302 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS. 



consideratibn ; that is to say, these figures include the study of 

 the relationships of adjacent leaf-hases. 



Observation shows that, as the rhombs are progressively ex- 

 tended along their transverse diagonal, the shorter construction 

 curves are " stepped " ; and since these diagonal paths were also 

 originally log. spirals, the curve of a value one stage lower in the 

 summation series than the numbers expressing the ratio of the 

 curves composing the system {i.e. their difference) may be con- 

 ventionally termed the " Spiral of Dorsiventrality." From this 

 standpoint a bifacial leaf is only flattened in a strictly horizontal 

 plane when it is produced in a symmetrical phyllotaxis system ; 

 in which case the paths of lateral extension are concentric circles : 

 in the more general case of asymmetry, structural dorsiventrality 

 becomes exaggerated along a spiral path, which has therefore no 

 direct relation to external environment, as, for example, the action 

 of vertical light, although it is the nearest approach possible to 

 a horizontal line in each rhomb. In other words, the architectural 

 scheme of each shoot is controlled by the growth-centre of the 

 axis, which is the fundamental growth-centre of the whole shoot- 

 system, and here, as in the case of eccentricity, the influence of 

 external environment, if this is the determining agent, must act 

 on the primary centre at the end of the shoot, and all subsequent 

 architectural details are worked out according to strict geometrical 

 principles. These construction diagrams further show that the 

 result of the sliding-growth effect is here to place this tangential 

 diagonal more and more in a horizontal line : a teleologist might 

 at this point even make the suggestion that the object of the 

 sliding effect was of the nature of a biological " adaptation " which 

 would render the surface of the leaf-lamina more strictly 

 horizontal; but such an explanation is wholly gratuitous. The 

 distinction which is here drawn between the geometrical plan of 

 leaf-base insertions and the geometrical properties of the free 

 portions of the primordia, as expressed in the lamina portion of 

 the leaf, will be further discussed from the standpoint of the 

 mathematical equations of the theoretical curves {cf. Mathematical 

 Notes, VIII.). 



A few interesting details are also more clearly exhibited by 



