71 



system of morphological description on the base of the symmetry- 

 principle. However, he could not succeed in this, because an 

 exact treatment of the symmetry-problem had not yet been made, 

 or at least was not known to him. Without wishing to belittle his 

 work, we feel compelled in the light of our more modern concep- 

 tions, to reject his system, and replace it by the one developed 

 here in detail. 



It should be remarked, that, of course, not only an organism as 

 a whole, but also every part of it may be morphologically described 

 by means of the principles here developed. Thus the corolla of a 

 flower can have a symmetry C 6 , its calyx that of group 5, its pistil 

 of C 3 , its ovary of C 5 ; etc. 



By simply writing down the symbol of its symmetry-group, as 

 adopted here, it is possible to characterise every form in the most 

 concise manner. l ) 



As instances of the symmetry D? and Z)f , in fig. 82 and <?j the 

 crystalforms are reproduced of chalcopynte: CttFeS 2 , and one of 

 the numerous forms of 

 calciie: CaC0 3 . In both 

 cases it may be seen 

 that really the princi- 

 pal axis, although as an 

 axis of the first order 

 only having a period 

 of 180, or 120 respec- 

 tively, is at the same 

 time an axis of the 

 second order with characteristic angles of 90 and 60. 



Moreover, it is also clear from these figures, that in the case of calcite 

 there is a real centre of symmetry, which on the contrary is absent 

 in the case of chalcopyrite. The case of Grovea pedalis, as evidently 

 belonging to the group Z)f , we have drawn attention to before. 2 ) 



x ) It must be remembered here that, from a historical viewpoint, the 

 zoologist Gust. Jager had before Haeckel already made such attempts in 

 this direction, without, however, publishing a complete system of classification 

 based upon the symmetry-principle. 



2 ) Of course the groups of the second order, which are related to D n can 

 be deduced as well from the groups C n of the second order, by combining 

 those with binary axes; just in the same way as in the previous chapter we 

 have derived D n from the cyclic groups C n . This, however, may be left to 

 the reader. 



Fig. 82. 

 Chalcopyrite. 



Fig. 83. 

 Calcite, 



