156 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS. 



Tt has been previously pointed out that the concentrated and 

 non-concentrated symmelrical conditions are only the limiting cases 

 of spiral constructions which vary in the degree of concentration, 

 all being concentrated to a certain extent in relation to the case of 

 superposed whorls ; the most concentrated asymmetrical system 

 being that in which the number of intersecting parastichies most 

 nearly approximates equality ; the least concentrated, that in which 

 they differ most widely. 



It is .thus clear that the least concentrated types must have one 

 of the members of the ratio unity, and the lowest members of the 

 normal phyllotaxis series (1 + 1), (1 + 2) may be therefore isolated 

 as representatives of such systems. In this construction other 

 contact parastichies are necessarily wholly absent {cf. Scheme B, 

 fig. 20) ; the one long curve iecomes the ontogenetic spiral, and the log. 

 spiral shorter curves become vertical spiral rows which may he con- 

 veniently described as " spires'' 



Thus the two-spired type occurs in Gasteria (figs. 57b, o8a), 

 and the three-spired type in Cyperus, Pandanus, Apicra spiralis 

 (fig. 59a, 6). Such two-spired plants occur in species of Gasteria 

 mingled, on the one hand, with specimens exhibiting normal ratio- 

 series (3 + 5) or (2 + 3), Gasteria ensifolia, G. candicans ; and, on the 

 other hand, with the special case of symmetrical (1 + 1) construction, 

 G. oUusifolia (fig. 57a). 



So closely are these connected that seedlings vary in the same 

 batch (fig. 58Z)). As the succulent dorsiventral leaves spread out, 

 the two spires become very pronounced; but any assumption of 

 torsion in one plant more than another, or, in fact, in any such 



obviously put out of court by tbe fact that the pbyllotaxis spiral is often anti- 

 dromous, and normal Fibonacci phyllotaxis phenomena may be found associated 

 with a two-sided apical cell. (Of. Schwendener, Botanische Miitheilungen, vol. i. 

 p. 156.) 



Nor was there ever any evidence in support of the older view beyond the 

 standpoint of the dominance of so special a mode of ceU-construction. On the 

 other hand, comparison of Equisetum and Aspidium show that whatever the 

 " growth-centre " may be, or whatever its nature, it is not localised in the nucleus 

 of the apical cell, but must be either a finite mass larger in these cases than a 

 single cell, or else represents a general function of the whole protoplasmic sub- 

 stance of the apex comparable with the somewhat allied conception of Polarity. 



