CONSTANT PHYLLOTAXIS. 105 



poverished (8 + 13) ; but as (6 + 10) would constitute a Ujugate type, 

 it will be considered under the special section. Taken in connec- 

 tion with the approach in these systems to the bulk ratio (4:1) 

 which is not provided for in the normal Fibonacci series, special 

 interest attaches to an exceptionally fine closed cone of Pinus Pinea 

 in which the parastichies, though somewhat irregular, were for a 

 considerable distance undoubtedly (7 + 10) (Broome, 1900). 



Although thus starting from a standpoint of a hulk-ratio 

 constant, it now appears that the number of parastichy cimrves iecomes 

 of increased practical importance, in that, while the bulk-ratio may 

 he expressed by fractional quantities, the actual ratio of the curves as 

 representing paths of distribution of growth must be expressed by 

 whole numbers. The very smallest corrections in any system must 

 thus be made in the bulk-ratio, the parastichy curves remaining 

 constant until some very considerable alteration becomes necessary. 



That is to say, so long as the plant is condemned by phylogeny 

 to build asymmetrically, one member at a time, and so produce a 

 spiral series, the optimum attempt at symmetrical growth, in- 

 volving symmetrical nutrition, would be given by the limiting 



ratio of the Fibonacci series ->^^ — . In such an ideal system 



the ratio of the parastichies which map out the orthogonal paths of 

 distribution of growth should therefore be ( ^5-1 : 2) ; but as a frac- 

 tional number of curves is impossible in practice, the nearest approach 

 to this ratio, as expressed in whole numbers, is selected in correlation 

 with the size of the lateral member required. The number of the 

 curves is therefore more important in practice than a perfect 

 oscillation angle of 137° 30' 27"-936. On the other hand,* the 

 approach to the " Ideal Angle" is wonderfully close even in low 

 ratios, being within one rainute for a (5 + 8) system. 



Constant phyllotaxis may thus be considered from two entirely 

 different points of view ; either a single growth-oscillation producing 

 members with a definite bulk-ratio is the determining cause, and 

 thus involves the fact that so long as one spiral is in operation the 

 numbers of the parastichy curves are only divisible by unity ; or else 

 the parastichy ratios are primary, and being normally successive 

 * Gf. Mathematical Notes on Log. Spiral Gonstructions, 



