382 C. K. SWARTZ PROPOSED CLASSIFICATION OF CRYSTALS 



1. The faces of the upper pyramid may be directly over those of the 

 lower pyramid. This type has a horizontal plane of symmetry and is 

 called the Orthoaxial class. 



2. The faces of the upper pyramid may be above the edges of the lower 

 pyramid — that is, the faces of the upper and 'lower pyramid alternate 

 about the vertical axis, which then becomes an alternating axis — Ameb- 

 axial class. 



3. The upper pyramid may be obliquely over the lower pyramid, being 

 turned through some angle intermediate between those of the two preced- 

 ing classes. Such rotation may be either to the right hand or the left, 

 giving two kinds of forms, termed right- and left-handed respectively. 

 They possess horizontal axes of symmetr} 7 — Poly axial class. 



II. Symmetry by reflection about a plane or planes. — Let several planes 

 intersect in a vertical axis. The period of the axis will then equal the 

 number of intersecting planes of symmetry. 



Any inclined crystal face will be reflected about the planes of symmetry, 

 producing a singly terminated pyramid whose faces are in pairs, the num- 

 ber of pairs being equal to the number of the vertical planes of symme- 

 try — Hedral class. 



Two such pyramids may be combined base to base, producing doubly 

 terminated pyramids. This may occur in two ways : 



1. The upper pyramid may be directly over the lower. A horizontal 

 plane of symmetry develops, while axes of symmetry are developed at the 

 intersection of the planes of symmetry, the axes being precisely like those 

 of the Polyaxial class — OrtliohedraJ class. 



2. The faces of the upper pyramid may be over the edges of the lower 

 pyramid — that is, the faces of the upper and the lower pyramid alternate 

 about the vertical axis. Horizontal axes of symmetry develop between 

 the vertical planes of symmetry. The vertical axis is therefore an alter- 

 nating axis whose least possible period is four, since the smallest possible 

 number of lateral planes pins axes is four (two of each) — Amebahedral 

 class. 



The intermediate position, corresponding to that of the Polyaxial class, 

 can not occur, since the planes of s}anmetry of the upper and lower pyra- 

 mid would not coincide, violating the law of Parallel Directions. There 

 are thus seven classes of symmetn r , and seven only, possible. 



Periods of the axes of symmetry. — It will be observed that three of the 

 classes — Polyaxial, Orthohedral, and Amebahedral — have many axes of 

 symmetry. All classes may develop a singular axis with periods of 2, 3, 

 4, 6, and the three classes with many axes may develop in addition three 

 equal axes having periods of 2 or 4, save that the alternating types have 



