ASYMMETRY. 11 



circles and radii. When asymmetrical, as in the more general 

 theorem, the structure is illustrated by the phenomena of a spiral 

 vortex, and all the lines of construction are orthogonally inter- 

 secting logarithmic spirals. 



In all eases the symmetrical must be regarded as due to second- 

 ary specialization of the asymmetrical case, just as the circle may 

 be regarded mathematically as a limiting case of an infinite log. 

 spiral curve. 



In both systems, "concentrated" and "non-concentrated" con- 

 structions may be possible: consideration of the schemes (B) and 

 (D) shows immediately that all spiral types are more or less con- 

 centrated constructions, and that (A) and (C) are only the limiting 

 cases in either direction. The same fact is illustrated by the 

 numerical relations of the curves contained in the capitulum of 

 Helianthus taken as a type (fig. 15), in which (34-1-55) give the 

 optimum concentration (1-|-1597), the minimum. The optimum 

 concentration, produced in an asymmetrical system by the approxi- 

 mation of the number of intersecting curves in either direction to 

 equality, being thus a secondary effect of an approach to symmetry, 

 actual equality gives the perfect symmetrical condition of scheme 

 (0). The fact that this is the normal symmetrical case found in 

 plants, while it is also regarded phylogenetically as a secondary 

 specialization, is satisfactory evidence that the " concentration " system 

 is after all not due to any hypothetical biological demands of hud- 

 construction, hut the natural outcome of its evolution from a spiral 

 series in which the claims of symmetry are expressed hy an approach 

 to equality in the numher of parastichies as indicating orthogonal 

 lines of equal action. 



The biological demand for a concentration construction is thus as 

 completely eliminated from the study of phyllotaxis as Bonnet's 

 original biological demand for equal transpiration space has already 

 been seen to be unnecessary. 



The terms " symmetrical " and " asymmetrical " are further prefer- 

 able to the older corresponding terms " whorled " and " spiral," in 

 that it will appear, as Sachs suggested, all spiral appearances are 

 subjective, and not the representation of any spiral aim on the part 

 of the plant; while the term "whorled" can only paradoxically 



