388 C. K. SAYARTZ PROPOSED CLASSIFICATION OF CRYSTALS 



The major divisions are seen to correspond to the isotropic and aniso- 

 tropic crystals respectively, while the latter are subdivided into the Tin- 

 axial (order 1), and biaxial (order 2) groups. His four minor divisions 

 thus correspond to the Isometric system, Hexagonal (including Trigonal) 

 system, Tetragonal system, and a fourth division, including the Ortho- 

 rhombic, Monoclinic, and Triclinic crystals. Hessel shows that these 

 divisions are fundamental, not only geometrically but also physically, and 

 harmonize with the optical and other properties of crystals which he 

 clearly describes. 15 



Hessel's discussion is obscure and very difficult to follow, not only 

 because of its length and involved character, but also because of the many 

 technical terms which he introduces. TThile his results are of the highest 

 order of importance, they were long forgotten and unappreciated. It is 

 only recently that they have been recognized and estimated at their true 

 worth. 16 



Bravais. — In 1849 A. Bravais published in the Journal de Mathe- 

 matique a discussion of the subject of symmetrical polyhedra entitled 

 "Memoire sur les Polyhedres de Form S} T mmetrique. 17 



Bravais was apparently without knowledge of Hessel's .previous work, 

 to which he does not refer. 



Like Hessel, he endeavors to develop all possible symmetrical pohy- 

 hedra. This he does by considering symmetry with respect to a center, 

 an axis, or a plane. He develops all possible types of symmetry in four 

 divisions. 18 



I. Asymmetric. Class 1. 

 II. Symmetric without axis. Classes 2, 3. 



( Period even. Classes 4-9. 



III. Symmetric with principal axis, j period odd Classeg 10m 



IV. Spheroidal symmetry, no principal axes. 



Quaterternaire. — Forms possessing four-fold axes. Classes 17-21. 

 Decemternaire. — Forms possessing ten three-fold axes. Classes 22-23. 



He presents a series of theorems concerning all possible combinations 

 of axes, planes, and centers of symmetry, developing finally twenty-three 

 classes of symmetrical polyhedra 19 in the preceding four divisions. Of 

 these the twenty-second and twenty -third classes, containing the decem- 

 ternaire forms of the above table, develop periods not possible in crystals. 

 There are thus twentv-one classes of symmetry occurring in the four 



13 Article Krystall, pp. 1277-1279. Reprint, vol. ii. pp. 93-94. 



16 See article by Sobncke. Zeitschrift fur Krystall.. bd. IS, pp. 486-498, 1891. 



17 Journal de Matbematique, Pures et Appliquees, yoI. 14, 1849, pp. 141-180. Repub- 

 lished in Ostwald's Klassiker der Exakten Wissenscbaften, no. 17, 1890. 



18 Journal de Matbematique, vol. 14, 1849, p. 145. Reprint, pp. 12-13. 



19 Ibid., p. 179. Reprint, p. 47. 



