8o 



RELATIONSHIPS OF SYMMETRY 



as well as with that at C. It is worthy of remark that the changes of position 

 associated with so rapid a decrease in size of the organs proceed in the path of the 

 recurrent series we have considered. The heads of Compositae especially supply 

 examples of this in the region where the first flowers are in contact with the 



FIG. 36. First transition-figure. After 

 Schwendener. 



FIG. 37. Second transition-figure. After 

 Schwendener. 



enveloping bracts; so also do the bases of cones of Coniferae at the point of 



connexion of the scales of the cone with the bracts of the stalk. 



We have concerned ourselves hitherto only with spiral positions, a few words 



are now necessary regarding cyclic positions. 



If we leave aside the relatively rare twisted whorls (the bijugate and trijugate 



systems of the brothers Bravais), as well as some irregular arrangements within 



flowers, we may lay down the rule that members of the several whorls alternate 



with one another, that is to say, the 

 organs of one whorl stand over the 

 gaps between the members of the pre- 

 ceding whorls. The contact-lines run 

 in this case to both sides with the 

 same angle; they form then a span- 

 roof with rafters of equal length which 

 can give no occasion to lateral dis- 

 placements. A change of the position 

 must on the other hand ensue so 

 soon as the relative size of the organs 

 changes in the direction of a para- 

 stichy. Displacements of this kind 

 are often seen in the spadices of 

 Aroideae. Fig. 38 is a diagram of 

 such a case. The lower part shows 

 five-membered whorls, the upper shows 



o 



a spiral arrangement with a -- diverg- 



9 

 ence. The number of the organs 



in a complete cycle is here less than 

 double the number of members in a 

 preceding whorl. In an analogous 

 manner by a corresponding increase of the relative size of the organs in a 



2 



four-membered whorl, there may result a - spiral; and generally upon whorls 



7 



FlG. 38. Transition of a pentamerous cyclic arrangement 

 into a spiral one with a divergence of $. Diagram based upon 

 observations on the spadix ot Aroideae. After Schwendener. 



with n members there follows a spiral arrangement with the divergence 



2 n i 



If 



