184 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS. 



greatest diameter a perfect (16 + 26) system. D.pilosus presents a more 

 primitive type than the preceding species, in that the stem is branched 

 freely to the third and fourth degree, and lateral branches continue the 

 structure of the main axis, also retaining the bijugate construction. The 

 capitula are smaller and contain relatively fewer flowers, the ultimate 

 heads, in fact, often producing so few that the parastichies are too ill- 

 defined to be counted: 



The plant produces a multitude of small capitula instead of specialising a few 

 large ones in the terminal region, and the type of construction is re- 

 markably constant. Thus the plant selected, producing branches to the 

 fourth degree from ten nodes, gave a total of 176 capitula sufficiently 

 well developed to be counted : the last small heads remain undeveloped 

 as the plant exhausts itself at the end of the summer. 



Of the 175 lateral capitula, 112 were acciu-ately (10 + 16) around the middle ; 

 30 were (6 + 10), the difference between these constructions being subject 

 to secondary error in counting adult structures, 8 were only one or two 

 curves out in either direction, and 25 were of the (8 + 13) type ; thus, in 

 aU, 80 per cent, were bijugate capitula, and about 15 per cent, reverted 

 to the normal Fibonacci ratio. 



The general phenomena of all multijugate systems can be readily 

 studied from their structural diagrams, and though in many cases 

 the systems are not necessarily constant for any considerable period, 

 it is only by expressing the construction geometrically that the sig- 

 nificance of a common factor to the ratio is made obvious. Thus 

 a (10 + 16) system, characteristic of the inflorescence of Dipsacus 

 pilosus and Cephalaria tartarica (fig. 68a), may be represented by 

 drawing the 10 and 16 log. spirals in the requisite ratio 5:8; and 

 since the mathematical fact that these curves plot the system is the 

 only definite statement that can be made with regard to it, it 

 follows that the system must be numbered by Braun's method, by 

 taking members as differing by 10 and 16 along their respective 

 paths : on so doing (fig. 70) it will be found that no interpretation 

 in terms of " genetic-spirals " is possible save that which admits the 

 presence of two equal and concurrent paths orientated at points 

 diametrically opposed. Taking one of these as No. 1, the 

 members are represented by odd numerals only, there are two 

 Nos. 3, for example, but no No. 2, and by taking a divergence angle 

 of 137° from 1, it will be found that each system has its own path 

 1, 3, 5, etc., and 1', 3', 5', etc., and these " genetic-spirals " work out in 

 a direction the reverse of that of a normal (5-)- 8) system. 



