NORMAL FIBONACCI PHYLLOTAXIS. 87 



there is no evidence that there is any real vortex movement in the 

 protoplasm, the hypothesis that the lateral primordia must be 

 necessarily produced in close lateral contact cannot be maintained 

 on these grounds, and it does not follow that the ratio of the 

 diameters of axis and primordium will adequately represent the 

 distance-ratio of the new impulses.* Thus in the broad flat apex 

 of Aspidiv/m Film-Mas, the new primordia arise obviously at points 

 spaced at a considerable distance from each other, and yet fall 

 along the well-defined paths of a (5-|-8) curve system. (3-|-5), fig. 

 35f ; cf. also Sempervivum, fig. 82; Nymphaea, fig. 94.) 



Although in such cases the elevation of the protuberances may 

 be imperceptible at their periphery, it is probable that each 

 primordium is strictly localized from the first at or below the 

 surface, even if this is not obvious to the eye, and in the great 

 majority of cases the actual close lateral contact of the primordia 

 is undoubted. The ratio of the diameter of the axis to that of 

 the primordium arising on it may therefore be conventionally used 

 as a constant and may be termed the Bulk-ratio. 



* It will be noticed that this view of hulk-ratio is an entirely artificial one, 

 and can only be useful so far as it ia regarded as a convention which may make 

 discussion easier. Widely differing results are given when comparison is made 

 between the area of the true ovoid curve and that of the circle drawn through 

 the centre of construction, which is the true centre of insertion. Two standpoints 

 are involved : one, that of the completed system in which the bulk reached by 

 the lateral members may acquire some relation to that of the axis on which they 

 are inserted ; the other, that of the similarity of growth-extension from all 

 growth-centres. In the former case, each lateral member would be regarded as 

 possessed of a certain relative size to begin with, and the same view would be 

 adopted so long as the growth may be considered uniform, since one part cannot 

 grow faster than another. In the latter, granted an increased rate in the lateral 

 members, each lateral growth-centre would continue to expand uniformly until 

 contact was made with adjacent members ; ultimate extension would thus be only 

 limited by the distance between adjacent impulses ; and this for a (1 -t- 1) system, 

 for example, is relatively enormous. 



The importance of such lateral contact has been emphasised by Schwendener, 

 and it obtains in the vast majority of constructions ; but it is also clear that in a 

 system for which the rate of growth is not uniform, the growth of a lateral 

 primordium might be so affected that contact may either never be established or 

 only be attained at a subsequent stage. [Aspidivm.) 



t Part I. Plate VIII. 



