RISING PHYLLOTAXIS. 117 



the curve, marking out sections of (3 • 2 • 3 • 2 ■ 3), etc., all round the 

 system (fig. 44). 



The same relation will be found to hold for other ratio-systems. 

 Formula IE. thus gives a special case of the law of subdivision of 

 paths of growth, corresponding to the "]"-shaped segmentation of 

 Algal membranes; while the graphic method of representation 

 furnishes a key to the construction of a normal transition system. 

 Deviations from it will imply irregularity in expanding phyllotaxis ; 

 while a guide is again provided for checking the anomalous addition 

 of new ridges as a consequence of the formation of new parastichies 

 in many Cactaceae. 



The areas on such a construction diagram are readily numbered, 

 the members of the original (8 + 13) system being marked out by 

 their differences of 8 and 13. Although the diagram was con- 

 structed empirically to begin with, the correspondence of such a 

 theoretical construction with the phenomena actually observed in a 

 capitulum (fig. 45) is so striking that the accuracy of the method is 

 beyond doubt, and its mechanism may be further analysed. Thus, 

 the system was originally (8 + 13); each of the first 13 new 

 members adds a new long curve and the system is thus gradually 

 changed to (21 + 13). 



It is important to note that when forking of the paths takes place, 

 it is the external or peripheral portion of the subdivided segment 

 which must always he regarded as the "member adding the new 

 curve." 



Thus 1-13 each adds a new long curve in 13 segments of the 

 original (8 + 13) system. 



The system is now practically (21 + 13), which, it must be noted, 

 is quite a different construction from (13 + 21), although the possi- 

 bility of an adjustment on the part of the plant is not to be 

 neglected, since the transition takes place in a growing system. 



No. 14 again adds the first short curve ; 21 of the same type are 

 required, and thus 33 adds the last one of the set, and at 34 the 

 new system is completed, and the Fibonacci ratio is again perfected 

 and may be continued indefinitely. But if expansion is to again 

 commence, it is clear that it cannot begin before the 34th member. 

 In a normal system, therefore, each member adds one new path to the 



