169 



ly planned with regard to Sachs' theory mentioned above. 1 ). 



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 (<P -f- ij) for that 

 of the bracts in the multiple fruit of Ananassa (fig. 129). Of these 

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

 spiral (see fig. 127). The symbols mentioned above, would be -| 

 or ^- 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: 

 (/ + 7), (/ + 2), (2 + j), (3 + 5), (5 + 8), etc. Here also the succes- 

 sive values approach gradually to a limit: / -7-5 - -> an d 



I,Ulo 2 



the ratios naturally adopted by the plant for its intersecting para- 

 stichies are the successive terms of the continuous fraction: 

 1 



1 +1 



1 + etc., 



In a great number of cases z is equal to unity in this fraction. 

 These values would for growing plants with a definite number 

 of leaves give the optimum approach to a symmetrical distri- 

 bution in such a spiral system. However, it may appear doubtful 

 whether the mechanical or physiological causes of this leaf-distri- 

 bution are really better explained by this mode of reasoning than 

 by previous views 2 ). 



The true "pentamery" as observed in the flowers of many Dicotyle- 

 dons and in many lower animals (Chapter /// and IV), is a special 

 case of this ideal arrangement, and in truth the most highly perfected 

 condition of phyllotaxis 2 ), expressed by the special symbol: (5 +5). 



In this respect a certain tendency of living nature to the ratio 

 expressed by the "aurea sectio" may be stated again, - - a fact 



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

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

 (Private communication to the author by Prof. ]. C. Schoute). 



2 ) G. van Iterson Jr., IOQO cit., p. 106, 108, and 144, (1907). 



