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 1 50 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- Braun 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 
1. The assumption of a special divergence- angle. 
K k 
