RISING PHYLLOTAXIS. 121 



the disk being (55 + 89), inner involucre (21 + 34), outer involucre 

 (8 + 13), and the upper foliage leaves (3 + 5). 



Comparison of the Daisy (fig. 47) again shows foliage leaves 

 (2 + 3), involucre of 13 members (5 + 8), and florets (13 + 21), the 

 rays being not yet determinable. 



A large (89 + 144) Sunflower would therefore be built up by the 

 regular expansion (2 + 3), (5 + 8), (13 + 21) foliage leaves, (34 + 55) 

 involucre, and (89 + 144) florets of the disk, thus giving five com- 

 plete transitions along the axis. The difficulty of giving a definite 

 phyllotaxis constant for the leafy stem is thiis rendered obvious. 



It now becomes possible to give a connected account of the pbyUotaxis 

 pbenomena of Helianthus, based on this remarkable limitation of the 

 involucre to the minimum number of members which represents the 

 transitional period between two cycles, and the following scheme is borne 

 out by the data tabulated in the Variation Curves of Weisse (loc. cit., 

 p. 478). The weakest Sunflower axis resembles the Daisy in presenting 

 two changes only : i.e., beyond the 3-4 pairs of decussating (2 + 2) leaves of 

 the seedling, the system assumes normal asymmetry by laying down (2 + 3) 

 curves.. After producing a total average of 20-22 foliage leaves (average 

 of 20 plants =22), the system expands to (5 + 8); since 8 new curves 

 are to be added, 8 members represent the minimum number of leaves 

 before another transition can be initiated. Such an involucre would 

 therefore normally contain 8 leaves and the second transition would give 

 (13 + 21) florets system, of which the first 13 would form the ray-florets. 



If, for example, after forming 8 members of the transition to (5 + 8) the 

 vegetative condition was still vigorous, a further rise to (13 + 21) would 

 be effected in 21 leaves, the involucre would now be 21, the total munber 

 of leaves about 22 + 8 = 30, and the parastichies of the disk (34 + 55) 

 (Weisse, p. 478). Again, with the same proviso another change in the 

 vegetative region to (34 + 55) would be effected in 55 leaves, and if these 

 represent the involucre the next rise (89 + 144) would give the largest 

 ■capitulum. Such a plant should show (22 + 8 + 21) = 51 foliage leaves 

 and 55 involucral scales, total 106. It is of interest to note that one 

 such plant counted by Weisse gave 46 of the former and 62 of the latter, 

 total 108 ; when allowance is made for the ill-defined character of the 

 involucral region, the correspondence is remarkably close. 



That Helianthus presents a progressively rising phyUotaxis involving the 

 minimum number of members with a remarkable degree of accuracy 

 appears therefore fairly clear. Similarly (3 + 5) as a stronger system 

 would give other terms of the capitulum series, and expanding systems 

 derived from variation derivatives (2 + 4), (3 + 4), would lead on to 

 capituta of the series 16, 26,, 42, 68, and 29, 47, 76 respectively. These 

 variations are however comparatively rare, the former representing the 



