196 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS. 



VIII. Anomalous Series. 



Under this heading may be included all ratios not divisible by 

 a common factor which are not included in the Fibonacci series. 

 The formation of imitation summation series has been previously 

 described, as for example: — 



3, 4, 7, 11, 18, 29, 47; 



4, 5, 9, 14, 23; 



5, 6, 11, 17, 28, etc. 



And it has been pointed out that such series differ from the 

 Fibonacci series in that the ratios of successive terms are neither 

 approximately constant, nor do they always approach 1 : 1-62, 

 although this ratio is approached as the series proceed. 



It has further been shown that the number of parastichy curves 

 is usually low, and it follows that among low numbers almost any 

 ratio must be capable of expression in one series or the other. 

 For example, in such a series as — 



6: 6\ 



one system would be symmetrical, 

 two bi jugate, one trijugate, and 

 one anomalous ; 



and the close relation of such forms as variation types, is seen 

 among Cactaceae. {Cf. special section.) 



But it does not follow that all the ratios of such hypothetical 

 series actually exist in plant structures. 



For example, (3 + 4) is found not uncommonly (Sedum, Ewphorbia, 

 Oereus), and (7 + 11) also occurs (Araucaria), but (4+7) is very 



