horizontal, as B. By comparing Fig. 26. with Fig. 

 27. in which the letters are placed as if all the angles 

 and edges had different functions, it will be easy to 

 conceive the distribution adopted in Fig. 26. and brought 

 to the symmetry of the true primitive form. For, in the 

 present case, we have Ez=A, D=C, G=F. 



The figure denoting the number of ranges, will be 

 placed under the letter, to denote decrements proceed- 

 ing from B. It will be placed at one side, or below, to 

 denote those proceeding from A ; according as their 

 effect respects the triangle A I A, or the triangle A IF. 

 It will be placed above or below, for those which pro- 

 ceed from C, according as their effect is produced on 

 the first or the second of these triangles. It will be 

 placed at one side for the decrements which proceed 

 from F. Finally, it will be placed above, below, or on ei- 

 ther side, for the decrements that proceed from I, accord- 

 ing as their effect takes .place towards B or towards F. 



The tetrahedron being always regular, when it be- 



Tiieory. 



comes the primitive form, it will be expressed as in 

 Fig. 28. To indicate, for example, a decrement by 



three ranges on all the edges, we would write B B ; CCXXIH. 



Plate 



Fig. 28. 



and to indicate a decrement by two ranges upon all the 

 angles, we would write A 2 A 2 , as in the case of the re- 



2 



gular octahedron. 



A simple inspection of Fig. 29. is sufficient to make p;„ 09. 

 us understand the symbols in the case of regular six- 

 sided prisms. The figures are written precisely in the 

 manner already decribed for the four-sided prism ; to 

 which, therefore, we refer the reader. But it happens 

 sometimes that three of the solid angles taken alter- 

 nately are replaced by faces, while the intermediate 

 angles remain untouched. In that case the prism is 

 distinguished as in Fig. 30. Fig. 30.. 



In the rhomboidal dodecahedron (Fig. 31.) each so- pj„ 3! 

 lid angle composed of three planes may be assimilated 

 to a summit of the obtuse rhomboid. Hence, it is only 

 necessary to give letters to one face, as may be seen in 

 the Figure. 



Hitherto there has been no occasion to use any sym- 

 bols for the dodecahedron with triangular faces, be- 

 cause it is more natural to substitute in place of it the 

 rhomboid from which it is derived, and which gives 

 simpler laws of decrement. 



We have still to explain the method of representing 

 a peculiar case, which sometimes occurs in some crys- 

 tals, where the parts opposite to those which under- 

 go certain decrements remain untouched, or are modi- 

 fied by different laws. This case belongs chiefly to the 

 tourmaline, and it is easy to indicate its peculiarity by 

 means of zeros. 



For example, in the variety of tourmaline represent- 

 ed in Fig. 33. the primitive form of which is represent- Figs 32, 

 ed in Fig. 32. ; the prism, which is nine-sided, has six 33. 

 of its faces, namely, s, s (Fig. 33.) produced by the 

 subtraction of one range upon the edges D, D (Fig. 32.) 

 and the three others, such as I, by the subtraction of 

 two ranges on the three angles e (Fig. 32.) only. Far- 

 ther, the inferior summit has only three faces parallel 

 to those of the nucleus ; while, on the superior sum- 

 mit, the three edges B are replaced each by a face n, n 

 (Fig. 33.) in consequence of a decrement which has 

 not reached its limit. This crystal is represented by 



i 2 2.0 



the following symbol : D e E P B b. The quantities 



1 1 .0 



si P n 

 2.0 

 E, b indicate, the one that the angles E (Fig. 32.) 



I -o 



opposite to e undergo no decrement ; the other, that 

 the edges parallel to B remain equally untouched. 



If these edges underwent a different law, which pro- 

 duced, for example, an abstraction of two ranges, the 



1 4- 2 o ... 



symbol would become D e EPBi. From this, it is 



1 2 



obvious, that it must be understood that the decrements 

 represented by a capital letter accompanied by any fi- 

 gure, do not implicitly include the similar decrements 

 represented by a small letter of the same name, or the 

 opposite, that is to say, that B does not implicitly in- 



2 

 elude b, or vice versa, except when the second letter 



2 

 does not enter into the symbol with a different figure, 

 or does not bear the same figure accompanied by a ze- 

 ro. In the first case, each of the two letters indicates 

 a decrement which is peculiar to the edge or angle in- 

 l 



