RISING PHYLLOTAXIS. 135 



made before the system was altered and distortion ensued was 

 thus 6-700. The large capitulum (89 + 144) (fig. 13), similarly 

 shows unbroken short curves for 13-14 members, giving a total of 

 nearly 1900 florets before the reduction set in. An anomalous 

 head (29 + 47) (fig. 54), is only constant for about seven members 

 along the shorter curves, or for a total of 320, the reduction taking 

 place about half way in from the edge of the disk. In rare cases 

 alteration may commence right on the edge and the parastichies 

 then become too irregular to count. Although the general plan of 

 reduction is clear, it does not appear to be sufficiently accurate to 

 warrant the construction of theoretical diagrams. It is possible, 

 however, that the actual change is still rapidly effected, and the 

 mechanism of transition should again be denoted by a reduction in 

 terms of the (2, 1, 2, 1, 2), etc., expression which marked the 

 transit from one Fibonacci ratio to another. 



ASYMMETKICAL CONSTRUCTIONS IN FLOEAL DIAGRAMS. 



Keferring back now to the general scheme for the orientation 

 of the cycles of the Schimper-Braun series (fig. 1), it becomes 

 increasingly obvious that such constructions and their inter- 

 pretations have no necessary connection with spiral systems, but 

 are merely the expression of the relationship of successive terms 

 of the Fibonacci series ; and as already noted, Schimper and Braun 

 added nothing to the Spiral Theory of Bonnet, but intercalated the 

 Fibonacci ratios, which thus constitute an entirely independent 

 generalisation. The fact that the tabulated orientations do agree 

 with phenomena observed in the plant is really the expression of 

 the rise of the Fibonacci ratios in the sequence 2, 1, 2, 1, 2, etc., 

 as presenting the most symmetrical approximation to equal 

 division of the new paths of growth. 



It is further apparent that it is impossible to construct any 

 accurate presentation of a spiral system in terms of circles ; and as 

 soon as circles are adopted, a source of error is introduced which 

 leads one on unconsciously to further fallacies. It is impossible 

 to interpret an asymmetrical system other than by spiral con- 



