164 



(i -\- i), (i + 2),(2 + j), (j + 5), (5 + 8), etc. Here also the succes- 



j % l/"^" 



sive values approach gradually to a limit : / Y6i8 = ~~2 ' an( * 

 the ratios naturally adopted by the plant for its intersecting paras- 

 tichies are the successive terms of the continuous fraction: 



1 



1 +1 



1 +1 



1 + etc., 



In a great number of cases z is equal to unity in this fraction. 

 These values would for growing plants with a definite number 

 of leaves give the optimum approach to a symmetrical distri- 

 bution in such a spiral system. However it may appear doubtful 

 whether the mechanical or physiological causes of this leaf-distri- 

 bution are really better explained by this mode of reasoning than 

 by previous views 1 ). 



The true "pentamery" as observed in the flowers of many Dicotyle- 

 dons and in many lower animals (Chapter /// and/F), is a special case 

 of this ideal arrangement, and in truth the most highly perfected 

 condition of phyllotaxis x ), expressed by the special symbol (5 + 5). 



In this respect a certain tendency of living nature to the ratic 

 expressed by the "aurea sectio" may be stated again, - - a fact 

 already pointed to in 1611 by Kepler in some of his botanical 

 speculations. 



But it must be clear from the above that in the light of this theory 

 all supposed analogy with the arrangement of the molecules in 

 crystals, as suggested by Wulff, vanishes completely now. Churcl 

 expressly points out that wo Archimedian spirals ever play a role 

 in natural phyllotaxis 2 ), and therefore the development of sucl 

 a spiral in a plane does not give a system of points endowed witl 

 the peculiarities of a Bravais' net-plane. 



The result in this case will rather be a system of logarithmic 

 curves, to which no reasonings as brought to the fore by Wulff, 

 can be immediately applied. Only more complicated and elongate( 

 relations exist between these logarithmic spirals 3 ) and the helic 



G. van Iterson Jr. loco cit., p. 106, 108, and 144. (1907). 

 Cf. G. van Iterson Jr., loco cit., p. 1. (1907). 

 A. H. Church, loco cit. 



