CON T E N T S. 



PREFACE VII. 



FIRST CHAPTER. 



Introductory remarks. Symmetry. Its character and aesthetic value. Defini- 

 tion of symmetrical figures. Symmetry-properties p. 1. 



SECOND CHAPTER. 



General considerations on change of position. Characteristic motions. Figures 

 and their mirror-images. Reflection and inversion. Finite and infinite figures. 

 Symmetry-properties and symmetry-elements of the first and second order. 

 Euler's theorem. Deduction of symmetry-character as a mathematical problem. 

 Geometrical centre. Periods of axes of the first and second order. Repeated 



reflections. General demonstration of symmetry-theorems p. 13. 



THIRD CHAPTER. 



Equivalent and non-equivalent operations. Definition of group. The problem 

 of the possible classes of symmetry as a problem of the theory of groups. The 

 groups with axial symmetry. General theorems. Cyclic, dihedral and endospherical 

 groups. Symmetry-centre and enantiomorphism. Applications in morphology, p. 33. 

 FOURTH CHAPTER. 



Remarks on axes of the second order. Relations between groups of the 

 second order and those with axial symmetry. General theorems. Deductions of all 

 possible groups of the second order. Summary. Applications in morphology, p. 49. 

 FIFTH CHAPTER. 



Limits of axial periods in crystalline matter. Hauy's law. Crystallographical 

 symmetry-axes. Groups and sub-groups. Holohedrism, hemihedrism,tetartohedrism. 

 Crystal-systems. Gadolin's projection. Symmetry-classes of the cubic system. 

 Groups with axes of isotropy. Symmetry of physical phenomena. The symmetry 

 of cause and effect. The symmetry of the electrostatic and of the magnetic field. 

 Centrically symmetrical phenomena in crystals. Superposition of different causes. 

 Symmetry and dissymmetry. General remarks. Problems of the future, p. 77. 

 SIXTH CHAPTER. 



Periodical repetition of identical repeats. Endless patterns. Homogeneity in 

 periodical arrangements. Homologous points. Distribution of points in a plane. 

 Net-planes; their significance in endless patterns. Symmetry of patterns and 

 their net-planes. Points regularly distributed in space. Space-lattices. The sym- 

 metry of Bravais' space-lattices. General symmetry-relations in endless 

 systems. The fundamental domain. Enantiomorphous structural units. The 



