The original pattern is to be seen most perfectly at the actual growing-apex ; 

 increased tangential extension of the members may give contact-series in lower 

 numbers of the summation-series ; just as diminished growth may imply that more 

 contact-lines are seen (cf. Pinus excelsa, cone (5 : 8) in the first year, with scales 

 broadening to (3 : 5) contact in the second season ; the latter case in cones of 

 Araucaria brasiliensis. In Abies nobilis the cone-scales, tangentially extended and 

 flattened, retain contact-parastichies of (5:8); but the reflexed bract-scale laminae 

 neatly fill rhomboidal areas of which the most obvious contact lines are (13 : 21)). 



A higher ratio is also seen in the case of ' open ' cones, as compared with the 

 original pattern of the ' closed ' green cone. On the adult cylindrical axis the system 

 may remain more or less obvious ; wiih enlarged scale-effects (Pinus Laricio, 5-10 

 years), or with the units more widely spaced (Araucaria imbricata for 50 years). 



Owing to the relatively small size of the photosynthetic members (?>ricrophylly), 

 except in Dammara and Ginkgo, as also the absence of secondary changes in size, 

 shape, dissection of the lamina, intercalation of petioles, &c., general in ' broad- 

 leaved ' Angiosperms, the original arrangement remains particularly clear throughout 

 the group, and is little interfered with ; hence such patterns attract attention, and one 

 asks why such constructions occur, and what they may mean. Fibonacci relations 

 represent the mathematical solution of the problem of building a centric shoot-system, 

 one member at a time, at successive angles of 137-2 (very approximately), as a con- 

 dition of balanced symmetry ; and any ratio of the Fibonacci series closely approxi- 

 mates this angle. The construction was not initiated specially for land-plants, though 

 it may have been modified and much improved in present mechanism ; but is again 

 inherited as an optimum method of older shoot-construction, probably from distant 

 marine organism since Fibonacci relations occur also in marine plants unless there 

 may be some reason for changing it. 



Variations on these numbers are relatively unimportant ; but the occurrence of 

 such ratios as (3 : 4), (7 : n), (10 : 16), which are found exceptionally at the apices of 

 shoots, or in cones, and are open to similar secondary changes along their cor- 

 responding summation-series, suffice to indicate that the mechanism of the curve- 

 pattern is now complex, and subject to deterioration, since no longer limited either 

 to any accurate spacing-angle, or to strict Fibonacci numbers. Such variants are 

 sufficiently uncommon to be regarded as anomalies in the construction-system, and 

 the retention of Fibonacci ratios is a characteristic feature of the groups Abietineae, 

 Taxodineae, Araucarineae, and Taxoids. 



In the Cupressineae alone whorled symmetry is attained as an alternative arrange- 

 ment, appearing expressed in terms of circles (whorls) instead of spirals, though still 

 following the lower numbers of the Fibonacci series (2, 3), and giving whorls in 

 alternating sequence (dimery, trimery). So strict is this feature within this group that 

 it serves to delimit an entire series of genera. Such reduction-specialization, clearly 

 secondary and derivative from Fibonacci-construction, may be regarded as a form of 

 xeromorphic adaptation ; since not only do such symmetrical constructions, by involving 

 increased superposition, imply a diminished light-utilization, and hence less chloro- 

 evaporation in the long run; but the lower members of the series (i, 2, 3) imply 

 a diminished output of leaves per node, which, other things being equal, again 

 expresses a diminished transpiration. In the special case of the decussate Cupressineae 

 (Thuya, Chamaecyparis, Libocedrus) the strict symmetry of the construction in planes 

 at right angles is utilized to extend the possibilities of shoot-construction preferably in 

 one plane, resulting in the production of characteristic ' phyllomorphs '. 



On the other hand the advantage of a symmetrical balance in any ratio of the 

 Fibonacci series is seen in the isolation of the members of a ' false-whorl ' of lateral 

 leaders from successive members of the spiral sequence, with optimum effect at the 

 Fibonnacci numbers themselves (3, 5, 8). 



II. The Branch-system may be based on the theory of the axillary-bud, which 

 suggests that every leaf-member may subtend a new shoot in its axil to continue the 

 ramification of the axis; and that, conversely, no ramification takes place in these 

 plants except by means of axillary buds. Dichotomy of the apex, and fasciation, 

 would be regarded as a freak, or the expression of failure in normal mechanism. 

 The origin of this mode of construction is still obscure ; i. e. it also undoubtedly 



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