194 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS. 



as was the original demand for a spiral line of growth as an expres- 

 sion of the plant aim. 



Inherent asymmetrical growth entails the phenomena of a spiral 

 system, and the number of the curved paths is determined by the 

 mathematical claims of radial symmetry in construction, limited by 

 the relative size of the new members. Individual or accidental 

 variations on such a theme will produce more or less definite modi- 

 fications ; and such, if markedly beneficial, may no doubt become 

 stereotyped as specific constants. There is so far no reason there- 

 fore why (2-1-4) as a variant of a (2 + 3) system should not be 

 almost as common as the symmetrical (2 -)- 2) ; it does not give the 

 symmetry which protects lower leaves from vertical light, but it 

 does give two opposite members which become localised at a node, 

 and this in Bipsacus and Silphium (sp.) appears to be a definite 

 biological advantage, although it is not apparent in Scabiosa ana 

 Oephalaria. Once given the (2-J-4) system, the expansion deriva- 

 tives follow rules as perfect as those deduced for Helianthus and 

 Gynara, while the descending system is again the most perfect yet 

 described. 



The phenomena of multijugate systems thus indicate even more 

 clearly than in the case of expansion systems and falling phyllotaxis 

 of the normal series, the weakness of the " genetic-spiral " hypothesis 

 as interpreting changes and variations either local or specific in 

 asymmetrical construction. 



How the asymmetrical system is actually originated in a shoot- 

 apex is not yet apparent, but the conventional standpoint of bulk- 

 ratio, in which a member is formed of a certain relative size at an 

 approximately accurate divergence angle, so far summarises the 

 facts. But once a working system is produced and the members 

 of a full cycle laid down, it becomes increasingly clear that the 

 subsequent history of the system is controlled much more by these 

 existing curves than by any " spiral line of growth.'' New paths 

 are added regularly according to the Fibonacci law, or quite 

 irregularly, with the result that the numbers indicated by the 

 contact-parastichies alone express the system, and if these happen 

 to vary so as to be divisible by a common integral factor, multi- 

 jugate systems result. 



