163 



tThus, in the case of "diagonal" orientation the asymmetrical 

 _ystem of fig. 130 appears; and how closely these constructions, - 

 which can moreover be performed in a simple geometrical way, 

 lead to arrangements corresponding with what is observed really 

 natuiv, may be seen by comparison of Church's fig. /?/, which 

 epresents the con- 

 figuration of fruits 

 in the capitulum of 

 Helianthus annuus, 

 s deduced from his 

 .eory, with the ar- 

 ngement of the 

 fruits really obser- 

 as represen- 

 ted in fig. 126. In 

 is construction 

 .e whole system 

 remains orthogonal, 

 as was originally 

 planned with regard 

 to Sachs' theory 

 mentioned above *). 



Fig. 131. 



Church's construction of the Configuration of the 

 fruits in the Capitulum of Helianthus annuus. 



32. As to the 

 numerical relations 

 expressed by this 

 theory, in which the 

 orthostichies have wholly lost their significance, the following 

 remarks may be made here. 



The system of intersecting parastichies is indicated by two numbers, 

 as for instance (34 +55} in the case of Helianthus for the inplan- 

 tation of the individuals in its capitulum-disk, or (8 + /?) for that 

 of the bracts in the multiple fruit of Ananassa (fig. 127). Of these 

 numbers the first refers to the longer, the second to the shorter 

 spiral (see fig. 125). The symbols mentioned above, would be f^ 

 or -fo in the Schimper-Braun-theory; and in an analogous way 

 as therein the series of fractions indicated previously is arrived at, 

 we can write the "normal" series in the new symbols as follows: 



1) Recently some doubt has arisen also about the correctness of Sachs' original 

 view, concerning the "orthogonal" intersection of newly formed cell-walls. 

 J. C. Schoute; private communication to the author. 



