Church. — Note on Phyllotaxis. 483 



path, was explained by an assumption which has exerted 

 a powerful influence on subsequent speculations, that the 

 plant in fact purposely destroyed the postulated mathematical 

 construction, in order that the assimilating members might 

 be given free transpiration-space without any overlapping. 

 Generally speaking, but little real advance has been made in 

 the investigation of the primary causes of phyllotaxis beyond 

 these original views of Bonnet published nearly 150 years ago. 

 It will be noticed that the fractional expressions of Schimper 

 and Braun repeat the hypothesis of Bonnet in a more 

 elaborated form ; the Fibonacci series of ratios is introduced 

 in full, but these are so associated as to still imply helices 

 wound on cylindrical axes. However, as pointed out by the 

 brothers Bravais, axes are commonly conical, dome-shaped, 

 or even nearly plane, and on such surfaces the helices would 

 be carried up as spirals of equal screw-thread, and thus 

 become curves which in the last plane case are spirals of 

 Archimedes. That is to say, by expressing the helix- 

 construction in the form of a floral-diagram, the position of 

 leaves being marked on concentric circles whose radii are 

 in arithmetical progression, the genetic spiral becomes a spiral 

 of Archimedes, and the orthostichies are true radii vectores of 

 the system. Such a geometrical construction is implied in 

 the Schimper-Bi-aun terminology which postulates the exis- 

 tence of orthostichies as straight lines. At the same time, by 

 drawing curves through the same points in different sequence, 

 other spirals appear in the construction, and these, distinguished 

 as parastichies, are similarly by construction spirals of 

 Archimedes. 



Such geometrical plans are given in textbooks, and are 

 used for instilling a primary conception of the arrangement 

 of lateral members ; the fact that they do not always agree 

 with actual observations is glossed over by the assumption of 

 secondary disturbing agencies, as for example torsion. 



On examination, these fundamental expressions are seen to 

 be based on : — 



I. The assumption of a special divergence- angle. 



