On Crystatlography. 293 



as the bevel with respect to those of the nucleus ; and be- 

 cause the cube has eight solid angles, the secondary crystal 

 xvill have twenty-four faces, which will tend to unite in 

 fours in the form of a pyramidal summit above each face of 

 the nucleus. But if we suppose that the decrement does 

 not attain iis limit, there will remain six faces parallel to 

 those of the nucleus, and we shall have the polyhedron 

 with thirty faces, or the triacoatahedron represented in 

 fig. 36. 



By comparing this figure with the 35rh, we shall easily 

 conceive tliat the faces k m Ir , k' m' I' /, k" m" I" r" (fig. 

 36), which answer to those of the nucleus, should be 

 rhombs; and because the nuu)her of ridges of molecyle* 

 subtracted lengthways from E O (fig. 35) is double of that 

 of the ridges subtracted lengthways from O I, and so with 

 the other sides ; the great diagonal of the rhombus will be 

 the double of the small diacronal, and the obtuse anajle will 

 be 126° 52 8", which is the measurement of the incidence 

 of the faces of the dodecahedron with twelve pentagons 

 (fig. 14) at the places of the ridjes t n, p q, &c. 



With respect to the faces viX' r o, or' k'r', Sec, (fig. 

 36) produced by the decrement, they will be all equal and 

 similar trapezoids ; and if wc take for an example the tra- 

 pezoid m I' r 0, we shall have the angle m of 57" O' 50", the 

 angle O of 1 1 6' 6' 13", the angle 7' of 1 1 1° 50' 44", and 

 the angle / of 75° 2' 13". 



This form is that of one of the varieties of sulphuretted 

 iron. Geometry has also its triacontahedron, all the faces 

 of which are equal and similar rhombs. This solid has se- 

 veral mteresting properties, w hich will be deniunstraied in 

 the part assigned to geonieiricil calculation. 



Let us now suppose intermtdiate decrements towards tlic 

 two lateral angles G, G' (fig. 31) of the faces of a rhomboid, 

 and always by ranges of double molecules, i. e. parallel lo 

 the lines um, xy, u' vi', x' y'. It is obvious that these 

 decrements will produce above each primitive rhombus, 

 such as S Go-" G', two faces, which, setting out from the 

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

 proceed to unite on a common ridge situated above the 

 diagonal S g'^ but inclined to this diagonal. We shall 

 therefore have, as the complete residt of the decrement, 

 twelve faces arranged by sixes towards each siuninif. 



Fig. 37 represents one of these solids, whicti results from 



a decrement by a simple range of double molecules, in 



.such a inanner that the edges of the lamina; of superposi- 



T 3 tior> 



