RISING PHYLLOTAXIS. 137 



Few systematists probably ever troubled about the question as to 

 whether the helix of Bonnet and Schimper became an Archimedean 

 spiral in the ground-plan ; they accepted the construction and 

 expressed the " orthostichies " of the genetic spiral as radii of the 

 circle, assuming therefore that they were also radii veotores of the 

 spiral.* 



So long as this is regarded as a simple method of constructing a 

 spiral system, and the convention is granted, no particular harm is 

 done, and for conventional floral-diagrams the same construction 

 may be retained so long as it is clearly understood that such a 

 diagram does not present the facts of the given floral-structure 

 accurately, but only a symmetrical version of them. Thus in a 

 floral-diagram of the Buttercup, it is possible to place 8 oblique 

 rows of stamens on a certain number of circles, or emphasise 13 

 rows on a smaller number, but the true (8-|-13) construction, 

 giving 8 oblique rows one way and 13 the other, requires a spiral 

 curve. On the other hand, if the circular plan be adopted and the 

 convention forgotten, error creeps in and may become magnified in 

 the course of further deductions. Thus, having placed three members 

 of a " I " spiral on one circle, and two on another, the next false 

 step was readily made in assuming that the spiral either actually 

 consisted of two such whorls in the plant, or might be interpreted as 



heading " Varying growth in lateral members " ; it thus becomes referable to 

 the laws which control the tangential extension of foliar members and 

 discussion must therefore be postponed until the angles normally subtended by 

 free and packed primordia have been tabulated for the different systems. (Gf 

 Mathematical Notes.) 



* It is of interest here to compare the views of Schleiden {Grundriss der 

 Botanik, Eng. trans., p. 264), who in his masterly analysis of the principles of 

 phyllotaxis as they were discussed in his day, stood alone among German 

 botanists in his support of the theories of the Bravais, in that they were 

 logically based on mathematical laws, and deduced from the properties of a 

 " mathematical spiral " ; the fact that an almost indefinite number of mathe- 

 matical spirals may be proposed appears to have been completely forgotten. 

 The spiral in question was a helix wound on a cylinder, which has a jsarallel 

 screw-thread, but also makes equal angles with vertical lines drawn on the 

 cylinder ; it may thus be continued up a cone as a curve, which would on a 

 projection give either a spiral of Archimedes or a logarithmic spiral, according 

 as the former or latter property was allowed to determine the curve. 



