308 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS. 



flatter and flatter at the periphery of the system, is also very 

 nearly approximated; while it is clear that such Archimedean 

 spirals would be replaced by helices on a cylindrical axis, and 

 thus constitute the ultimate spirals observed on the adult plant, 

 in which growth may be considered to have wholly ceased. A 

 curve-tracing for the expression of these secondary growth- 

 phenomena may thus be constructed by taking the central portion 

 as normally log. spiral, while at the periphery the curves grade 

 into the corresponding spirals of Archimedes. "With such an 

 empirically constructed curve-pattern a system of dorsiventral 

 primordia may be plotted, which, when due allowance is made 

 for the phenomena of sliding-growth adjustments along the spiral 

 of " dorsiventrality," presents a most accurate imitation of the 

 phenomena observed in the macroscopic view of a plant which 

 presents only these modifications of its construction system in 

 the adult condition (c/. Sempervivum spinulosum, fig. 4). 



On the other hand, it is equally clear that, so long as any 

 growth persists, the curves will never really become spirals of 

 Archimedes, although the approximation may be very close to 

 the eye, and the previous construction will not correctly interpret 

 the phenomena observed in the section of a growing apical system, 

 as seen, for example, in a transverse section of the apex of 

 Euphorbia Wulfenii (fig. 90) or Podocarpus japonica (fig. 42) 

 which comprises young growing members only. Since "dorsi- 

 ventrality" may be regarded as the expression of a decrease 

 in the radial growth of the primordia, the log. spiral construction 

 curve may be modified by giving it a radial retardation ; and for 

 present purposes it may be sufficient to assume that such retarda- 

 tion may be uniform. A curve of this form (Type II.) may 

 therefore be used to plot a system which is still growing, but 

 at a progressively slower rate (fig. 100), and by adding the 

 sliding adjustment which steps the shorter curves, a very close 

 approximation is afforded to the Euphorhia section of fig 90 — 

 at any rate, one so close that the amount of error is not appreciable 

 to the eye, the actual rate of retardation not being known.* 



* Thus, fig. 100 represents a simple geometrical construction in which 

 uniform growth at the hypothetical growth-centre undergoes subsequently 



