128 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS. 



type of plant in which very considerable expansive changes take 

 place, and the optimum succession of the transitions may be very 

 closely approximated. 



That such perfection of transition is not the general rule is seen 

 in Gynara, in which a change was only effected after a considerable 

 number of members of the different system had been laid down. 

 The case of the Daisy is also of interest, in that 13 involucral 

 members are retained in a capitulum which would in Helianthus 

 average 8 ; this involucre of 13 being practically constant for the 

 Daisy (Ludwig).* 



Again, the essential feature of such transitions lies in the fact 

 that, given a ratio of the Fibonacci series, the change is rapid, and 

 when completed gives another member of the .Fibonacci system. 

 From this point of view, the ratios of the Fibonacci series may be 

 regarded as stations of stable equilibrium, in that they give the 

 optimum working angle and set of curves plotting the system, and 

 any alteration of such a system produces a state of instability 

 which as rapidly as possible resumes the Fibonacci relationship. 

 The changes may take place with the minimum number of 

 members intervening {Helianthus), or they may be separated by a 

 larger or even variable number {Bellis, Gynara) ; but the change 

 when it does take place is in these types rapidly negotiated and 

 the " stable equilibrium " of a Fibonacci ratio regained. 



So long, therefore, as it is regarded as a mere convention, which 

 describes phenomena without explaining them, it is convenient to 

 regard a normal plant as possessed of what may be termed a 

 Fibonacci sense, by which any alteration in the phyllotaxis system, 

 whether due to alterations in the bulk-ratio or not, is controlled 

 and corrected to a system of the normal optimum series. 



In other plants, there may be no evidence of any such control- 

 ling power : examples being afforded by the stems of tree-ferns and 



* The relation between these constructions may be obtained from the contact 

 relations of dorsiventral members. It consists in the fact that while 8 members 

 of a (5-1-8) system form a minimum single investment to an axis, 13 will form 

 a double one. The limitation of the calyx of a pentamerous flower to 5 members 

 of a (3-t-5) system is thus curiously repeated in the case of the similarly pro- 

 tective involucre of the Simflower capitulum. 



