CONSTANT PHYLLOTAXIS. 95 



when mature plants are alone considered, the fact that the total 

 number of foliage leaves and sterile involueral scales borne on a 

 stem is with few exceptions less than one complete cycle of floral 

 members,* as indicated by the parastichies of the disk, shows how 

 difficult such an idealistic conception may be to disprove. 



On the other hand, seedlings of Helianthus,-]- well supplied with 

 reserve food materials, evidently lay down their primary curve 

 system under very similar conditions, and a section of the terminal 

 bud of a young plant shows a low ratio of parastichy contact-curves, 

 while definite contact curves of higher ratios, but yet lower than 

 those of the disk, are seen on cutting across the involueral scales of 

 the capitulum. 



It remains, therefore, to consider the preceding curve-systems 

 from the standpoint of — 



(1) Transition from one series to another. 



(2) The phenomenon already indicated as possibly representing 

 a " rising phyllotaxis " characteristic of the involucre of Composites 

 (ffelianthus, fig. 14 ; Oynara, fig. 9). 



To take an example of the former, the difference between a 

 (34-1-55) construction and that of a (21 -|- 34) might be expressed 

 by retaining the 34 long curves and reducing the short curves to 

 21 — that is, by dropping out 34 of the latter parastichy lines of equal 

 gi'owth. If this be attempted on a diagram, the parastichy curves 

 and the ontogenetic spiral still follow the same direction, but the 

 orthogonal areas are obviously no longer " squares," and would not 

 therefore contain the homologues of circles — that is to say, in the 

 plant the circular primordia would be distorted to broad ellipses. 

 The (21 -|- 34) system can thus only be represented by 21 long curves 

 and 34 short. By dropping out 11 long and 21 short from the 



* Cf. Weisse, Tables of Helianthus, Prings. Jahrb., xxx. p. 474. 



t Seedlings of Helianthus annuus produce, beyond the cotyledons, usually 

 3-4 pairs of decussating foliage leaves. It is not clear why such a symmetrical 

 construction should be found in the Sunflower and subsequently converted into 

 a spiral system ; in allied species, however, the decussate system is continued 

 throughout the whole of the assimilating region, but ceases at the branching of 

 the inflorescence region (cf. H. strumosus and JS. rigidus, garden varieties). 

 When the secondary change takes place the curves (2H-2) become typically 

 (2-1-3) or (3 -J- 5), rarely (3-1-4) or (2-1-4). 



G 



