VIII 



number of papers which are accessible in all greater libraries 

 to every one who wishes to go further into these subjects. Hence 

 it appeared needless to repeat such an exhaustive treatment of 

 these questions in this book, which is intended rather to fascinate 

 the more vivid and impulsive imagination of the observer and 

 experimenter, than to satisfy the more slowly working and quiet 

 mind of the mathematician. The whole treatment of the necessary 

 theorems and deductions of the general doctrine of symmetry has 

 therefore been condensed into four chapters of this book, in which 

 at the same time even its applications to morphology have been 

 inserted. Notwithstanding this the author hopes that he has given a 

 sufficiently complete deduction of the theorems, so that even for 

 those students who wish to go further into the mathematical theory 

 itself, the general way of reasoning may be found clearly indicated. 

 After seriously testing the methods of argumentation hitherto 

 elaborated, the writer has in many places finally adopted that of 

 Schoenflies, chiefly because in his opinion it offers, from a teaching 

 standpoint, undeniable advantages over the often not less happy 

 and concise ways of treating the problem employed by authors such 

 as Von Fedorow, Wulff, Viola, Barlow, Boldyrew, and others. 

 However many alterations and extensions have been occasionally 

 made, chiefly with the intention of keeping the deduction as general 

 as possible, even for cases which are of no special crystallographical 

 interest, though doubtless important for biologists. The author is 

 convinced that Mobius' definition of symmetrical figures has some 

 logical advantages above the somewhat dualistic definition of Von 

 Fedorow and Schoenflies adopted here; and also that, from a 

 mathematical standpoint, the methods of demonstration of Wulff 

 and Viola,- and more especially that of Boldyrew, may perhaps be 

 considered more homogeneous. But he is convinced also that con- 

 fusion is more readily created in the mind of students of these 

 subjects, when all symmetry-properties are reduced to mere 

 "reflections" in planes of different functions, than when the "axial 

 symmetry" is considered as well. Attention is however occasionally 



