RISING PHYLLOTAXIS. 127 



or the 13 members required to add the long curves of the system 

 now momentarily (21 + 13). At 43 (the internal member of the 

 22 fork), a short curve is added and 21 new short ones are put in 

 between 43 and 63, the system being completed at 64. This 

 latter change may be better checked on a construction diagram 

 similar to that of fig. 44, in which No. 1 is replaced by 30. The 

 correspondence is again exact, and the actual construction of the 

 diagram and the addition of the new curves one at a time, as 

 they are observed on the specimen, is required to fully comprehend 

 the symmetrical relations of the construction. The ray-florets, it 

 will be noted, should extend from 30-50 inclusive; a minor 

 variation is of interest in that in this specimen an extra ray-floret, 

 since 22 are actually present, is formed from a disk-member, while 

 in the previous capitulum of fig. 44 an extra one was produced by 

 an involucral member. Thus the floret in the axil of 53 forms 

 a ray, and the member flattens out to an involucral scale, while 

 51, 52, and 54 are normal disk-florets. 



The identity of the expansion mechanism, which is thus twice 

 repeated in one section, with the construction previously postulated 

 is very striking, and nothing is more remarkable than this rhythmic 

 interchange in the addition of the curves in the two directions. 



It will be noted that in the case of the production of (3-1-5) 

 from (2-f2) the transition was very rapid, and not conformable 

 to the rule for asymmetrical expansion unless the decussate con- 

 dition be regarded as a variation of (l-f-2), in which case expansion 

 should be completed in 5 members. In such case the next ex- 

 pansion commences immediately at 6, while a wide gap separated 

 (8-1-13) transition from the (21-1-34), 10 members being laid down 

 with constant phyllotaxis. The previous estimates for the phyllo- 

 taxis phenomena of Helianthus are thus subject to an error of a 

 variable number of members between the systems, and the involucre 

 does not necessarily include a definite transitional period. The 

 limitation of the involucre of Helianthus to a minimum number 

 of leaves has nothing to do with the expansion mechanism, but 

 represents that number of members which makes a complete cycle 

 of contact around the axis before the last transition commences 

 at the region of ray-florets. Helianthus may then be taken as a 



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