118 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS. 



system ; the difference in the number of curves betvjeen two transition 

 stages gives the number of members involved in the change, thus : — 



(3 + 5) passes into ( 8 + 13) in 13 members. 



(5 + 8) „ (13 + 21) in 21 



(8 + 13) „ (21 + 34) in 34 „ etc., etc. 



The law of normal expansion is so simple, and works in Helian- 

 thus capitulum with such remarkable accuracy, that there can be 

 little doubt that it represents in some way a mechanical distribution 

 of growth-energy which is a common property of all plants grow- 

 ing under conditions in which these mechanical relations are allowed 

 free scope. 



As soon as the Fibonacci ratio is disturbed, the system gradually 

 and uniformly proceeds, one member at a time, to put it right again. 

 The addition of all the long curves before the short curves are put 

 in is perhaps the most curious feature. It may be also noted that 

 though the initiation of the change makes for symmetry, by raising 

 the lower number of the ratio first, the ratio is equalised, then again 

 rendered unequal, and again equalised at one point in putting in the 

 short paths ; but the system does not remain stationary at these 

 points of symmetry, it passes on and only rests in the condition of 

 equilibrium of the completed ratio. Although the acquisition of a 

 Fibonacci ratio may be regarded as the optimum attempt at sym- 

 metry in an asymmetrical system, the plant does not so far show 

 any preference for a symmetrical relation attained dm-ing a period 

 of transition. Nor is it clear that such a point of symmetry, al- 

 though isolated in the construction by considering one new curve 

 at a time, is at all comparable with a true symmetrical construction. 

 The whole system is growing together in a correlated method, and 

 the metaphor of crystallisation is perhaps the only one which fits 

 the phenomena observed. 



Now that the number of members involved in making any 

 given normal expansion is known, it remains to see to what extent 

 one expansion can rapidly succeed another. Thus in the Helianthus 

 capitulum taken as a type, it appeared that a new system could 

 only commence at the 34th member or beyond it. The data given 

 for Cyrvara show that the number of members of each system was 



