74 KELATION OF PHYLLOTAXIS TO MECHANICAL LAWS. 



The exceptional occurrence of such parastichy ratios has the 

 greatest interest when taken into account with the possible 

 existence of intermediate conditions between the case of normal 

 asymmetry and true symmetry, since the fact that such may be 

 found might throw light on the causes which tend to induce sym- 

 metry; but, clearly, nothing is to be gained by formulating 

 hypothetical fractional series to contain them. 



Symmetrical construction is quite definite and stands by itself ; 

 the normal asyrometrical series, again, selects the optimum 

 ratios for the construction of an asymmetrical system to the best 

 advantage within certain restrictions : other ratios may requisition 

 a more symmetrical set of curves, but at the expense of an inferior 

 working angle (c/. fig. 63, (6 + 7). 



Examiuation of the higher members of such series which necessarily 

 approach the ratios 1 : (1-62) shows that they may be regarded as 

 composite systems in which two or more ratios of the normal series 

 are compounded : e.g. 



29 



47={^^:{g} n:28={ii}:{i*} 



23:S7={^^}:{^} 



l«^31={il}:{i^}e.c. 



It is true that such manipulation of ratios is open to the objec- 

 tion that it is " playing with figures " ; but from such relations it 

 follows that these systems might be expected to occur in plants 

 which also exhibit bijugate constructions; and, in fact, most of 

 them have been described in Bipsacvs, the typical multijugate in- 

 florescence, while they occur as exceptions in Helianihus (fig. 54), 

 which again presents occasional bijugate capitula.* 



The idea that there might be only one normal phyllotaxis series, 

 from which aU others were derived by slight deviations, -occurred 

 first to the brothers Bravais ; and, when one recognises the possi- 

 bility of the addition of new parastichy lines, one at a time, or the 

 elision of one, as in the inflorescence of Dipsacus (figs. 38a, 6), just as 

 Cacti may add or lose ridges according to circumstances of nutrition, 

 it is clear that many of these so-called phyllotaxis constants must be 

 of local and secondary origin. Further discussion of such forms may 

 be left until multijugate types have been more fully considered. 

 * Gf. Weisse, Pringsheim's JaMyiicher, xxx. p. 474. 



