310 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS. 



original contact-relations may remain largely unaffected, and the 

 recognition of the phyllotaxis constant for any given bud may be 

 rendered easy, however much the members may be apparently 

 extended tangentially. As a general rule, the original contact- 

 phenomena remain unaffected near the points of insertion, and the 

 clean-edged long curves and " stepped " shorter curves are readily 

 distinguished. As noted previously, however, and as in the case of 

 hexagonal facetting, such sliding effects always bring a third set of 

 contact-curves into view ; so that, when excessive, some confusion 

 may be produced in the primary system (c/. Sedum acre, fig. 102). 

 Again, as soon as the amount of " dorsiventrality " and the 

 accompanying sliding-growth becomes considerable, the original 

 spiral " orthostichies " become extremely vague, owing to the 

 difficulty of judging the centres of construction, to which the 

 vascular system does not always afford a sufficient guide ; and 

 although theoretically the curvature of this spiral increases with 

 progressive "dorsiventrality," the superposition of the extended 

 members is so close, to the eye, that any deviation from the 

 superposition demanded by the Schimper-Braun hypothesis is 

 inappreciable. It is thus evident that the Schimper-Braun 

 formulae for estimating and describing adult phyllotaxis continue 

 to hold with a considerable amount of accuracy for shoots with 

 markedly dorsiventral members in which the rate of growth is 

 considerably lessened, which constitute, in fact, the normal type 

 of foKage-shoots ; but the appearances regarded by Bonnet and 

 Schimper and Braun as primary are now seen to owe their 

 existence to a series of secondary growth-phenomena. 



Contact-cycles. — The empirical constructions given in figs. 107 

 and 109, for systems plotted by Archimedean spirals of the second 

 and third intersection, further suffice to bring into prominence 

 a valuable indication of the relation of the individual members of 

 one cycle of a phyllotaxis system, from the standpoint of their 

 overlapping to form continuous investments of the axis. These 

 relations necessarily hold whatever may be the nature of the 

 spirals used to plot the system ; but by using a form of curve 

 which exaggerates the tangential lines of contact, in the manner 

 seen in section of a foliage or flower bud, the relations become 



