390 fi. K. SWARTZ PROPOSED CLASSIFICATION OP CRYSTALS 



III. Forms possessing several axes of period more than 2 : 



1. Tetrahedral forms (p. 653). Symbol T. 



2. Hexahedral forms (p. 664). Symbol H. 



3. Octahedral forms (p. 672). Symbol O. 



4. Dodecahedral forms (p. 679). 



5. Icosahedral forms (p. 692). 



All possible groups of crystals are found by substituting the periods 



I, 2, 3, 4, 6 in the axes of the above divisions save the last two (Dodeca- 

 hedral and Icosahedral), which produce no crystallographic groups, since 

 the periods developed are not possible in crystals. In this manner Moe- 

 bius arrives at twenty-eight groups of crystals only, four of the thirty-two 

 groups being missing. (See table II, p. 396.) His discussion, while clear 

 and simple is therefore incomplete. 



Gadolin. — The preceding authors have endeavored to develop all possi- 

 ble forms of symmetrical polyhedra. Alexis Gadolin differed from them 

 in restricting himself to the development of forms possible in crystals. 

 This he did in a memoire published in 1871, entitled "Memoire sur la 

 Deduction d'un Seul Principe de tous les Systems Crystallographiques 

 avec leur Subdivisions." 25 His work possessed such elegance and fullness 

 that it attracted widespread attention to the new conceptions concerning 

 crystallography. 



Gadolin first 26 develops all possible forms of symmetry about an axis, 

 or a combination of axes, producing eleven groups of symmetry. He then 

 develops symmetry about combinations of planes with the preceding axes, 

 distinguishing three types. 27 His results may be summarized as follows : 



I. Forms possessing axes of symmetry only : 



1. Many axes, 6 groups. 



2. One axis, 4 groups. 



3. No axis, 1 group. 



II. Forms possessing axes and planes of symmetry : 



1. Faces parallel, 11 groups. 



2. Faces nonparallel ("Plane of symmetry"), 9 groups. 



3. Alternating ("Sphenoidal") symmetry, 1 group. 



Thirty-two groups are shown to be possible in this manner, which are 

 precisely the groups developed by Hessel. Gadolin next ■ refers these 

 groups 28 to the ordinary six systems of crystals, classifying the groups as 

 holohedral, hemihedral, hemimorphic, and tetratohedral. The entire dis- 

 cussion is based upon the law of the Eationality of Parameters, to which 

 he devotes especial attention in an appendix. 29 



25 Acta Societatis Scientiarum Fennicse. Helsingforsise, vol. 19, 1871, pp. 1-71 (read in 

 1867). Reprinted in Ostwald's "Klassiker der Exakten Wissenschaften," no. 75, 1896. 

 2«Ibid., pp. 11-15. Reprint, pp. 12-18. 

 «Ibid., pp. 16-25. Reprint, pp. 19-31. 

 28 Ibid., pp. 25-41. Reprint, pp. 31-49. 

 » Ibid., Appendix A. 



