ANOMALOUS SERIES. 209 



was due to an entire misapprehension of the phenomena of proto- 

 plasmic growth, as was only natural when protoplasm was still un- 

 known (1754-1835). By regarding growth as the addition of 

 layers of equal thickness in equal times, as in the conventional 

 representation of the addition of annual rings to a tree, expressed 

 in terms of concentric circles with equal increments on the radii, 

 a conception of arithmetical progression was introduced, which 

 naturally resulted in the adoption of the spiral of Archimedes. 

 A clearer recognition of the interstitial growth of a mass of proto- 

 plasm throughout its whole substance, by becoming expressed as a 

 series of concentric circles in geometrical progression which may 

 contain a network of similar figures, leads equally naturally to the 

 assumption of a log. spiral as the actual curve of asymmetrical 

 growth. 



Finally, it must be pointed out that the whole of the observations 

 and deductions hitherto given for phyllotaxis constructions, in- 

 cluding systems expanding and falling according to the Fibonacci 

 law, are the expression of the geometrical properties of intersecting 

 spiral curves, without necessarily adding any further information 

 with regard to the character of the spirals ; and almost any pair of 

 unequal curves will give approximate results. The appearance of 

 log. spirals will be produced subjectively by arranging any collection 

 of similar figures in spiral series ; and it is thus necessary to keep 

 in mind Sachs' original observation that the subjective appearance 

 does not necessarily tell anything of the mode of formation of a 

 given construction. The log. spiral theory demands orthogonal 

 intersection, and this has so far not been proved, although it might 

 be legitimately hypothecated from the analogy of the orthogonal- 

 intersection theory of cell-formation proposed by Sachs ; since it is 

 sufficiently clear that if the segmentation of the plant-body in 

 terms of cells and cell-layers can be expressed by orthogonal 

 trajectories, there must be some law behind these phenomena 

 which controls the distribution of growth-energy, and this may 

 prove to be in some way comparable to that which governs more 

 strictly physical phenomena.* 



* " Sections tlirougli growing, and especially tlirougt young parts of the plant, 

 always show arrangements of the cells which are quite definite, and in the 



