BASIS OF CLASSIFICATION 3?3 



(&) Several planes of symmetry intersecting in one line produce an axis 

 of symmetry at their intersection whose period equals the number 

 of intersecting planes. Thus, in figure 4 it is readily seen that 

 the intersection of the three planes will repeat the point a at a', 

 and this pair of points will appear in three similar positions — 

 that is, they are repeated three times about the line of intersec- 

 tion, which becomes, therefore, a three-fold axis. 



(c) The intersection of alternating planes and axes in one line produces an 

 axis of alternating symmetry at their intersection whose period 

 equals the number of intersecting axes and planes. An ordinary 

 axis of half that period will coincide with the alternating axis. 

 This law is readily seen by the repetition of the point x in fig- 

 ure 5. 



We may express the theorems a, &, c by the statement that 

 the intersection of several axes or planes of symmetry produces 

 an axis of symmetry at their intersection whose period equals the 

 number of intersecting elements. If axes or planes only intersect, 

 it is an ordinary axis ; if alternating axes and planes intersect, it 

 is an alternating axis. 

 5. Possible periods. — The only periods axes of symmetry can possess in crystals 

 are 2, 3, 4, 6. This springs directly from the law of the Ration- 

 ality of Parameters. Its proof is too full to be given here. (See 

 Groth's Physikalische Krystallographie, fourth edition, 1895, page 

 313.) 



DEVELOPMENT OF CLASSIFICATION 



Basis of classification. — The basis of classification is symmetry. Crys- 

 tals have been classified upon the basis of the crystallographic axes, as is 

 done in the ordinary definition of the systems of crystals. These are, 

 however, imaginary mathematical lines, existing only subjectively in the 

 mind of the observer and developed by him for convenience sake. They 

 are not present objectively in the crystal and can not, therefore, form a 

 natural basis for the classification of crystals. That they are arbitrarily 

 determined is seen by the fact that the same crystal may be referred to 

 several different axes, as in the use of the axes of Miller and Bravais in 

 the Trigonal system. 



Symmetry affords a natural basis for the classification of crystals, being 

 expressed by the elements of symmetry which are objectively present in 

 the crystal. Moreover, symmetry is expressive not only of the geometrical 

 form, but of the relations of the physical properties as well, and probably 

 springs from the arrangement of the atoms and molecules in the crystal. 3 

 It is, moreover, the basis upon which the modern classification has been 



3 See recent articles of Barlow and Pope in Journal of the Transactions of the Chemi- 

 cal Society of London, vol. 89, 1906, p. 1675, and vol. 97, 1907, part ii, pp. 1150-1214. 



See also abstracts in American Chemistry Journal, vol. xxxvil, 1907, p. 638, and vol. 

 xlii, 1909, p. 158. 



