CRYSTALLOGRAPHY. 



thefe deciemcnls take place by two ranges, then inftead 

 of twenty -four faces, only fix would be formed ; and if we 

 conceive thefe prolonged till they meet each other, they 

 would conjointly form a very obtufe rhomboid {fg. 39.) en- 

 ciichngthe nucleus. The fummits A, O', coincide with the 

 fummits of the nucleus, from which the decrements com- 

 menced, and each of the faces Aeio correfponds witli one 

 of the faces of the nucleus ; fo that the diagonal which pafles 

 through the points ei is parallel with the diagonal E I of 

 the nucleus, but fomevvhat more elevated. This kind of 

 cryftal is found among the fecondary cryftals of oligifte 

 iron-ore. 



As a further illuilration of the different variations of 

 which the decrements, whether of the fuperior or inferior 

 angle, are fufceptible, let G ^ be any rhomboid whatever 

 [Plate IV. /?f. 4;.), the fummits of which are S s. Let 

 S^", s G", [fig- 46, be a quadrilateral figure formed by cut- 

 ting through the rhomboid G^ in the direftion of a plane, 

 formed by the two oblique diagonals S^", s G",Jig. 45, and 

 the edges S G, S^". This quadrilateral figure, termed by 

 Haiiy the principal feftion of the rhomboid, is divided in 

 the figure into a number of fmall quadrilaterals reprefcnting 

 the principal feftion of as many molecules. Let SG,^" G', 

 fg. 47, be the face of the rhomboid, Jg. 45, marked with 

 the fame letters, and fubdivided into the bafes of the mole- 

 cules of which it is compofed. If we fuppofe that the 

 angle g" undergoes a decrement by a fingle row of mole- 

 cules, the fmall rhomboid reprefented by n a g" will be 

 wanting ; hence it is obvious, that the edges of that plate 

 will have the direftion z, and that the diftance between 

 the angle g", from which the decrement fets out, and the 

 edge z, will be meafured by the femidiagonal of a molecule, 

 or rg. If the decrement took place by two ranges, the 

 edge ij, the firft plate of fuperpofition, will correfpond with 

 c //, and the diftance between it and the angle g" will be 

 meafured by the diagonal molecule gn. Hence we may 

 conclude, that in the decrements on the angles, the diftance 

 between one plate and the fucceeding one, which is the 

 fame as between the point of departure and the edge of the 

 firft lamina, is equivalent to as many femidiagonals of a 

 molecule as there are ranges taken away, as will be further 

 evident by infpefting ^g. 37. But in decrements on the 

 edges, the diftance between two fucceflive plates is equal to 

 the breadth of as many molecules as are taken away. 



This being underftood, let us fuppofe a decrement of two 

 rows upon the angle g". In that cafe, the quadrilateral 

 neap. Jig. /\.6, being a feftion made in the firft plate of 

 fuperpofition, the decreafing edge of this plate will coincide 

 with the httle edge e n, fince ^ n is the fame diagonal as in 

 fig. 47 ; therefore if we draw the ftraight line g" e h, it will 

 coincide with the face produced by the decrement. But 

 g" h being, in this cafe, parallel to the axis S s, as may be 

 demonftrated geometrically, hence it follows that the 

 fecondary faces conftitute the faces of a prifm. If the 

 decrements went on more rapidly, the faces of the fecondary 

 cryftal would have different angles of inchnation to the 

 axis, as will be evident from infpefting Plate IV. jig. 46. 

 If the decrement were by four ranges, for inftance, the edge 

 of the firft plate of fuperpofition would coincide with the 

 line yg ; then the line g q S' indicates the pofition of the 

 face produced by the decrement. What has been ftated, 

 where the plane is parallel to the axis, holds true with refpeft 

 to ^U pofiible varieties of the primitive rhomboid. 



Mixed Decrements are thofe in which the number of ranges 

 taken away in breadth and height give ratios, the two terms 

 of which furpafs unity. As, for example, decrements by 

 two ranges of molecules in breadth, and three in height, or 



by three ranges in breadth and two in height, &c. It is eafr 

 to fee, that the theory may with facility be reduced to that 

 of decrements, in which there is only one row of molecules 

 taken away in one of the two direftions. 



Intermediate Decrements. — Thefe will be beft explained 

 by references to the figures. 



Let A A', Plate IV. Jig. 48. be a parallelepiped of any 

 kind, which undergoes a decrement by two ranges on the 

 angle E O I of its bafe A E, O I. It is obvious, that the 

 edges of the plates of fuperpofition will have the direc- 

 tions !> c, r s, parallel to the diagonal E I, and fo fituated that 

 thefe will be upon the fides O E, O I, two rows of mole- 

 cviles comprehended between the angle O and the line be, 

 and likevvife between b c and r s. But, as has been already 

 faid, the plates apphed upon the adjacent faces I O, A' K, 

 E O, A' H, undergo likewdfe auxihary decrements, whicii 

 continue the effeft of the decrement upon the angle E I. 

 But fuch, in this cafe, are the effefts of thefe decrements, 

 that the edges of the plates applied upon I O, A' K have the di- 

 reftions f^,j/,'and thofe of the plates applieduponEO, A'H, 

 the direftions bg,nt; for fince the lower edge of the firft 

 plate applied upon E O I coincides with b e, and the height 

 of this plate correfponds to that of a fingle molecule, a little 

 attention will fatisfy us, that the plane b eg, which, on one 

 part, coincides likewife with b c, and, on the other, feparates 

 from the bafe A E, O I by a quantity meafured by O^, the 

 height of a fingle molecule, is neceflarily parallel to the face 

 produced by the decrement. The fame holds with the plane 

 rts. From this it follows, that if we fupprefs the part 

 fituated above rts, we fhall have a folid on which the face 

 rts will reprefent tlie effeft of the decrement which we are 

 confidering. 



Now the direftion eg, st, of the plates applied upon the 

 face I O, A' K, (and the fame maybe faid of the face 

 E O, A' H,) in confequence of the auxihary 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 parallelifm will become ftill greater, if we fuppofe the 

 decrements upon the angle of the bafe E O I to take place 

 by three or four ranges. This is the kind of decrement to 

 which the name of intermediate has been given. It is ob- 

 vious, that it may take place in an infinite number of dif., 

 ferent direftions, according as it deviates more or lefs from 

 its two hmits, the parallelifm with the edge and the dia^ 

 gonal of the face. 



In cafes fimilar to thofe of Plate IV. Jig. 48. we avoid 

 the complication mtroduced by thefe intermediate decre- 

 ments, by fuppofing them comprehended under the principal 

 decrement. But certam cryftals exift in which all the three 

 decrements roimd the fame folid angle are intermediate. In 

 fuch a cafe, the fimpleil of the three is chofen as the prin- 

 cipal decrement, and the other two confidered as auxiliary. 

 Fig. 49. reprefents a cafe of this kind : c n, which is the edge 

 of the firft of the plates applied upon A E, O I, is fo fituated, 

 that on the fide of O I there are three molecules fubtrafted, 

 while on the fide O E there is only one : np, which is the 

 edge of the firft plate applied upon I O, A' K, indicates 

 three molecules fubtrafted from O I, and two from O A' : 

 cp, which is the edge of the firft plate apphed upon 

 E O, A H, ftiews the fubtraftion of two molecules on O A', 

 and only one on O E. 



It is eafy to fee, that the decrements take place relatively 

 to the different faces fituated round the angle O, as if the 

 molecules that compofe the different plates of fuperpofition 

 being united invariahly feveral together, compofe other mole • 

 cules of a higher order, and as if the fubtraftion took placy 

 by fingle ranges of thefe compound molecules. Thus there 



v/il} 



