REVIEW OP DEVELOPMENT OP THIRTY-TWO GROUPS 387 



over the lower, producing enantiomorphous forms (ebenbildlich). He 

 recognizes thus four major types of axes: (1) singly terminated, (2) 

 doubly terminated direct, (3) alternate, (4) oblique. Each of the first 

 three types may coincide with a plane of symmetry or not, producing 

 faces which are in pairs (zweifach) or single (einfach). 



All possible symmetrical forms are next developed about the seven 

 types of axes, by which means he arrives at all possible symmetrical fig- 

 ures. 10 



Hessel now develops the law of the Eationality of Parameters, 11 which 

 he terms Das Gerengesetz, and shows that according to this the only axes 

 possible in crystals are those possessing periods of 1, 2, 3, 4, 6. Applying 

 this law to the preceding forms, he shows that all crystals fall into 32, 

 and only 32, groups of symmetry, of which a summary is given on pages 

 1280 to 1284 of his article. 12 



He divided all the groups, save those of the Isometric system, into 

 seven types, according to the character of their dominant axis. Unfor- 

 tunately he failed to carry his classification into the Isometric system, but 

 was led to adopt another principle in it, namely, the number of similar 

 radii in the crystal. The Isometric system was thus termed the eight- 

 rayed (8-strahlig) system, its groups being as follows: 13 



Zweifach 8-strahlig (Hexakisoctahedral of Groth). 



Einfach 8-strahlig (Pentagonal Icositetrahedral of Groth). 



Zweifach 4-strahlig (Hexakistetrahedral of Groth) . 



Einfach 4-strahlig (Tetart. Pent. Dodecahedral of Groth). 



Zweimal 4-strahlig (Dyakisdodecahedral of Groth). 



Had he succeeded in carrying his seven divisions into the Isometric 

 system he would have largely anticipated the author's classification. Un- 

 fortunately, however, he failed to do so. 



Larger divisions of crystals. — Hessel does not refer the thirty-two 

 groups to the usual six (or seven) systems, but united them in four divis- 

 ions, which he classifies as follows : 14 



I. Class without principal axes (Hauptaxenlos). 



Order 1. With four three-fold axes. 

 II. Class with a principal axis (Hauptaxig). 

 Order 1. Possessing one principal axis. 



Family 1. "One and three dimensional." 

 Family 2. "One and two dimensional.'* 

 Order 2. Possessing several different axes. "One and one dimen- 

 sional." 



10 Article Krystall, pp. 1062-1157. Reprint, vol. 1, pp. 48-145. 



11 Ibid, pp. 1232-1276. Reprint, vol. ii, pp. 45-91. 

 "Reprint, vol. ii, pp. 95-98. 



13 Article Krystall, p. 1280. Reprint, vol. ii, p. 95. 

 « Article Krystall, p. 1277. Reprint, vol. ii, pp. 92-93. 



