142 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS. 



IV. The Symmetrical Concentrated Type. 



This, the most highly perfected condition of phyllotaxis, is met 

 with more especially in the mechanisms of flowers, and also in 

 specialised assimilating shoots (^Hippuris, Zabiatae, Oleaceae, 

 Equisetineae, Dasycladeae, Characeae) presenting the case of " alternat- 

 ing whorls." In the theory of Schimper and Braun it can only be 

 derived from a! spiral system, by complicated processes of "Pro- 

 senthesis " ; while, on the other hand, very curious hypotheses that 

 spiral arrangement is secondary and derived ontogenetically from 

 primitive whorled conditions have been put forward by Henslow 

 and Airy.* 



It is sufficiently clear, however, that these latter views ignore the 

 normal facts of spiral development as expressed by Hofmeister; 

 and previous considerations of the normal asymmetrical concen- 

 trated type indicate that the " concentration " of the system is one 

 of the surest marks of its origin from the corresponding asymmetri- 

 cal case, by the assumption of true circular symmetry as a special 

 ease of log. spiral construction. The system is thus simply defined 

 by the number of members in the alternating series, and accurately 

 planned in a diagram by symmetrical pairs of mutually orthogonally- 

 intersecting log. spiral parastichies ; i.e., when the number of 

 members is very large, the construction is checked by counting an 

 eqiial numher of contact-parastichies in either direction (fig. 55). 

 It may be derived from the preceding type either phylogenetically 

 or ontogenetically, by the growing zone becoming at any sta»e 



* Trans. Linn. Soc, ser. ii. vol. i. 1875. Proc. Roy. Soc, vol. xxiL, 1874 

 297-307. 



