4 87 
Church . — - Note on Pkyllotaxis . 
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 then 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 
modem lines of thought (cf. Tait, 4 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 phyilotaxis-plan of Fig. 2. 
It is difficult to avoid the conclusion that the log. spiral 
