INTRODUCTION 371 



defined them by his seven types of axes and who referred to them all 

 crystals with a principal axis. While he developed this classification 

 consistently in crystals with a principal axis, he did not succeed in fol- 

 lowing it in the Isometric system, where he adopted another principle. 

 Had he recognized these types in that system also, he would have largely 

 anticipated the results here given. The author's development is, however, 

 independent of that of Hessel, with whose methods he was not acquainted 

 at the time of its origination, while the method used differs from Hessel's 

 in its simplicity and brevity. It is, however, encouraging to find so large 

 an agreement between the results obtained by Hessel and by the writer. 



We are especially indebted to Schoenflies, from the reading of whose 

 work we were led to the conceptions here presented and with whose 

 method of development that proposed has some features in common. 



Certain conclusions of the author are related to those of Miers. They 

 were, however, developed before the appearance of that author's work. 



These types have been more or less clearly recognized by many in closely 

 related systems, such as the Trigonal, Tetragonal, and Hexagonal, but 

 their occurrence in all systems does not seem to have been shown by others 

 before this.* 



The discussion will consist of two parts : 

 I. A proposed classification of crystals. 



II. An historical review, giving an outline of the earlier development 

 of the thirty- two groups and their relation to the author's results 



Part I: Proposed Classification of Crystals 



PRELIMINARY CONSIDERATIONS 



Before discussing the proposed classification it is desirable to present 

 certain preliminary considerations. 



Definitions — Singular direction. — A singular direction in a crystal is 

 one about which the arrangement of parts and properties differs from 

 that about every other direction in the crystal. Thus, in a crystal of the 

 Hexagonal system, the c axis is a direction about which the properties 

 differ from those about every other direction in the crystal — that is, it is 

 a singular direction and is not duplicated by any other line. 



Elements of symmetry. — It is often convenient to have a word which 

 includes both axes and planes of symmetry. We will define an axis or 

 plane of symmetry as an element of symmetry (the "basis of symmetry" 

 of Moebius). 



* This classification is published only after it has been in use in teaching for 

 ber of years and its value shown by experience. 



