484 Church. — Note on Phyllotaxis. 



2. The existence of accurate orthostichies : these latter 

 following from the construction as being radii vectores of 

 a spiral of Archimedes, the spiral again being derived from 

 Bonnet's helix with parallel screw-thread. 



Since helices and spirals of Archimedes are also commonly 

 the result of torsion-action, the way becomes paved for the 

 addition of theories of lateral displacement or torsion-effects, 

 which are expected to produce secondary alterations in the 

 original simple system of Schimper and Braun. 



It becomes therefore necessary to test the basis of these 

 generalizations, and to examine the possibility of checking by 

 direct observation either the divergence-angle or the ortho- 

 stichies themselves ; and finally to compare the plane construc- 

 tions by spirals of Archimedes and see how far these really do 

 interpret the appearances seen in a transverse section of the 

 developing system in the plan,t. 



Such investigation shows that the hypotheses have no true 

 basis, while the construction by spirals of Archimedes is 

 a conspicuous failure. Thus, the divergence-angle is hope- 

 lessly beyond the error of actual observation on the plant, 

 since the points from which the angles have to be taken must 

 be judged by the eye ; when, therefore, the divergence-angles 

 are expected to be true to a matter of minutes and seconds in 

 fairly high divergences, this becomes a matter of impossibility ; 

 and the Bravais showed in 1835 that it was in fact impossible 

 to disprove the standpoint that there was only one angular 

 divergence in such cases of normal Fibonacci phyllotaxis, 

 namely Schimper's 'Ideal Angle' of 137°, 30', a7"-936. 

 Similarly, it is equally impossible to judge straight lines by 

 the eye alone, and the existence of orthostichies in spiral 

 phyllotaxis as mathematically straight lines thus becomes 

 as hypothetical as the Schimper-Braun divergence-angles. 

 In neither of the two methods used for the practical deter- 

 mination of phyllotaxis-constants is there then any possibility 

 of accurate mathematical demonstration. Although the 

 tabulation of appearances as judged by the eye may be 

 taken as an approximately accurate version of the real 



