94 RELATION OP PHYLLOTAXIS TO MECHANICAL LAWS. 



a fact the secondary parastichy curves are a necessary consequence ; 

 and the ratios that these exhibit present a remarkable analogy 

 to the well-known phenomena observed in root-apices, in which the 

 number of the protoxylem points which here subsequently deter- 

 mine the arrangement of the lateral members is also an inherent 

 property of the protoplasm, determined for each root-apex, constant 

 within a certain range for the species, but varying in actual branches 

 with the amount of nutrition : for example, the constant for the 

 typically tetrarch root of Hanunculus rejpens sinks to 3 and 2 

 in a weak lateral root, or rises to 5 or 6 in a strong one. In the 

 same manner, the foliage leaves of a shoot of a given plant may 

 be laid down with a low number of curves, e.g. (2 + 3), but in 

 strong axes the ratio may rise to (3 + 5), and in reduced axes may 

 fall to (1-1-2). The commonly observed fact that the main axes 

 of many plants present a " | " phyllotaxis, while secondary lateral 

 ones reduce to " §," as first noticed by Bonnet, is thus explained 

 by the assumption that the protoplasm of any asymmetrically grow- 

 ing shoot possesses a definite curve-system, usually of a low ratio, 

 and ranging for the great majority of plants between (2 + 3) 

 and (8 + 13). 



In certain cases, however, as in the capitula of Helianthus and 

 other Composites, the phyllotaxis ratios reach very high numbers, 

 and these exhibit the same phenomena in correlation with the 

 nutrition of the axis in which they obtain. Thus, the medium 

 capitulum of Helianthus taken as a type, presents (34 + 55) par- 

 astichy curves, but in weaker plants these reduce to (21 + 34) and 

 (13 + 21), while in exceptionally well-nourished plants the con- 

 struction curves are (55 + 89) and (89 + 144). The same progressive 

 reduction in the curve system is also noticed in the capitula 

 terminating lateral branches of the first and second degree. 



Either, therefore, these high ratios are determined by the apex 

 at its first formation, or a transition must take place from one 

 series to another; the former alternative might be homologised 

 with the production of a polyarch root in many plants, and from 

 this point of view, the shoot of Helianthus might be regarded as 

 containing a large number of potential curves, of which, if con- 

 ditions were unfavourable, only a few would be utilised; and, 



