190 RELATION OP PHYLLOTAXIS TO MECHANICAL LAWS. 



ing of the expansion system of the involucral region (fig. 

 74,1). 



The terminal capitulum of Cephalarim, tartarica may be taken 

 as typical of a definite (10 + 16) system. The vegetative shoot is 

 normally (2 + 4), as shown by the paired leaves, and the terminal 

 head presents the six-parted pattern (fig. 75) characteristic of 

 Silphiv/m, etc., and is much flatter than that of Dvpsacus. A 

 section of such a capitulum may be taken in the bud condition, 

 8 mm. in diameter, just at the level of the insertion of the 

 last formed bracts, to include every leaf on the head, owing to 

 the close imbrication of the well-developed peripheral members. 

 The subtending bracts are markedly dorsiventral, and the slight 

 amount of sliding growth has operated normally, with the result 

 that the longer paths become more pronounced; and where the 

 florets are 'cut, the curves, as in -Sg^ww^Aws, approach the ortho- 

 gonal construction more obviously owing to the similar character 

 of the more or less circular florets. Such a section (fig. 75) affords 

 a beautiful example of rising and falling phyllotaxis, and this 

 particular capitulum shows a descending system with the accuracy 

 of the diagram of Dvpsacus pilosus, the terminal members being 

 two sterile scales orientated in the same plane as the first invol- 

 ucral pair. 



On such a diagram every leaf may be numbered by taking an 

 approximate oscillation-angle of 137° from 1 and 1', whichever 

 end of the system be taken as a starting-point ; the figure is thus 

 numbered from the outer involucral scales 1 and 1' to 137 and 

 137', the capitulum thus including 136 members. Owing to the 

 marked dorsiventrality of the members and slight sliding-growth 

 across some of the curve-paths, it is not possible to accurately 

 follow the interposition of new paths, according to the convention 

 adopted in the previous cases of ffelianthus and Dipsacus. The 

 system, however, commences as (2 + 4), and 3 and 3' do not make 

 complete contact, but open up room for 5 and 5' : thus according to 

 the convention, each 5 may be said to add a new curve to the 

 system. That the maximum attained is really (10 + 16), as shown 

 in the section, appears to be fully warranted by the comparison 

 of other sections, although it is true that only the tips of the 



