370 C. K. SWARTZ PROPOSED CLASSIFICATION OF CRYSTALS 



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Schoenflies 393 



Miers 394 



Relation of the divisions proposed to those of preceding authors 394 



Summary 397 



Introduction 



The modern classification of crystals into thirty-two groups is a highly 

 important addition to our knowledge of the subject of crystallography, as 

 well as a valuable contribution to the field of natural science. 



We are indebted to many investigators for the development of this 

 classification, chief among whom are Hessel, Bravais, Moebius, Gadolin, 

 Curie, Fedorow, and Schoenflies. While Hessel was the originator of the 

 classification, his work, unfortunately, was long neglected, and it was not 

 until Gadolin, Schoenflies, and others had given independent develop- 

 ments of the subject, that crystallographers came to recognize the impor- 

 tance of their work. 



While the classification of crystals into thirty-two groups rests on a 

 sound and philosophical basis, it is probable that many have found its 

 presentation, especially to elementary students, attended by certain diffi- 

 culties, chief among which are the multiplicity of the types of symmetry 

 of the thirty-two groups and the consequent lack of a. clear conception on 

 the part of the student of the relations which exist among them. 



Though developed, as so elegantly expressed by Gadolin, by the applica- 

 tion of a single principle, the method by which this has been done histor- 

 ically is too intricate to be employed in an elementary presentation of the 

 subject. If, however, the use of this principle is abandoned and the 

 groups of symmetry are treated as separate units, the student fails to 

 comprehend one of the most important elements in the classification of 

 crystals, namely, the development of the thirty-two groups by the applica- 

 tion of a single principle — a result which seems to the author most unfor- 

 tunate and which leads to embarrassment, because of the failure to per- 

 ceive the larger relations which exist among the groups of symmetry as 

 well as the unity of the subject. 



It is the purpose of the author to give, in the following communication, 

 an elementary development of the thirty-two groups of crystals which 

 leads to the recognition of seven fundamental types of symmetry of crys- 

 tals. It is believed that the recognition of these seven types of symmetry 

 not only greatly simplifies the discussion, but emphasizes relations among 

 the groups of crystals which have not hitherto been clearly recognized. 



These seven types of symmetry were recognized in part by Hessel, who 



