CRYSTALLOGRAPHY. 



will be on the bafe A E, O I, a decrement of triple molecules 

 by two ranges in height, fuicc on one part, the quadrilateral 

 figure f O n z, which reprefents the bafe of a compound 

 molecule, is equivalent to the bafes of three fimple molecules ; 

 and on the other, the line Op, which correfponds to the 

 height of a plate of fuperpofition, is equivalent to the height 

 of two fimple molecules. It is eafy to conceive likewife, 

 that the decrement relative to the face E O, A' K, takes 

 place by two ranges in height of double molecules ; becaufe 

 eOpx contains the bafes of two fimple molecules, and On 

 is equal to the length of three fimple molecules. In the 

 decrement which takes place upon I O, A' K, there is a fub- 

 traftion of one row of molecules, triple in one diredtion, and 

 double in the other. 



Among thefe three decrementSj the one which it appears 

 natural to adopt as the principal, is the fecond which takes 

 place upon the face E O, A' H ; becaufe it is the one whofe 

 direiflion deviates the leaft from that of the diagonal E A', 

 or becaufe it takes place by double molecules, which is a 

 more fimple decrement than the other two. 



To give fome further examples of intermediate decre- 

 ments, let us fuppofe O I, I' O', one of the faces of cubic 

 nucleus (fee Plate IV. Jig. 50. Cryjlallography) ; and that 

 the decrement took place on the angles by the fubtraftion 

 of double molecules : in this cafe, the edges of the laminae 

 of fuperpofition will be in the diredlion of the lines dn, 

 km, a b, c h, &C. 



Let EI', ^^.51. be a cubic nucleus, and fuppofe the 

 decrements are made parallel to the lines h m, Im, k r, Ir, 

 always by fubtraftton of double molecules, but in fuch a 

 manner, that there fhall be tlu-ee ranges taken away in the 

 direftion of the breadth, and one in that of the height : in 

 this cafe, the decrements wiU be both intermediate and 

 mixed. Suppofe alfo that the edges of the laminae of 

 fuperpofition, fituated around one folid angle O, had 

 direftions which crofled, fo that with refpeft to the face 

 O I, I' O', the greateil: number of the faces[of the molecules 

 Ihould be taken away on the fide O I' ; but on the face 

 E O, O E, it (hould'be on the fide O O', and with refpeft 

 to face E A, I O, it Ihould be on the fide E O ; the effed 

 of thefe different decrements would produce three faces 

 round each folid angle, which would be fituated in an 

 inchued manner with refpeA to the faces of the nucleus ; 

 and becaufe the cube has eight folid angles, the fecondary 

 cryllal would have twenty-four faces, which would tend to 

 unite four and four, and form the fummit of a p\Tamid 

 round each face of the nucleus. But if we fuppofe the 

 decrement to ceafe before thefe are completed, there will 

 ■ remain fix faces parallel to thofe of the nucleus, and we 

 fhall have a polyhedral cryllal of thirty faces, as reprefentcd 

 jfy. 52. The angles km, I r, correfponding with thofe 

 of the nucleus, are rhombs ; and the faces m I', r 0, are 

 equal and fimilar trapeziums. This form is that of one 

 of the varieties of iron pyrites. 



Suppofe intermediate decrements on the two lateral 

 angles of a rhomboid G G', Plate IV. fg. 47, and that 

 thefe decrements take place by ranges of double molecules, 

 parallel to the faces u m, xy, u' m', x'y'. It is evident, that 

 thefe decrements will produce above each rhomb of the 

 primitive nucleus S G, g" G', two faces, wliich, commencing 

 at the antrles G G', will converge towards each other, 

 and come°in contadl in a line fituated above the diagonal 

 S g", but inclined to that diagonal ; fo that the complete 

 refult of the decrement will be the formation of twelve 

 faces, difpofed fix and fix towards each fummit. Plate IV. 

 fig. Si- reprefents one of thefe folids, with its nucleus in- 

 fcribed. It is a variety of calcareous fpar which fome- 



VoL. XXXIX. 



times occurs. The lines aij fhew the direftion of :> 

 fraaure parallel to the face G,", G' S, ollT^rJu:. 

 nucleus. It appears from this figure, that the nucleus 



does not touch the fecondary cryftal, except by its lateral 

 angles, which are fituated in the edges B S', D /', C s' &c 

 P/'l' I'r" 'i"^ dodecahedron of Bergmann, reprefented m 

 Plate 11. Jg. 22, and called by Haiiy metajalic calcareous 

 Jpar, the lateral edges of the nucleus coincide with thofe 

 edges of the fecondary cryftal that conftitute the common 



o, rt . ^^^° pyramids, as is evident from infpeaine 

 Plate II. Jig. 23. * 



Hitherto immediate decrements have been obferved only 

 m a fmall number of inftances, but they lead to forms as 

 iimple as the other, and give fome curious refults, which 

 deferve to be ftudied in a mathematical point of view, 

 without any reference to cryilallography. 



Compound fecondary Forms Simple fecondary forms are 



thofe which proceed from a fingle law of decrement, the 

 effeft of which covers and conceals the nucleus, which 

 only touches the fecondary cryftal by certain angles or 

 edges. Compound fecondary forms are thofe which are 

 produced by feveral fimultaneous laws of decrement, or by 

 one law which has not reached its limit ; fo that £aces 

 remain parallel to the original faces of the nucleus, whkh 

 further modify the faces of the cryftal. 



Suppofe, for inftance, the law which produces the ofta- 

 hedron from the cube [Plate IVL. fg. 36.) Ihould concur 

 with tliat from which refults the dodecaiedron with penta- 

 gonal faces. {Plate III. Jg. 31.) The firft of thefe laws 

 would produce eight faces, which would have for centres 

 the eight angles of the cubic nucleus. Each of thefe faces, 

 as, for inftance, that whofe centre coincides with the folid 

 angle O, fg. 31, will be parallel to the equilateral triangle, 

 whofe fides pafs through the points p s t, Jig. 34 ; in like 

 manner, the centre that coincides with the front O' will be 

 parallel to the equilateral triangle, whofe fides pafs through 

 the points s np, Plate IV. fg. 54. But the fecond law pro- 

 duces faces fituated as the pentagon, cut by the fides of the 

 triangles pst, snp'. Now the fedion of thefe triangles 

 upon the pentagon / O .r, O' n, fg. 3 1 , reduces the pentagon 

 to an ifofceles triangle, which has the fine t n for the bafe : 

 the two other fides are thofe which pafs through the 

 points t s, ns. The fame takes place with the other pen- 

 tagons. Hence it follows, that the fecondary cryftal 

 produced will be an icofahedron, bounded by eight 

 equilateral triangles, and twelve ifofceles triangles. (Sec 

 Plate IV. fg. 54.) This icofahedron occurs in iron 

 pyrites ; it is different in its form from the regular icofa- 

 hedron of geometers : the latter form does not exift among 

 cryftals, and cannot be produced by any law of decrement. 

 The fame remark apphes to the dodecpJicdron, boimdcd 

 by twelve regular and equal pentagons. 



Another illuftration of a compound fecondary form 

 is offered in the regular fix-fided prifm of calcareous fpar. 

 ( Plate 11. fg. 1 7. ) From the manner of differing this prifm, 

 (fee Crystal,) it is eafy to conceive that the rhomboidal 

 nucleus, ^^. 21, has fix of its fohd angles E, O, I. K, G, H, 

 fituated in the middle of the lateral faces of the prifm : 

 hence it follows, that thefe angles are the points from 

 whence the decrements fet out from the three plane angles 

 of the rhomb E O I, EGA', 10 A', which form tJie 

 fohd angle O ; but it is only ncceffary to conCder the 

 decrements on one of them, fuppofing the fame decrement 

 extends on the two adjacent planes that form the folid 

 angle. Let us then refer all the decrements to the fix 

 angles E O I, E H G. I K G, H G K, O I K, H G 0, 

 the firft of which are turned towards the fummit A, 



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