450 



CRYSTALLOGRAPHY. 



Theory. 



Purr. 

 CCXXfl. 

 Fig. 9. 



Plate 

 CCXXII. 

 •Fig. s. 



not parallel to the faces. This is the case with the 

 rhomboid, which constitutes the primitive form of the 

 tourmaline. It may be divided by planes passing 

 through the axis and the oblique diagonals. The re- 

 sult is the production of tetrahedral molecules, such as 

 are represented in Fig. 0. 



Thus, besides parallelepipeds, there are two other 

 shapes which the integrant molecules assume ; namely, 

 the tetrahedron, and the triangular prism. Now, it de- 

 serves attention, and it is a point of considerable con- 

 sequence in the theory of crystals, that the tetrahedral 

 and prismatic molecules are always arranged in such a 

 manner in the interior of primitive and secondary crys- 

 tals, that, taking them in groups of 2, 4, 6", Or 8, they 

 compose parallelopipeds ; so that the ranges subtracted 

 by the effect of decrement, are nothing else than these 

 parallelopipeds. 



In order to conceive the better how this may be, let 

 us suppose for an instant that the molecules of calca- 

 reous spar are divisible into tetrahedrons, as is the case 

 with the rhomboid, which constitutes the primitive form 

 of ^the tourmaline. This supposition will change no- 

 thing in the explanation of the different forms of which 

 calcareous spar is susceptible ,• that is to say, that in 

 determining the forms of this mineral, aided by the 

 theory, we may always satisfy ourselves with consider- 

 ing decrements by one or more ranges of rhomboidal 

 molecules. 



What is only a hypothesis with respect to calcareous 

 spar, is a reality with regartl to the tourmaline. But 

 although the rhomboids, to which we arrive by mecha- 

 nical division in this species, are themselves divisible 

 into tetrahedrons, still the decrements which produce 

 the secondary forms take place by the subtraction of 

 rhomboids similar to the primitive form ; so that we 

 may suppose, in the calculations relative to the deter- 

 mination of these forms, that the tetrahedrons which 

 constitute the true molecules are united together in an 

 invariable manner, in each rhomboid. 



Let us take another example from those crystals 

 whose primitive form is a regular six-sided prism. Let 

 AD (Fig. 8.) be the base of such a prism, divided into 

 -small triangles, which constitute the bases of the inte- 

 grant molecules. It is evident, that any two neigh- 

 bouring triangles whatever, such as Ap i, AO i, com- 

 pose a rhomb, and of course the two prisms to which 

 they belong form by their union a prism with a rhom- 

 boidal base, which is a species of parallelopiped. If 

 we conceive that the two triangular prisms, which con- 

 stitute elements of the parallelopipeds, are invariably 

 united together, it is obvious that we may consider the 

 six-sided prism as composed of rhomboids instead of 

 • triangular prisms. Now, if we conceive a series of 

 plates piled upon the hexagon ABCDFG (Fig. 8.), 

 -which undergo, for example, upon their different edges, 

 a subtraction of one row of parallelopipeds similar to 

 those that we are supposing here, these edges will suc- 

 cessively correspond with the lines of the hexagon 

 i Imnrh, kuxyge, &c. ; from which we see, that the 

 quantity that each plate passes the other is a sum of 

 parallelopipeds or prisms with rhomboidal bases ; and 

 it is easy to judge, that the result of the decrement, 

 supposing it to reach its limit, will be a right hexangu- 

 lar pyramid, which will have for its base the hexagon 

 ABCDFG. 



All the other primitive forms different from the pa- 

 rallelopiped, give analogous results. We might even 

 substitute for each of these forms a nucleus similar to 

 ■the little parallelopipeds, which are formed by the 



ry. 



ocxxiir. 



Fig. 8. 



union of the tetrahedrons or trwuagula • pi-isms, and wri Tin 

 would succeed equally in explaining the secondary '■"""" 

 forms by laws of decrement applied to that nucleus, 

 which would be obtained likewise by mechanical di- 

 vision. 



The Abbe Hauy, to whom we are indebted for the 

 whole theory of crystals, calls these parallelopipeds, 

 composed of tetrahedrons or triangular prisms, snbtrac- 

 tive molecules. They are always substituted in place of 

 the tetrahedrons or triangular prisms, in considering 

 the decrements which produce the secondary forms in 

 these cases. Thus, as far as the theory of crystals is 

 concerned, we have nothing to do with the integrant 

 molecules, but may conceive all crystals composed of a 

 congeries of parallelopipeds. 



Though we have extended this Chapter to a consi- 

 derable length, there is still another particular which 

 requires explanation, before we proceed to the mathe- 

 matical theory. The Abbe Hauy has invented parti- 

 cular symbols to denote the particular laws of decre- 

 ment which produce the secondary forms. As these 

 symbols occur constantly in his writings, and as they 

 are useful, by greatly shortening the account of the 

 formation of secondary crystals, it is proper they 

 should be understood. W T e shall endeavour, therefore, 

 to explain them in this place. 



Let Fig. 8. represent an oblique parallelopiped, the H.tuy's 

 faces of which have angles with different measures, and symbols. 

 let it be the primitive form of some mineral ; as, for Plate 

 example, of felspar. 



The vowels are adopted to represent the solid angles. 

 The four first, A, E, I, O, are placed at the four angles 

 of the superior base following the order of the alpha- 

 bet, and that of ordinary writing, namely, beginning 

 at the top, and going from left to right. 



The consonants are chosen to denote the edges. The 

 six first, B, C, D, F, G, H, are placed on the middle 

 of the edges of the superior base, and upon the two 

 longitudinal edges of the lateral faces, which occur first 

 in going from left to right. These consonants are like- 

 wise arranged in the alphabetical order, and according 

 to the usual mode of writing. 



The letters P, M, T, which are the initials of the 

 syllables of which the word primitive is composed, are 

 placed each in the middle of the superior base, and of 

 the two lateral faces exhibited to view. 



Each of the four solid angles, or of the six edges 

 marked by letters, is susceptible in the present case, on 

 account of the irregular form of the parallelopiped, of 

 undergoing particular laws of decrement. Hence the 

 reason why they are marked each with a different let- 

 ter. But as the laws of decrement act with the great- 

 est symmetry possible, every thing which takes place 

 with respect to the angles and edges marked with let- 

 ters, takes place also with respect to the opposite an- 

 gles and edges which are not marked, or are not visi- 

 ble. It was only necessary to mark the number of 

 solid angles or edges which undergo distinct decre- 

 ments, because these decrements include likewise im- 

 plicitly all those which take place upon analogous an-* 

 gles or edges. 



In some cases, however, it is necessary to indicate 

 these last angles or edges. In such cases, the small 

 letters, having the same names as the capitals, are em- 

 ployed for the purpose. The angles analogous to A, E, 

 I, O, are denoted by a, e, i, o ; and the edges analo- 

 gous to B, C, D, F, G, H, are denoted by b, c, d,f,g, h. 

 But it is very seldom necessary to mark these small let- 

 ters on the Figure ; it is sufficient to introduce the» 



