RHYTHM. 231 



than potential energy, a similar distribution of energy will be 

 found in the two-dimensional motion of an incompressible 

 fluid.* 



But it must always be borne in mind that such hypotheses of 

 equal energy-distribution only deal with the hypothetical region 

 included under the conventional expression " growth-centre." 

 Away from this region, which represents a more or less gratuitous 

 conception, and which, being beyond the range of actual observa- 

 tion, must also always remain hypothetical, retardation of growth 

 ensues, and tends to produce rapid deformation of the log. spiral 

 systems. Similarly, in the case of eccentric growth, deformation 

 immediately sets in at different rates on different sides. Hence 

 any theory of energy - distribution involving equal amounts of 

 energy on every square must still remain hypothetical, though 

 the quasi-square system, whether deformed by retarded or unequal 

 growth - rates, will continue to indicate equivalent growth-areas ; 

 and such areas mapped by the intersecting curves, whatever the 



* Again, even the homology of vortex construction is open to objection, 

 since, although it was expressly stated (Part I. p. 36) that the terminology of 

 spiral and circular vortices was introduced as a metaplwr to make clear what 

 was implied by a certain type of geometrical construction, the idea of a 

 spiral vortex appears to carry with it an impression of spiral movement. 



It cannot be too strongly insisted that no spiral growth-movement either exists 

 in the plant or is implied by the log. spiral theory. 



The theory may be a spiral one, the phyllotaxis may be justly termed 

 spiral, since the pattern seen may be expressed as spirals, but the growth-move- 

 ment is absolutely radial. {Gf. Weisse, Prings. Jahrb., 1904, p. 419.) 



It is in this sense that the suggestion of Sachs is so valuable and correct, 

 that " all the spirals are subjective " ; and as a purely psychical phenomenon 

 it is interesting to note how the spiral pattern of a moving mass insensibly 

 leads many observers on to the interpretation of a spiral motion (c/. Goethe) 

 just as phyllotaxis has been for a similar reason inundated with torsion 



It is, in fact, one of the best points of the log. spiral theory here put forward 

 that not only is the growth-movement regarded as radial, but it can be shown 

 mathematically that even in a centric spiral system such lateral primordia are 

 bilaterally symmetrical about the radius along which they travel away from 

 the growing-point. {Cf. Mathematical Notes, Form of the Ovoid Curve.) 



Further, in order to avoid the repetition of a " spiral " standpoint, the 

 expression asymmetrical is definitely adopted as a better mathematical mode 

 of expression. 



