60 RELATION OP PHYLLOTAXIS TO MECHANICAL LAWS. 



spadix (8 + 13) of Anthv/rmm, crassmervium, so widely is this type 

 of construction distributed among flowering plants. 



The diagrams are less satisfactory in the lower ratios, owing to 

 the fact that but few members can be represented on a circle, this 

 being correlated with the production of relatively large lateral 

 members over a greater vertical extent of a narrow axis ; but they 

 are sufficiently clear to show (1) the approximate alternation of 

 successive cycles, (2) the fact that the first member of each successive 

 cycle falls on a spiral line and that there are no radial orthostichies 

 present. On the other hand, when parastichies are drawn through 

 the points of intersection of radii and circles as demanded by the 

 Schimper-Braun construction, these curves, which irritated Sachs,* 

 are seen to be neither log. spirals nor mutually orthogonal, and the 

 essential points of their construction are lost.f 



The series of common phyllotaxis expressions can therefore only 

 be represented in terms of the intersecting contact-parastichies, in 

 the form : Ps. = (1 + 1), (1 + 2), (2 + 3), (3 + 5), (5 + 8), (8 + 13), etc., 

 in which the first number (the lowest of the pair) gives the long " 



* Sachs, he. dt., pp. 497-498 : " Among the errors of this (Spiral) theory is the 

 one that the spiral arrangement of all organs on a common axis must necessarily 

 follow from its so-called parastichies." " Even ordinary wall-papers show such 

 parastichies, and in the same way the arrangement of scales on the bodies of 

 fishes, of the hairs on the skin of mammals, and of the tiles on a roof, exhibit 

 such parastichies clearly enough." 



t Van Tieghem, Traits de Botanique, vol. i. p. 63. A construction of a | 

 system with the genetic spiral represented as a Spiral of Archimedes gives points 

 along 5 radii vectores which are the orthostichy lines of Schimper. Curves 

 drawn through the points differing by 2 and 3 respectively are again by con- 

 struction Archimedean spirals in the ratio 2 : 3. Such a simple spiral construc- 

 tion was evidently present in the minds of Bonnet and Calandrini in proposing 

 the original quincuncial system, and the fact that they observed that leaves did 

 not obey such a construction accurately was thus glossed over as a secondary bio- 

 logical phenomenon. Similarly all the divergence fractions of Schimper and 

 Braun clearly imply constructions by Spirals of Archimedes, and these spiral 

 systems are thus based on the fact that orthostichies are often fairly accurate to 

 the eye. 



The Archimedean spirals, it is important to note, are based on hypothecated 

 orthostichies, and not the orthostichies on postulated Spirals of Archimedes. 

 Since, then, these spirals are usually associated with torsion phenomena and the 

 formation of screws, various torsion-hypotheses become superimposed on the 

 original unproven premises. 



