206 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS. 



retained, and new paths are added symmetrically (as in Squisetum) 

 by the bifurcation of old ones, although the change of bulk-ratio 

 which requires to be negotiated in adding two new curves is very 

 considerable. 



To sum up, anomalous ratios are rare in normal plants, but are 

 especially characteristic of specialised inflorescences and xerophytic 

 assimilating shoots of such plants as Sedum, Ev/phorhia, Lycopodium, 

 Cactaceae. 



So long as they are primary constructions, they imply a reduction 

 of the Fibonacci sense ; but with the loss of the Fibonacci ratios, 

 there is correlated a general independent attempt at symmetry 

 as expressed by equality of the ratios, with the general result 

 that these are less than (1 : 1-62) ; while in extreme cases the 

 approximation is so close that the anomalous system may often 

 be regarded as the exception. 



They represent modifications of the faormal phenomena of 

 phyllotaxis, and occur as local, individual, or even specific variations. 

 Taken in connection with multijugate systems, they may be re- 

 garded as a second case of a break towards adult symmetry, as 

 opposed to a symmetrical building mechanism. The loss of the 

 Fibonacci series is more complete, and the capacity for independent 

 approximation to actual equality in the ratio is correspondingly 

 increased. 



Special interest attaches to the case of Lycopodium Selago in that 

 here the " Fibonacci sense " appears to be entirely lost, and the- 

 approximation to a construction which involves a nearer aproach 

 to adult symmetry is so close that strictly symmetrical examples 

 are as general as the asymmetrical approximations. Viewed from 

 the standpoint of a plan of building, it is clear that the hypothesis 

 of an oscillation-angle can no longer explain the mechanism {cf. fig. 

 78), since the system is built on a distinct spiral path ; and on the 

 other hand, the view that the " genetic-spiral " is the determining 

 factor, while it gives an interpretation of the asymmetrical cases, 

 only exaggerates the gap which has been held to exist between 

 asymmetrical and symmetrical constructions. That such con- 

 structions may be really separated by only very trivial distinctions 

 appears to be shown by the occurrence of cases like that of the twin 



