28 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS. 



Further observation shows again that while as in Pinus and 

 Ewphorhia Wulfenii, a formula may be obtained from the curves, 

 no obvious orthostichies are present, and the genetic spiral cannot be 

 traced without numbering the members. An alternative hypothesis 

 thus immediately presents itself, as in preceding cases, namely, that 

 the single genetic spiral of Schimper is not a primary cause of the 

 formation of parastiehies, but that the parastichies are the primary, 

 as they are the only constant feature ; and that the genetic spiral 

 is of no real significance, but an appearance produced on elongated 

 axes by pulling out such a system as that here found in an arrested 

 condition. 



Further examination of the head brings out the interesting detail 

 that the sterile ray-florets are closely related to the long curves, and 

 typically equal to them in number, while their three-angled 

 ovaries form the " half-bricks " which fill out the mosaic of the disk 

 to the circular outline of the involucre (fig. 11).* 



Other Composites, especially those with bracteate inflorescence 

 receptacles, are equally constant, though in usually lower series ; 

 good examples being afforded by Aster, Chrysanthemum, a,nd the 

 monstrous forms of florists' Dahlias (fig. 16). 



To sum up, the fruit-heads of the Sunflower present a persistent 

 phyllotaxis system in which the members still retain, as in the 

 Pine cone, the actual lateral contact they had when they were 

 formed, unmodified by longitudinal extension. Growth has 

 operated so symmetrically that the structure of the capitulum is 

 practically the same as when the flowers were being first laid down 

 on the apical cone. That such is the case may be readily checked 

 by transverse sections of the young inflorescence, in which the 

 circular outline of the flower primordia is clearly defined. The 

 primary members, it is true, are reduced to supporting frameworks 

 enclosiag cylindrical florets ; but allowing for this peculiarity, the 

 Sunflower-head presents on a large scale the actual conditions 



this seedling would be described as f, or 144° angular divergence. According 

 to the theory subsequently put forward, the angular divergence of the system 

 approximates 138°'5, as measured on a geometrical construction. 



*Gf. Weisse, he. cit: Ludwig, Bot. Oentralb., Ixiv. p. 100 : "Ueber Varia- 

 tionskurven nnd Variations flachen der Pflanzen." ^ 



