262 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS. 



All these hexagonal structures are, however, secondary pro- 

 ductions in the phyllotaxis system ; another example of primary 

 hexagonal leaf-facetting is afforded by Euphorbia, mamillaris 

 (fig. 92). 



This plant is of further interest in that, as in analogous ridged Cactoid 

 forms, the phyllotaxis is anomalous : e.g., the specimen figured was 

 7-ridged = (3 + 4) system at the level of the soil, but after the pro- 

 duction of about 300 members, two new curves are added quite 

 irregularly (fig. 926), and the system becomes (4-1-5) or 9 -ridged, this 

 formation being continued to the apex for over 500 members. The 

 new curves are interpolated without rule, and the lateral branches 

 commence as (2-1-3), which again soon rises to (3-1-4). 



The case of the Pinus cone (fig. 7) is of interest, again, in that 

 the rhomboid scales are not leaves but secondary structures obeying 

 the same laws of uniform growth as their reduced parent members ; 

 but these cone-scales are not elongated radially like the Helianthus 

 ovaries, but tangentially. The converse phenomenon will there- 

 fore be observed in the phyllotaxis pattern; that is to say, the 

 long curves will now remain clear, while the short curves will 

 exhibit stepping. The readjustment of a tangentially elongated 

 member in the square meshwork of the growth-system thus tends 

 to change the shape but not the position of the member. Such 

 tangential extension, again, becomes the normal condition in all 

 " dorsiventral " foliage-leaves, and will be considered again from 

 another standpoint. The essential point at present is to note that 

 such readjustments, and sliding-growth effects, do not imply any 

 displacements of the growth-centres. The primary phyllotaxis 

 relations are unaffected, and any secondary appearances which 

 may be involved in the pattern follow geometrically from the 

 properties of intersecting spiral curves. All contact-pressures 

 must be growth-pressures, and must be studied, therefore, from 

 the standpoint of growing systems. 



Schwendener's Dachstuhl theory, as a working hypothesis, thus 

 disappears as completely as the original one of Schimper and 

 Braun it was designed to correct. Apart from the mathematical 

 conception of helices inherited from these botanists, it was founded 

 on two perfectly definite facts of general observation : — 



