110 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS. 



stichy ratio of the disk-florets. {Of. fig. 54a, showing the anomalous 

 ratio (11 + 18), the zone of transition being marked by the line of 

 ray-florets, at which the parastichy curves are uniformly increased 

 to form those of the disk-florets (fig. 546). 



With these data it is now possible to construct the diagram for 

 any capitulum of Helianthus, and to exhibit the relations between 

 the disk-florets and the sterile involucral scales, as well as the 

 position and number of the ray-florets, for any particular phyllo- 

 taxis ratio. The compensatory allowances for the inevitable dis- 

 tortion cannot be completely shown, since these are necessarily 

 corrected as the new members are formed, one at a time, and the 

 curve-construction must therefore exhibit the asymmetry of either 

 the former or latter condition. Since secondary changes are very 

 marked in the former system, which produces dorsiventral members 

 only, the curve-system may be constructed from the ratios pre- 

 sented by the parastichies of the disk. 



The diagram (fig. 44) is thus drawn for a small capitulum in 

 which the disk-florets have the phyllotaxis (214-34), and the inner 

 sterile involucral scales (8 -Hi 3). 



By selecting the proper (8-1-13) curves from the (21-f-34) 

 system, it will be seen that the amount of malformation in the 

 diagram is not large. 



Since a complete cycle of (8-1-13) contains 21 members, the 

 transition to (21-1-34) will be most economically effected, as already 

 indicated, by an approach to quadrant division in each area ; while, 

 to effect the transition, it is necessary to add 13 long curves and 21 

 short ones. If then one long one and one short are added in each 

 of the first 13 members of the cycle, the last 8 will only require to 

 add one short one each to complete the requisite number. 



This is done in the diagram (fig. 44), and the whole construction 

 is closely comparable with the segmentation of a layer of cell- 

 tissue into new lines of cells, as seen in the familiar examples of 

 the cell layer of Melohesia or Goleochaete. The diagrams here work 

 centripetally instead of centrifugally, but the method of segmenta- 

 tion is identical. Each area is thus seen to be subdivided by two 

 new lines, constituting the new paths in either direction, and these 

 are directly homologous with the characteristic T-shaped waU of 



