ANOMALOUS SERIES. 203 



still greater approximation to adult symmetry than that afforded by 

 the Fibonacci series, 



Similarly a still closer approach to symmetry may be indicated 

 by the assumption of such ratios as (6 + 7), (7 + 8), (8 + 9), (9 + 10), 

 etc., and these are to be observed more especially among the 

 Cactaceae, in which any biological effect implied in decreasing the 

 leaf surface exposed to light is nil. (Cf. special section.) 



IV. Once it is granted that a new row of members, implying the 

 opening up of a new spiral path, may be initiated at any point on 

 any expanding axis, or again dropped out on a decreasing one, 

 without necessarily implying the corresponding change all round 

 the system, it is obvious that a vast number of anomalous systems 

 may be secondarily produced, as in the case of Bipsacus taken by 

 the Bravais. Among the variety of ratios thus obtained, some, as 

 soon as they happen to be divisible by a common factor, would be 

 classed as multi jugate; so that it now becomes clear that the 

 multijugate condition is only a special case of an anomalous con- 

 struction, and often no doubt produced by the same causes. 



While, however, the multijugate primary condition has been re- 

 garded as a break in the direction of symmetry consequent on the 

 loss of the Fibonacci series, it does not follow that such will always 

 explain anomalous secondary systems. The very fact that new curves 

 may be added singly, without compensation, throughout the rest of 

 the system, shows that the sense of symmetry has deteriorated. 



In dealing with any given case, therefore, it becomes of interest 

 to see what alteration is made at any given change of system. 



Does the change, that is to say, make for symmetry, or the 

 reverse ? 



In other words, is a long curve added or a short ? Similarly ia 

 reduction, the loss of a short curve makes for symmetry, as expressed 

 by equality in the ratios ; the loss of a long curve renders the con- 

 struction more asymmetrical. 



Eemarkable examples are afforded among the Cactaceae, in 

 which any alteration in the phyllotaxis system is rendered obvious 

 by a corresponding addition or loss of a vertical ridge. The change 

 will often be observed to make for asymmetry ; the following ex- 

 amples suffice : — 



