372 ON THE THEORY OF VEGETABLE SPIRALS. 



respective element of all three fulfilling Goethe's law of alternate 

 contraction and expansion. 



The ordinary form of the edge of a dicotyledonous leaf, sup- 

 posing the jags filled up, is sufficiently well represented by the 

 projection on a vertical plane of an equiangular spiral, chased 

 on an inverted cone (I mean of course the outline of the half 

 leaf, commencing at the base and going thence to the apex). 

 The curve in question cuts the axis at a greater angle at the 

 base than at the apex, and has its maximum ordiriate nearer 

 to the former than to the latter. Something may be said on 

 theoretical grounds in favour of taking this as the typical leaf 

 curve in cases in which the petiole and lamina lie in the same 

 plane, since when they are at right angles, or nearly so, there 

 is an approximation towards a circular form, as if the plane 

 of projection had now become parallel to the base of the cone, 

 and the slope of the spiral were diminished. In truth this 

 principle of projection might be applied to explain many varia- 

 tions of form in the flower as well as in the leaf. 



7. The preceding remarks can scarcely be free from mis- 

 takes, and yet they may contain a portion of important truth, 

 namely, the principle of tracing numerical relations in the un- 

 symmetrical as well as in the symmetrical aspects of vegetable 

 growth. Just as spirals are less symmetrical than whorls, and 

 yet enable us to understand the latter better than we otherwise 

 could have done, so likewise unsymmetrical portions of spirals 

 and of whorls may throw new light upon both *. 



1 For an account of recent speculations on the theory of vegetable spirals, see 

 Braun, Betrachungen uber die Ersckdnung der Verjiingung in der Natur, p. 125. 



