328 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS. 



such ratios for one reason only, is in any way aiming at all 

 the mathematical consequences of its choice, although they must 

 all necessarily follow ; and it thus becomes increasingly difficult 

 to draw the line between the tabulation and interpretation of 

 actual observations and the pursuit of abstract mathematical 

 functions, which rapidly degenerates, as Sachs pointed out, into 

 mere " playing with figures." 



So, again, the introduction of equiangular spirals as indicating 

 curves of growth necessarily brings with it all the mathematical 

 properties of these curves. One admittedly makes no further 

 advance toward the interpretation of the causes of phyllotaxis by 

 the mere introduction of equiangular, ontogenetic, and parastichy 

 spirals, 



But if the primary mathematical conception is based on a 

 legitimate foundation, such as that of uniform growth-expansion 

 appears to be, the properties of log. spiral systems become 

 increasingly important as indicating symmetrical or asymmetrical 

 cases of perfect growth, although such spirals may never be 

 measured or even really exist in actual phyllotaxis phenomena, 

 since the modification of the primary construction spirals may be 

 made the subject of subsidiary hypotheses. 



While, therefore, the purely mathematical investigation of log. 

 spiral constructions can add nothing to the explanation of the 

 phenomena, it becomes of interest to tabulate the properties 

 of intersecting systems of these curves, in that functions may 

 be deduced mathematically which are not readily apparent in 

 geometrical constructions, just as geometrical constructions, on 

 the other hand, may confirm or make more obvious a mathematical 

 generalisation. 



It remains, therefore, to consider what the properties and 

 appearance of such abstract ideal phyllotaxis systems would be, 

 ■the relation of their parastichies and orthostichies, as also the 

 form of the curves which would represent the homologues of 

 circles inscribed in the orthogonal meshes, and the angles sub- 

 tended by these in the different systems : the whole set of 

 phenomena thus affording a view of an ideal uniformly expanding 

 system of lateral appendages on a growing axis, which may then 



