ASYMMETRICAL LEAST-CONCENTRATED TYPE. 155 



In Filicineae and Equiaeta, however, considerable departures have 

 been made from the type so far as the origin of the lateral foliar 

 members is concerned. Thus, while in Eguhetum circular symmetry 

 is apparent almost immediately behind the apex, and the number 

 of members in a whorl of leaves is by no means necessarily a number 

 of which three is a factor, nor bears any relation to the series of 

 Fibonacci ; on the other hand, in Ferns, a specialised concentrated 

 system may be in full operation, and thus Aspidium Filix-Mas, with 

 a three-sided apical cell, produces foliar members and a correlated 

 stelar meshwork in the system (5-f 8), (34-5), or (2 + 3) (fig. 35).* 



independent of ita histological composition. The limitation, for the present, of 

 the term "leaf" to such a massive protuherance avoids the difficulty of dis- 

 tinguishing between leaves, branches, or mere hairs in Algal forms. Special 

 interest attaches to the three-sided cell of the Musdneae, since this directly cuts 

 off the segments which become lateral members, and in that in the majority of 

 forms, the arrangement of leaves becomes discussed in terms of the Fibonacci 

 series. Thus Fontmalis antipyretica, with a tetrahedral oeU like that of Equisetum, 

 gives keeled leaves in three well-marked spires, which straighten out on elongated 

 shoots, but on short thick ones compare with the spires of Pandanus (Goebel, 

 Leitgeb.). In other cases (Polytrichum formosum type) the apical cell divides by 

 oblique septa in a constant manner, giving three series of oblique segments, the 

 three spires being so much exaggerated that " orthostichies " may be expressed, 

 in high ratios, of both Fibonacci and anomalous series (Hofmeister, Miiller, 

 Lorentz, Goebel). Torsion is admittedly absent (Goebel), and the leaf -traces in 

 the stem follow the same coiled three-spired series. It thus becomes a question 

 as to whether this oblique segmentation is really the cause or a consequence of 

 the formation of new growth-centres in a definite manner within the substance 

 of the apical cell, and that the whole mechanism of asymmetrical growth, which 

 in more massive plants produces a Fibonacci system of cell-aggregates, is not here 

 enchained by the necessities of cell-segmentation, so that the new lateral growth- 

 centres are never sufficiently free to assume the homologue of a spherical form, 

 correlated with a centric distribution of growth-energy ; and the exigences of 

 histological division may thus effectually mask the true asymmetrical construc- 

 tion. It may be noted that the oblique leaf segments ultimately produce very 

 fairly symmetrical leaf -forms, and that the space-form of the adult shoot com- 

 pares very favourably with that of ordinary leafy stems. (Gf. Groebel in SchenKs 

 Hamdhuch, vol. ii. p. 373 ; MuUer, Prings. Jahrb., vol. v. p. 247 ; Engler and 

 Prantl, Nat. Pflanz. Farm., Musci, p. 178). 



* De Bary, Gomparative Anatomy, p. 285. 



Hofmeister attempted to derive the j^j phyUotaxis of Aspidium Filix-mas from 

 the segmentation of the three-sided apical cell, although the segment waUa were 

 clearly parallel to the sides. Hofmeister's view that the genetic spiral was 

 necessarily homodromous with the cell-spiral and each segment gave a leaf, is 



