INTEGRITY OF THE CLASSES 381 



class. All are very closely related and form a natural unit. The same 

 is true of the members of other classes. The Isometric groups do not 

 differ in any essential way from the other members of the classes to which 

 they are referred, the only difference being the repetition of the faces 

 about the three-fold axis. 



3. Again, all the members of one class are closely related physically. 

 Thus all the Polyaxial groups manifest circular polarization, the members 

 of the Hedral class show hemimorphic physical properties, etcetera. 



4. The members of each class possess a single type of symmetry, as 

 seen in the spherical projections of plate 31. 



Definition of class. — For the unit so described we propose the term 

 class. It may be denned as follows : A class of crystals is the sum of all 

 crystals having similar combinations of elements of symmetry. Thus all 

 the members of the Axial class have one axis of symmetry; members of 

 the Orthoaxial class, one axis and a plane normal to it, etcetera. 



The discussion given above is seen to lead to an elementary development 

 of the thirty-two groups, the recognition of seven classes of crystals, the 

 development of the seven systems, and the recognition of the larger and 

 more fundamental divisions which correspond with the physical and geo- 

 metric properties of crystals. 



ELEMENTARY DEVELOPMENT 



It is possible to give a 'still simpler development of the preceding classi- 

 fication, for the use of elementary students, which it may be desirable to 

 outline here, in addition to that already presented. Although certain of 

 its features have been stated in the foregoing discussion, they will be re- 

 stated here for the sake of clearness and brevity. 



Symmetry has been defined as the repetition of similar parts and prop- 

 erties in a crystal. It may be of two types : 

 I. Symmetry by rotation about an axis. 



II. Symmetry by reflection about a plane or planes. 



In the first type planes occur singly, about the axis of rotation. In the 

 second type planes occur in pairs in the general form (nPn). 



I. Symmetry by rotation. — If an inclined plane be rotated about a 

 vertical axis, it will produce a singly terminated pyramid, the number of 

 whose sides equals the period of the axis. It may be rotated into any 

 position about the vertical axis-producing pyramids of the first, second, 

 and third orders. Axial class. 



Two such pyramids may manifestly be joined base to base, producing 

 double pyramids. This combination may be effected in three different 

 ways: 



