484 Church . — Note on Phyllo taxis. 
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 plant. 
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', 2p'^ 36. 
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 
