Mathematical Notes on Log. Spiral Systems and 

 their Application to Phyllotaxis Phenomena. 



By 



E. H. HAYES, M.A., 



Fellow of New College, Oxford ; 



AND 



A. H. CHUE,CH. 



Any application of mathematical methods to such a subject as 

 that of Phyllotaxis must necessarily be limited by the hypotheses 

 taken as the basis of any conception of the relationship of the 

 phenomena observed, and clearly no further information can be 

 deduced than follows from the original premises. 



Thus, as already seen (p. 6), the mathematical conception of 

 a helix winding on a cylinder, which was assumed by Bonnet to 

 be a satisfactory interpretation of the facts observed on adult 

 shoots, — although it did not hold for younger ones, — forms the 

 basis of the Schimper-Braun formulae ; and the assumption of a 

 spiral with parallel screw-thread led to the adoption of spirals of 

 Archimedes when the phenomena were required to be represented 

 as a plane circular system. 



Similarly, the introduction of the Fibonacci series of ratios by 

 Schimper and Braun naturally brought with it all the mathematical 

 properties of this curious series of numbers, and other observed 

 ratios were readily fitted into similar summation series. But in 

 deducing all the mathematical properties of such combinations, 

 which necessarily follow from the presence of the numbers 

 themselves, it does not follow that the plant, in possibly selecting 



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