RHYTHM. 233 



it, so long as " growth-movement " and " growth-energy ' are re- 

 cognised as being in some way comparable, though not necessarily 

 identical, with more strictly physical phenomena. While, again, 

 the application of the strictly mathematical conception of a 

 uniform distribution of growth-energy around an initial growth- 

 centre must remain necessarily in the condition of a working 

 hypothesis, since it can only apply to a region which is itself 

 somewhat hypothetical, in which the rate of growth is conceivably 

 uniform, there can he no doubt that such an hypothesis must con- 

 tinue to form the basis of all considerations of the geometrical repre- 

 sentation of the growth-phenomena presented by the plant-body ; 

 and before passing on to the discussion of the numerous conditions 

 which may be superimposed on such an elementary phyllotaxis 

 system, it may perhaps be as well to sum up the points which so 

 far appear definitely established. 



Thus, in Part I. {Construction by Orthogonal Trajectories), it 

 appeared increasingly evident that the general method of accumula- 

 ting phyllotaxis data by the observation of orthostichies was 

 hopeless, not only from the standpoint of actual observation, but 

 a consideration of the mathematical propositions of Schimper and 

 Braun showed that helical constructions had become applied to 

 something they were never intended for, i.e. to the developing 

 systems at the growing-points, in which, since the spirals are 

 obviously neither helices nor spirals of Archimedes, the postulated 

 helical mathematics no longer held, and the systems of orthostichies 

 as vertical lines vanish for theoretical reasons, as also for practical 

 purposes. The study of orthostichies thus became eliminated 

 from phyllotaxis, while the value of parastichies and the genetic- 

 spiral remained unassailed. 



In Part II. (Asymmetry and Symmetry), on the other hand, a 

 general consideration of the phenomena of the phyllotaxis systems 

 most commonly exhibited in the plant-kingdom clearly brought 

 out the fact already noted in the preceding chapter, that mathe- 

 matical systems of intersecting curves presented different 

 phenomena, with the result that the genetic-spiral only held for 

 one out of three possible cases, and this again only so long as 

 the system remained constant. Since the genetic-spiral conven- 



