ASYMMETRICAL LEAST-CONCENTRATED TYPE. 161 



standpoint, it may be noted that the effect of progressive dorsi- 

 ventrality in a growing system will be to exaggerate the curvatures 

 of all the spiral paths. Thus the attainment of a degree of dorsi- 

 ventrality sufficient to make a member about twice as broad as 

 thick, as in the leaves of AUes, etc., will result in the fact that the 

 " orthostichy " lines or "spires" become as curved as the shorter 

 paths of the normal curve tracing, while these latter become as 

 markedly curved as the normal longer paths. With a still greater 

 degree of dorsiventrality the spires become still further pronounced, 

 so long, that is to say, as the system is either still growing, or else 

 has stopped altogether. 



The difficulty in the case of Gyperus and Pandanus is, however, 

 not to prove that the curvature of the so-called " orthostichies," 

 which is sufficiently clear in a section of the apex, may be due to 

 torsion* since in theoretical construction they should be curved 

 and not straight ; the question is why, with so great an assumption 

 of dorsiventrality, these lines are not much more curved? This 

 may be possibly very largely due to the special mode of folding the 

 strap-shaped leaves into one another ; as they grow they slip over 

 each other in such a way that they must form three rows in the 

 bud, and the assumption of a divergence angle of 120°-126'' 

 (Schwendener) may be thus quite secondary. For example, in 

 Gyperus (fig. 51), the last leaves being rudimentary do not fold, and 

 in a section cut apparently quite transversely the divergence angle 

 between 6 and 7 was 134°; beyond these members the angles 

 vary owing to change of system, while other irregularities are 

 observable in the last folded members. There is no real necessity 

 to postulate torsion, nor is there any ready method of proving it. 



So great is the alteration in such systems owing to the effects 

 of rapid retardation in the rate of growth behind the apex, that 

 the log. spiral construction, founded on theoretical uniform growth, 

 completely fails to represent the results attained in the plant. 

 One fact alone remains clear : in a construction in which growth 

 is rapidly slowing down, and the members acquiring approximately 

 equal radial depth, but still elongating tangentially, the appearance 



* For torsion theory cf. Schwendener, Botanische Mittheilungen, vol. i. p. 163, 



