CRYSTALLOGRAPHY. 



457 



Mathema« 



tical 



Theory. 



Theory of 

 the paral- 

 lelopiped. 



Fie. 



II. Theory of the Parallelopiped. 



Let AG (Fig. 35.) be a parallelopiped, the faces of 

 which have any dimensions, and the angles any size 

 at pleasure. Let ns conceive this solid to be subdivid- 

 ed by planes parallel to its different faces into a multi- 

 tude of elementary parallelopipeds, which constitute its 

 integrant molecules. Each of its faces will be divided, 

 of course, into a multitude of small parallelograms, 

 which constitute the bases of as many molecules. If 

 we choose any two of the six faces of this solid, pro- 

 vided that they be opposite, we may consider the solid 

 as an assemblage of plates divided from each other by 

 planes parallel to these faces. 



Suppose, now, that new plates formed of small pa- 

 rallelopipeds, similar and equal to the preceding, are 

 applied to the different faces of the parallelopiped, so 

 that the faces in contact exactly coincide, just as is the 

 case in the interior of the solid. There are three se- 

 parate cases, which may be distinguished. First, The 

 plates may extend by their edges in such a manner as 

 exactly to inclose the nucleus, which will thus increase 

 in size without altering its shape. Secondly, The plates 

 may continue of the same size as the face of the nu- 

 cleus to which they are applied ; in which case, it is 

 easy to see, that re-entering angles would be formed at 

 the edges DC, BC, CG, &c. Thirdly, The plates may 

 progressively decrease in certain directions, so that 

 each will be passed by the preceding plate by a quan- 

 tity equal to one or more ranges, either in breadth or 

 height. 



Of these three cases, the first relates to the primi- 

 tive forms given immediately by crystallization, and is 

 attended with no difficulty. The second is excluded 

 by the laws of crystallization, no example of it ever 

 occurring. The third constitutes the object of the 

 theory. 



Let us suppose, first, that the decrements take place 

 in breadth on all the edges, by the subtraction of an 

 equal number of ranges j and let us confine ourselves to 

 consider the effect of the decrement which takes place 

 parallel to the edge BC, upon the face ABCD. 



If we suppose that the form of the integrant mole- 

 cule (which is similar to the nucleus) is determined, 

 and that the law of decrement is known, it will be 

 easy to find the angle which ABCD makes with the 

 face produced in consequence of the decrement. Let 

 ag (Fig. 36'.) be one of the molecules, Avhose faces, ana- 

 logous to those of the parallelogram Fig. 37, are mark- 

 ed by the same letters. From the point c, draw c s and 

 c r perpendicular to b c. But the ratio between these 

 two lines is given by hypothesis, as is also the angle 

 r c s, which measures the inclination of the faces abed 

 and beg h. 



Now, let op (Fig. 35.) be the distance between the 

 edge BC and the first plate of superposition, which dis- 

 tance is conceived to be measured upon the plane 

 ABCD. It is evident, that op is equal to c r (Fig. 36\) 

 multiplied by the number n of ranges subtracted, or 

 op — ny.cr. From the point p (Fig. 36.) raise pu 

 upon the lateral face of the first plate of superposition, 

 and equal to that plate in height. We shall have 

 p u=c s (Fig. 36.), and op uz=scr. Complete the tri- 

 angle upo (Fig. 35.) It is evident, that the line ou 

 will coincide with the secondary face of the crystal 

 produced upon the edge BC, and that the angle pou 

 will measure the incidence of that face upon the paral- 

 lelogram ABCD. Now, as in the triangle up o, we 

 know the two sides op, pu, and the included angle 



VOL. VII. PART II. 



op u, it will be easy to find the angle pou, and there- Mathema- 

 fore to obtain the incidence sought for. The triangle 

 pou is called the measuring triangle by Hauy, and the 

 same name is applied to all triangles performing the 

 same function. 



Let us now consider the effect of the decrement pa- 

 rallel to the same edge BC, but upon the parallelogram 

 BCGH. Let oih be the measuring triangle, in which 

 o i is the distance between the edge BC and the first 

 plate of superposition, ih coincides with the lateral 

 face of the same plate, and is equal to it in height, and 

 o h coincides with the new face produced by the de* 

 crement. 



Let n be the number of ranges subtracted. We will Pi»tf 

 have oi (Fig. 35.)=n xcs (Fig. 36.) ; and ih (Fig. 35.) p^^! Ir ' 

 =cr (Fig. 36.) ; and o i h—rcs. Hence it will be easy 

 to determine the angle which the face produced by de- 

 crement makes with BCGH (Fig. 35.) 



It may happen, that the two decrements which act 

 upon the sides of BC, have such a relation to each 

 other, that the two faces resulting from them coincide 

 in the same plane, so that the side o h of the triangle 

 o i h is a continuation of the side o u of the triangle 

 o p u, as is represented in Fig. 37. To prove this, let 

 us observe, that in this case the two triangles up o, 

 o i h are similar, both on account of the equality of the 

 angles o pu, h i o, as of the parallelism of the sides op, 

 i h, and the coincidence of the sides o u, h o, in the same 

 direction. Hence 



Figs. 



3fi. 



Fi". 8i 



pu : op : : oi 

 cs (Fig. 36.): 



n 



: ih; or, which is the same thing, 

 n X c r : : k'xcj; c r. This gives us Fig. 3$, 



That is to say, that the two faces will be in the same 

 plane whenever the decrements in the direction BC to 

 GH are inversely, as those in the direction BC to AD ; 

 or, which comes to the same thing, when there is on 

 one side a decrement in height equal to what takes 

 place in breadth on the other. It is obvious also, that 

 the two faces will be in the same plane when the de- 

 crement proceeds on both sides by one range. 



Hence in all such cases, we may abstract one of the 

 decrements altogether, and consider the face as a con- 

 tinuation of that which proceeds from the other decre 1 - 

 ment. 



From what has been said, the method of proceeding 

 to determine the incidences of the secondary faces upon 

 all the other faces of the nucleus is obvious. 



The greatest number of faces which a secondary solid 

 can have from such decrements is 24, since the nucleus 

 has twelve edges, each of which is capable of giving 

 origin to two faces. These new faces will be all tri- 

 angles, or partly triangles and partly trapeziums, ac- 

 cording as the nucleus is or is not more or less elonga- 

 ted in one direction than in the other ; or according as 

 the decrements parallel to certain edges follow a more 

 rapid law than the other decrements. The smallest 

 number of faces which the secondary crystal can have 

 is 12. In that case, all the decrements proceeding 

 from the same edge are the inverse of each other. The 

 simplest case is that in which the nucleus is a cube, 

 and we have «=], n'=l. In that hypothesis, the se- 

 condary crystal is a rhomboidal dodecahedron, as was 

 shewn in the last Chapter. 



Let us now determine, in the same manner, tne 

 decrements on the angles. The secondary faces form- 

 ed by such decrements are called by Hauy lateral 

 faves. Let us suppose, that decrements take place in 

 3 m 



