IDEAL ANGLES. 69 



possible of observation, being contained by spiral curves, but also 

 of representation on a diagram when the curve equations are not 

 given. Similarly, the " orthostichies " cannot be represented on the 

 diagram until the form of the log. spiral is known. It has further 

 been shown that each of the determining ratios of the Schimper 

 series comprises two log. spirals which have, as a rule, no simple 

 relation to each other, so that neither can be drawn while each 

 is imperfectly defined. 



The system can only be accurately planned by the parastichy 

 ratios, which, on the other hand, are much more readily observed 

 than an ambiguous orthostichy ; while, in addition, the fact that 

 the curves used form a mutually intersecting orthogonal pair admits 

 of a simple method of geometrical construction. 



The method of presentation by means of angles of divergence 

 and " orthostichies " must therefore be placed wholly on one side, 

 and it is, at the same time, clear that all observations on phyllo- 

 taxis constants, in which this method has alone been used to 

 determine them, are open to considerable error. 



The parastichy ratios will therefore be alone used to define any 

 given system, and the normal system thus becomes: — 

 Ps. = (l + 1), (1 + 2), (2 + 3), (3 + 5), (5 + 8), (8 + 13), (13 + 21), etc. 



By tabulating these as simple ratios, the idea of angular 



divergence is eliminated and a further fact is brought into 



prominence : — 



1 • 1 • ■ 1 • 1 



z 



and the limiting ratio 



2 : 75-1 : : 1 : 1'618 



The simplest summation series thus implies a practical constancy 

 of parastichy ratios in its higher terms, while the axis and the 

 lateral primordia may be variable quantities. 



Expressing this practically, in terms of the spiral-vortex 



