320 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS. 



case of the first pressure of the primordia to fill their quasi- 

 square areas. Thus, in the specimen figured, the first primordia 

 along a spiral path subtended angles of 47"5°, 52°, 53° (the theor- 

 etical angle for a quasi-square of a (5-1-8) system being 51 '5°); 

 but nearer the periphery the series fall off to 47°, 40°, 40°, 38°, 

 35°, respectively. (2) Diminution of radial growth implying 

 progressive " dorsiventrality " is relatively slight, and the sliding 

 effect small, the preceding data suggesting 2° only ; and this is 

 due to the fact that the leaves tend to round off at the angles 

 instead of sliding over each other by sharp-edged laminae. (3) 

 Growth of the axis pulls the older members down out of the plane 

 of section ; this growth being much more rapid in the seedling of 

 Pinus Pinea than in a lateral shoot of Araucaria, the compensating 

 effect of a longer needle in the former is not noticed. 



Further modification of the primary phyllo taxis pattern ensues 

 with the advance of " dorsiventrality " in the members ; in cases 

 in which the leaf-member is markedly dorsiventral, but also 

 elongated, thus constituting a strap-shaped member, the optimum 

 effect will be observed in bud-section (c/. Podocarpus japonica, fig. 

 42). In the more general case, however, the '' dorsiventrality " of 

 the leaf-lamina is not regular throughout the whole extent of the 

 member, but it may present the following cases : — 



(1) Very great at the base (type of the sheathing leaf-base). 



(2) Small in the petiole, or absent. 



(3) Considerably exaggerated in the assimilating lamina. 



The distinction between lamina and midrib may be small or 

 wanting (Sempervivum, fig. 83) ; or, again, very marked (Campanula, 

 fig. 101 ; Verhascum, fig. 36) ; and in the limit, the former may be 

 complicated by compound segmentation. In such case, the seg- 

 ments of each leaf are restricted to its own rhomboid area, and 

 each leaf is packed or crumpled independently within its own 

 rhomboidal domain {Ranunculus repens, Nigella, PJieum). Similar 

 phenomena are observable on a smaller scale in the case of the 

 simple leaves of Verbascum and Campanula (fig. 101); in the 

 figure of the latter instances of anomalous sliding effects have also 

 been indicated. When the sliding of dorsiventral leaves is ex- 

 cessive, it is clear that new contact-relations will be established in 



