124 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS. 



may be continued (fig. 48c, five pairs), and in other cases the (3 + 5) 

 system may be produced directly beyond the decussating members 

 (fig. 48e). This last example is of interest in that the (3 + 5) 

 system immediately commences a further rise, and examination of 

 the method shows that the leaf nxmibered 8 is the first member 

 definitely concerned, in that it does not fall accurately between 3 

 and 5, in the manner that 7 fell between 4 and 2, but beyond its 

 normal station, so that 13 afterwards falls behind it, in the manner 

 tabulated in fig. 44 as the type expansion. 



Another example (fig. 49) shows a similar transition only taking 

 place after a formation of 13 members of the asymmetrical system 

 (Nos. 14 and 19 falling in the gap between 9 and 11). The fact 

 that older leaves in Helianthus are spaced out owing to a great 

 development of hairs, militates against accurate observation of the 

 contact lines in sections of large area, but by remembering that 

 each leaf fills its own rhomb, and that the development of hairs 

 thus compensates the diminution in the rate of growth of the leaf 

 itself, the contact-lines of the hairs of adjacent leaves may be taken 

 as a very approximate representation of the theoretical members. 



Since the sudden change from a decussate (2 + 2) to an 

 asymmetrical (3 + 5), implying the sudden intercalation of four new 

 curves, is much less easy to understand than the change to (2 + 3) 

 which only adds one, a capitulum of (21 + 34), similar to the one 

 for which a diagram has already been constructed, may be selected, 

 in that it should present a double expansion of the type (3 + 5), 

 (8 + 13), (21 + 34). 



Such a capitulum in the bud condition, the whole 6 mm. in 

 diameter, is sectioned in fig. 50; every member being accounted 

 for beyond the uppermost pair of decussating leaves. 



As a (21 + 34) capitulum should present 21 ray-florets, 13 

 involucral scales, and an average of 25 foliage leaves, including the 

 primary decussating pairs (Weisse), the bud in question is evidently 

 well within normal range in that it shows a total of 27 vegetative 

 members between the uppermost pair and the first ray-floret. By 

 adding 8 for an average number of four pairs, the total number of 

 leaves should have been 35, which is again sufficiently close to the 

 theoretical number 25 + 13 = 38; the specimen is thus a normal 



