24 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS. 



from one cycle to the next would have been clear. In this actual 

 specimen the ultimate stage reached, as shown by the parastichies 

 of the disk-florets, would have been ^ (cp. fig. 53) : the scales, 

 however, pass on to this from the low divergence of the axis with 

 absolutely no visible line of demarcation between ascending 

 members of the series. That is to say, there is no visible "pro- 

 senthesis " anywhere ; the spiral, as in Sempervivwm, can be given 

 any fractional quality according to the point at which the observer 

 or the specimen stops. In fact, just as orthostichies cannot be ' 

 proved, so prosenthesis is a purely idealistic conception ; the angles 

 subtending the scales pass on by imperceptible gradations, so that, 

 whichever scale be numbered 1 as starting point, no difference in 

 the construction can be observed. 



Clearly, then, there is only one uniform genetic spiral as indicated 

 by the numerical order of development, and no manipulation of 

 phyllotaxis-fractions will explain the system: as pointed out by 

 Bravais,* the error of observation is such that it is impossible to 

 disprove that the angle is not constant for every member, and 

 might in fact be the " ideal angle " throughout. 



From the preceding example of Euphorbia, the conclusion was 

 reached that phyllotaxis is a function of the first zone of growth 

 in which no elongation takes place beyond the equal growth of all 

 the isodiametric initial-cells isodiametricaUy. 



All further investigations, therefore, demand the elimination of 

 the secondary elongation of the second zone of groiMh, either by 

 looking vertically down on the growing point, or by resolving 

 all forces into their transverse components. The following example 

 of a type in which this elimination of the longitudinal extension 

 has been combined with transverse extension on a large scale may 

 be considered. 



5. Phyllotaxis of Helianthus anntjus, L. 



The most perfect examples of phyllotaxis easily obtainable are 

 afforded by the common sunflower, so frequently selected as a typical 



* Loc. cit., p. 71, 



