MATHEMATICAL NOTES ON LOG. SPIRAL SYSTEMS. 343 



of which standpoints have been so frequently built up by a use of 

 the post hoc ergo propter hoc line of argument. 



II. It will be seen to follow from the remarkable property of 

 the Fibonacci ratios — that the ratios of any successive pair are 

 almost constant, and that 3 : 5 : 8 : 13 : 21, etc., with a considerable 

 degree of accuracy, so far as integers are alone concerned, as again 

 is the case in the curve-systems of phyllotaxis — that in the case of 

 all expansion systems derived from an initial pair with a view to 

 lessen the relative size of the lateral appendage, these numbers 

 alone give a minimum loss of regularity at every step in the change ; 

 while with any other series, such as 3:4:7:11, etc., the transition 

 would involve a large step and a small step alternately. In other 

 words, any aim on the part of the plant at uniformity of con- 

 struction in a system which is liable to change by the addition or 

 loss of paths, as in cases of very active growth {cf. Helianthus) in 

 which new curves are continually being added to reduce the 

 relative size of the lateral member as the growing-point gains in 

 bulk, can only be satisfied in one of two ways. Either the plant 

 acquires true symmetry and maintains it by adding curves in 

 either direction simultaneously (cf. Uquisetum), or that asym- 

 metrical system must be adopted in which the expansion 

 transitions can be effected with the least loss of regular construction. 

 The system which fulfils these demands is the Fibonacci series ; 

 and from merely numerical reasons there appears to be a balance 

 in favour of the chance of the initiation of curves in these ratios 

 to begin with. So that, granted the asymmetrical condition of 

 phyllotaxis is the primitive one, the general occurrence of curve- 

 ratios in the Fibonacci series would be mathematically expected 

 to occur. The choice of the plant for optimum phyllotaxis relations, 

 in fact, lies between true syvimetry and the Fibonacci type of 

 asymmetry; hence when true symmetry obtains the special 

 numbers of the latter sequence are no longer to be noticed as more 

 usual than others, and all other systems become rightly classed as 

 anomalous, in that they deviate from the two optimum conditions. 

 One thus becomes mathematically justified in regarding anomalous 

 variations, including the peculiar bijugate constructions, as ex- 

 pressive of a state of degeneration in the mechanism of shoot 



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