CRYSTALLOGRAPHY. 



447 



Theory. 



the part situated above ris, we will have a solid on 

 which the face rts will represent the effect of decre- 

 ment that we are considering. 



Now the directions eg, st of the plates applied upon 

 the face IOA'K (and the same may be said of the face 

 EOA'H) in consequence of the auxiliary decrements, 

 are neither parallel to the edge, nor to the diagonal of 

 the face, but intermediate between the one and the 

 other. This want of parallelism will become still great- 

 er if we suppose the decrements upon the angle of the 

 base EOI to take place by 3, 4, &c. ranges. This is 

 the kind of decrement to which the name of interme- 

 diate has been given. It is obvious that it may take 

 place in an infinite number of different directions, ac- 

 cording as it deviates more or less from its two limits, 

 the parallelism with the edge, and the diagonal of the 

 face. 



In cases similar to those of Fig. 4-. we avoid the 

 complication introduced by these intermediate decre- 

 ments, by supposing them comprehended under the 

 principal decrement. But certain crystals exist, in 

 which all the three decrements round the same solid 

 angle are intermediate. In such a case, the simplest 

 of the three is chosen as the principal decrement, and 

 the other two considered as auxiliary. Fig. 5. repre- 

 sents a case of this kind ; c n, which is the edge of the 

 first of the plates applied upon AEOI, is so situated, 

 that on the side of OI there a v e three molecules sub- 

 tracted ; while on the side OE there is only one : 

 np, which is the edge of the first plate applied up- 

 on IOA'K, indicates three molecules subtracted from 

 OI, and two from OA'; cp, which is the edge of the 

 first plate applied upon EOA'H, shews the subtraction 

 of two molecules on OA', and only one on OE. 



It is easy to see that the decrements take place rela- 

 tively to the different faces situated round the angle O, 

 as if the molecules that compose the different plates of 

 superposition, being united invariably several toge- 

 ther, compose other molecules of a higher order, and 

 as if the subtraction took place by single ranges of 

 these compound molecules. Thus there will be on the 

 base AEOI a decrement of triple molecules by two 

 ranges in height, since on one part the quadrilateral 

 figure cOnz, which represents the base of a compound 

 molecule, is equivalent to the bases of three simple 

 molecules ; and, on the other, the line Op, which cor- 

 responds to the height of a plate of superposition, is 

 equivalent to the height of two simple molecules. It 

 is easy to conceive, likewise, that the decrement rela- 

 tive to the face EOA'H takes place by two ranges in 

 height of double molecules, because cOpx contains the 

 bases of two simple molecules, and O n is equal to the 

 length of three simple molecules. In the decrement 

 which takes place upon IOA'K there is a subtraction 

 of one row of molecules triple in one direction, and 

 double in the other. 



Among these three decrements, the one which it 

 appears most natural to adopt as the principal, is the 

 second, which takes place upon the face EOA'H, be- 

 cause it is the one whose direction deviates the least 

 from that of the diagonal EA' ; or because it takes 

 place by double molecules, which is a more simple de- 

 crement than the other two. 



Suppose intermediate decrements on the two lateral 

 angles G, G' (Fig. 3.) of the face of a rhomboid, and 

 that these decrements take place by ranges of double 

 molecules, that is to say, parallel to the lines urn, xy, 

 n'm', x'y'. It is evident that these decrements will 

 1 



produce above each rhomb of the primitive nucleus, Theory, 

 such as SG g"G', two faces, which commencing at the "™ ""Y~ w ' 

 angles G, G', will converge towards each other, and 

 come in contact in a line situated above the diagonal 

 Sg", but inclined to that diagonal ; so that the com- 

 plete result of the decrement will be the formation of 

 twelve faces disposed six and six towards each summit. 

 Fig. 6. represents one of these solids, with its nucleus Plate 

 inscribed. It is a variety of calcareous spar which CCXXIJt, 

 sometimes occurs. The lines aba shew the direction Vl S- 6< 

 of a fracture parallel to the face G g" G'S of the primi- 

 tive nucleus. It appears from this Figure that the nu- 

 cleus does not touch the secondary crystal, except by 

 its lateral angles, which are situated in the edges BS', 

 T)s, Cs', &c. while in the dodecahedron of Bergman, 

 represented in Figs. 6. and 7, and called by Hauy, Plate 

 Chaux carbonate metastique, the lateral edges of the CCXXII. 

 nucleus coincide with those edges of the secondary S s " 6 ' '" 

 crystal that constitute the common basis of the two py- 

 ramids, as is evident from inspecting Fig. 7. 



Hitherto intermediate decrements have been obser- 

 ved only in a small number of instances, but they lead 

 to forms as simple as the other, and give some curious 

 results, which deserve to be studied in a mathematical 

 point of view, without any reference to crystallogra- 

 phy- 



5. Compound secondary forms. 



Simple secondary forms are those which proceed from Compound 

 a single law of decrement, the effect of which covers secondary 

 and conceals the nucleus, which only touches the sur- t0^Ins • 

 face of the secondary crystal by certain angles or edges. 

 Compound secondary forms are those which are pro- 

 duced by several simultaneous laws of decrement, or 

 by one law which has not reached its limit, so that 

 faces remain parallel to the original faces of the nu- 

 cleus, and which concur with the faces produced by 

 decrement, to modify the form of the crystal. Sup- 

 pose, for example, that the law which produces an octa- 

 hedron from a cube (described above) should combine 

 with that from which results the dodecahedron with 

 pentagonal faces. (Fig. 12.) The first of these laws Plate 

 would produce eight faces, which would have, for cen- CCXXIL 

 tres, the eight angles of the cubic nucleus. It is easy F 'S *-• 

 to see that each of these faces, that, for example, whose 

 centre coincides with the solid angle O, (Fig. 12.) will 

 be parallel to the equilateral triangle, whose sides pass 

 through the points p, s, t. In like manner, the face 

 whose centre coincides with the point O', will be paral- 

 lel to the equilateral triangle, whose sides pass through 

 the points s, n, p. But the second law produces faces 

 situated as the pentagons, cut by the sides of the tri- 

 angles p'st, snp. Now the section of these triangles 

 upon the pentagon t O s O' n, reduces the pentagon to 

 an isosceles triangle, which has the line tn for the base, 

 and the two other sides of which pass through the 

 points t, s, and n, s. The same thing takes place with 

 the other pentagons. Hence it follows that the se- 

 condary crystal produced will be an icosahedron, bound- 

 ed by eight equilateral triangles, and 12 isosceles tri- 

 angles. 



Fig. 7- represents this icosahedron, in which the let- Puts 

 ters correspond with those of Fig. 12. PlateCCXXII. and CCXXill. 

 shew to the eye the relation between the two solids. But '*=■' - 

 this icosahedron has dimensions much greater than those 

 of the icosahedron Avhich would be obtained by making 

 sections of the eight solid angles of the dodecahedron 

 (Fig. 12.), which are identified with those of the nu- 



