NORMAL FIBOKACCI PHYLLOTAXtS. 83 



I. Normal Fibonacci Phyllotaxis. 



The type corresponds to the case of cycles in spiral series derived 

 by Schimper and Braun from the fractional series of divergences 

 by the assumption of slight hypothetical " Prosenthesis," and by 

 Schwendener from the same fractional series by equally hypo- 

 thetical " contact-pressures." 



It can only be strictly defined by the number of intersecting 

 parastichies, the ratios of which mark successive values of the 

 stages of the continuous fraction 1 



m 



1+i 



1, etc., and is figured dia- 



grammatically by the corresponding number of log. spirals drawn 



with the appropriate curve tracing of the series 1, 1, 2, 3, 5, 8, 13, 



21, 34, 55, 89, 144, etc. 



From the fact that this is the system found in the Sunflower, 

 which was regarded as 'par excellence a normal plant, it may be 

 regarded as the normal type for all Phanerogams, without 

 necessarily implying that it is also the phylogenetically primitive 

 one. 



Since the construction diagrams are correct well within the 

 error of drawing, and far within that of any actual observation on 

 the plant, geometrical plans may be utilised for the further investi- 

 gation of the properties of such a system. 



Since also the spiral construction (Scheme D*) was derived 

 geometrically from the symmetrical case (Scheme C), and that for 

 all mathematical deductions from the latter case, homologous pro- 



* Part I., figs. 22, 23, p. 51. 



