152 



CRYSTALLOGRAPHY. 



Plat* 



oexxin. 



Fiiji 8. 11. 



Theory, in part by tlie effect of the decrement. The face T 

 (Fig. 11.) is parallel to T (Fig. 8.) The pentagons 

 (Fig. 11.) comes from a decrement by two ranges on 

 the angle I (Fig. 8.) parallel to the diagonal AO. As 

 this decrement does not reach its limit, the summit ex- 

 hibits a second pentagon P (Fig. 11.) parallel to the 

 base P (Fig. 8.) All this description may be exhibit- 



z 



«d in symbolic language as follows: G 2 M T I P. 



In order to prevent beginners from finding any thing 

 ambiguous in this symbolical mode of writing, especi- 

 ally in complicated cases, Hauy is in the habit of pla- 

 cing under the different letters which compose the sym- 

 bol, those which correspond to them in the figure. If 

 we adopt this mode, which is a considerable improve- 

 ment, the symbol denoting bibinaire felspar will be as 



follows: G'MTIP, 

 * M T * P. 

 These letters thus written below, enable us to com- 

 pare the symbol with the figure, and thus to decypher 

 the meaning with facility, how complicated soever it 

 should be. But some more observations will be neces- 

 sary, in order to understand fully the way in which 

 Hauy employs these symbols. 



Let us now then turn our attention to parallelopipeds 

 of a more regular form than that which constitutes the 

 primitive crystal of felspar. But let us suppose them 

 at first not to be rhomboids. They are nothing else 

 than what is represented in Fig. 8. but the form has va- 

 ried so as to render them symmetrical. In consequence 

 of this alteration, certain angles and edges which dif- 

 fered from each other in the first parallelopiped, have 

 become equal in this. Hence,everythingthattakesplace 

 on one of them is repeated on the other. They ought 

 therefore to be denoted by the same letter. Thus, in al- 

 gebra, certain general solutions are simplified in parti- 

 cular cases, when a quantity at first supposed to be dif- 

 ferent from another becomes equal to it. 



Let us suppose, for example, that the primitive form 

 is a rectangular prism, having oblique angled parallel- 

 ograms for its bases, one side of which is longer than 

 the other. In that case, we have OzrA (Fig. 8.), I=E, 

 &c. In such a case, the first letter of the alphabet 

 will be substituted for the other, as is clone in Fig. 

 12. 



If we pass through the different kinds of parallelo- 

 pipeds, we shall find them acquire different degrees of 

 simplicity, which occasions new equalities in the angles 

 and edges, and of course new substitutions of letters. 

 We shall have successively, 



The oblique prism with rhomboidal bases represent- 

 ed in Fig. 13. 



The rectangular prism, with rectangular bases, re- 

 presented in Fig. 14. 



The rectangular prism, with rhomboidal bases, re- 

 presented in Fig. 15. 



The rectangular prism, with square bases, represent- 

 ed in Fig. 16. 



The cube represented in Fig. 17. Here only the su- 

 perior base is marked with letters, because what takes 

 place with respect to it may be applied indifferently to 

 any of the other faces. 



The same mode is followed in writing the symbols 



for these different forms, only the letters that have the 



same name and the same figures, are not repeated. 



Fig. 18. An example will render the method evident. Fig. 18. 



represents the most common variety of the cymophane, Theory, 

 the nucleus of which is a rectangular parallelopiped, such s ""~i'~~"'' 

 as is represented in F'ig. 14. The symbol of the se- 

 condary crystal will be M T'GG'B A T T A. Hauy 



MT 5 i o 

 has called this variety annular cymophane. 



To understand the preceding expression better, let 

 us mark each angle and edge with a particular letter, 



as if the parallelopiped were oblique angled. See Fig. 19. P LATE 



i i CCXXIII. 

 In that case the symbol would become MT J G H 2 B P Fig. 19. 



1 i 



E^ *0. But if we compare Fig. 19. with Fig. 14. we 

 see that H=G, F=B, 0=A, &c. Hence, if we sub- 

 stitute, instead of the first letters, their values, we get 



MT2GG 2 BBA* 'A, which becomes the same with 



the one given above, when the useless repetition of B 

 is suppressed. 



From the preceding statement, it is evident that we 

 must take care not to confound, for example, 2 G G 2 with 

 G 2 "G. The first symbol indicates the decrements which 

 take place on the face T (Fig. 14.) and on the face op- 

 posite to it, going from the edges G towards those that 

 correspond with them behind the parallelopiped. The 

 second indicates the decrements which take place upon 

 the face M, and which meet each other in the middle 

 of that face. If these two decrements took place si- 

 multaneously, their symbol would be 2 Gs. 



In the preceding symbols, each letter, such as *G or 

 G 2 can only be applied to a single edge, situated to the 

 right or the left, as that letter itself. But 2 G 2 applies 

 indifferently to the one edge or the other. Hence, it is 

 needless to repeat that letter. 



Let us take the Figure 20. as another example.* If pig. 20. 

 we suppose Fig. 15. to represent its primitive form, 

 we will have for the symbol of the variety of crystal 



here represented, 'ffMBBEEP 

 o M r s z u P 

 In this symbol 3 G 3 indicates two distinct faces form- 

 ed on each side of each edge G. But it is not neces- 

 sary to place two letters under that symbol, because all 

 the faces situated in the same manner being distinguish- 

 ed by the same letter in the figure, it is sufficient to 

 point out that the symbol 3 G 3 applies to the faces mark- 

 ed with the letter o, and this requires only to write the 

 letter o once under the symbol. 



From the same principles, it follows, that the rhom- 

 boidal dodecahedron derived from the cube, (Fig. 17.) 



i 

 is expressed by the symbol B B. The octahedron de- 



i 



i 



rived from the cube is expressed thus, A "A 1 . 



The rhomboid, supposing it placed in the most na- 

 tural aspect, that is to say, so that the two solid angles 

 composed of three equal plane angles, are in the same 

 vertical line, has properly speaking no base, but mere- 

 ly summits, which are the extremities of its axis. Its 

 angles and edges are marked as in Fig. 21. The letter 

 e denotes that the angle marked by it is similar to that 

 which is marked with a capital E\ So that if all the 

 lateral angles were indicated by letters, the three near- 

 est the superior summit would have the letter E, and 

 the three nearest the inferior summit the letter e. 



As the rhomboid has its six faces equal and similar, 

 it is only necessary to consider the decrements relative 





* This Figure represents a variety of the topaz; of course, our supposition respecting the primitive crystal is not accurate, 

 that does not injure the illustration. 



But 



