20 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS. 



(fig. 6) shows eight right- and thirteen left-handed spirals. (Fig. 5 

 vice versa) Eeversing this expression for the position seen when 

 looking down on the apex of the cone instead of up from below as 

 in the figure, in order to express the cone in terms of a floral 

 diagram, the constant becomes (8 L+IS E), this being in fact the 

 actual terminology proposed by the brothers Bravais. The direc- 

 tion of the spirals, however, may vary from cone to cone, and the 

 more essential point is that the lower number of the expression 

 gives the number of the longer curves, so that the formula (8 long 

 -f 13 short spirals) remains the only cone-constant about which 

 there is no mathematical doubt. Braun's method marks a real 

 advance in that it replaces the long genetic spiral, quite impossible 

 of observation in high divergences, by short intersecting curves 

 which may be readily and accurately counted in the highest series. 



The closer the members, the more difficult the question of 

 orthostichies becomes, the method of transferring a system of 

 parastichies to fractional terms expressing an accurate angular 

 divergence becomes conventional, and one is unavoidably brought 

 into agreement with Bravais, that in such a case as Pinus, the 

 parastichies are all-important and the orthostichies imaginary. 



The tapering of the Pine cone, however, militates against its being 

 taken as a type of a cylindrical system, and the method, as appli- 

 cable to the whole series of divergences, requires to be tested on 

 cylindrical axes. 



3. PHYLLOTAXIS OF EUPHORBIA WULFENII, Hoppe. 



Ewphorbia Wulfenii, a handsome Spurge growing 5-6 ft. high, as 

 cultivated in the Oxford Botanic Garden, affords excellent cylin- ' 

 drical stems on which, owing to delayed. formation of cork, the leaf- 

 scars remain and are well marked. 



The flowering-shoots, bearing evergreen leaves, persist for 3-4 

 years, terminating in a compound inflorescence, without, however, 

 producing a terminal flower. 



Beyond the two cotyledons of the seedling, or the two prophylls 

 of a lateral shoot, the phyllotaxis rises to a medium elevation, 

 indicated roughly by ^, and produces vegetative leaves on a 



