328 HISTORY OF MINERALOGY. 



calcspar may be placed with one of its obtuse corners uppermost, so 

 that all the three faces which meet there are equally inclined to the 

 vertical line. In this position, every derivative face, which is obtained 

 by any modification of the faces or edges of the rhombohedron, 

 implies either three or six such derivative faces ; for no one of the 

 three upper faces of the rhombohedron has any character or property 

 different from the otner two ; and, therefore, there is no reason for the 

 existence of a derivative from one of these primitive faces, which does 

 not equally hold for the other primitive faces. Hence the derivative 

 forms will, in all cases, contain none but faces connected by this kind 

 of correspondence. The axis thus made vertical will be an Axis of 

 Symmetry, and the crystal will consist of three divisions, ranged round 

 this axis, and exactly resembling each other. According to Weiss's 

 nomenclature, such a crystal is " three-and-three-mernbered." 



But this is only one of the kinds of symmetry which crystalline forms 

 may exhibit. They may have three axes of complete and equal sym- 

 metry at right angles to each other, as the cube and the regular octo- 

 hedron ; or, two axes of equal symmetry, perpendicular to each other 

 and to a third axis, which is not affected with the same symmetry 

 with which they are ; such a figure is a square pyramid ; or they 

 may have three rectangular axes, all of unequal symmetry, the modifi- 

 cations referring to each axis 'separately from the other two. 



These are essential and necessary distinctions of crystalline form ; 

 and the introduction of a classification of forms founded on such rela- 

 tions, or, as they were called, Systems of Crystallization, was a great 

 improvement upon the divisions of the earlier crystallographers, for 

 those divisions were separated according to certain arbitrarily-assumed 

 primary forms. Thus Rome de Lisle's fundamental forms were, the 

 tetrahedron, the cube, the octohedron, the rhombic prism, the rhombic 

 octohedron, the dodecahedron with triangular faces : Haiiy's primary 

 forms are the cube, the rhombohedron, the oblique rhombic prism, tliu 

 right rhombic prism, the rhombic dodecahedron, the regular octohe- 

 dron, tetrahedron, and six-sided prism, and the bipyramidal dodecahe- 

 dron. This division, as I have already said, errs both by excess and 

 defect, for some of these primary forms might be made derivatives from 

 others; and no solid reason could be assigned why they were not. 

 Thus the cube may be derived from the tetrahedron, by truncating the 

 edges ; and the rhombic dodecahedron again from the cube, by trun- 

 cating its edges; while the square pyramid could not be legitimately 

 de.ntitied with the derivative of anv of theso forms ; for if we were tr 



