Church. — Note oh Phyllotaxis. 487 



into Archimedean spirals, which differ by a constant along 

 each radius vector, if they represent the limiting planes of 

 members which grow to a constant bulk and thea remain 

 stationary, in the manner that lateral members do on the 

 plant. The appearance of Archimedean spirals on adult 

 shoots is thus secondary, and is merely the expression of the 

 attainment of uniform volume by members in spiral series ; it 

 has nothing to do with the facts of actual development, during 

 which lateral members arise as similar protuberances, which 

 may be indefinitely produced without the possibility of the 

 system being closed by a terminal member. 



In other words, the genetic spiral must be regarded mathe- 

 matically as winding to infinity, and being engaged in the 

 production of similar members. That is to say, the possibility 

 is at once suggested that the genetic spiral can only be repre- 

 sented by a logarithmic or equiangular spiral which makes 

 equal angles with all radii vectores. 



Not only is this a mathematical fact there is no gainsaying, 

 but the introduction of log. spirals into the subject of Phyllo- 

 taxis at once opens up wide fields for speculation, in that 

 these spirals are thoroughly familiar to the mathematician 

 and physicist ; representing the laws of mathematical asym- 

 metrical growth around a point, they constitute in Hydro- 

 dynamics the curves of spiral-vortex movement, while their 

 application to Magnetism was fully investigated by Clerk 

 Maxwell. The possibility that the contact parastichies may 

 be also not only log. spirals but log. spirals which intersect 

 orthogonally, and thus plot out a field of distribution of energy 

 along orthogonally intersecting paths of equal action, is so 

 clearly suggested that it may at once be taken as the ground- 

 work of a theory of phyllotaxis more in accordance with 

 modern lines of thought (cf. Tait, ' Least and Varying 

 Action,' article Mechanics, Enc. Brit., vol. 15, p. 723). 



A geometrical construction in terms of such spirals in the 

 ratio (8 : 13) (Fig. 3) may be taken as a representative system 

 corresponding to the preceding phyllotaxis-plan of Fig. a. 



It is difficult to avoid the conclusion that the log. spiral 



