154 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS. 



V. Asymmetrical Least-Concentrated Type. 



In its simplest form, as expressed in terms of single cells, this is 

 the condition which obtains in the derivatives of the three-sided 

 apical cell of Ferns, Equiseta, and Muscineae, where the three series 

 of segments form superposed series ; a line joining their centres of 

 construction becomes the ontogenetic log. spiral, while the three lines 

 passing radially through the centres of construction of the super- 

 posed segments also form three log. spirals, so that no two members are 

 mathematically superposed, within any limit of construction. The 

 system is thus defined by the number of these " vertical " spiral 

 rows. 



In the case of the cell-segments of Pteris root-apex, these log. 

 spirals were not obvious, owing to the fact that only a few members 

 are shown in one transverse section, although, owing to their 

 rectangular construction closely approximating 1 : 5, more members 

 were seen than can be plotted out in a normal orthogonal curve 

 system. The fact that the arrangement naturally follows from the 

 presence of a three-sided cell, in which each segment produces a 

 foliar outgrowth, while the presence of the three-sided apical cell 

 may itself be a sign of a primitive method of concentrating the 

 terminal ramifications of a filamentous Algal type, lends consider- 

 able weight to the view that this method may be phylogenetically 

 one of the oldest constructions, ,so far as it occurs in Mosses.* 



* Cases in wMoli the relative size of the cell constituents of the plant-body is so 

 great that the arrangement of the lateral members is apparently within the control 

 of single cells, may be conveniently left for the present, and the discussion of 

 phyllotaxis confined to those cases in which the space form of the organism is 



