74 RELATIONSHIPS OF SYMMEl^RY 



of the mechanical hypothesis of phyllotaxy and of its empirical ground- 

 work. I was therefore desirous to have the principles of it explained 

 from the other side. 



SKETCH OF THE 

 MECHANICAL HYPOTHESIS OF LEAF-POSITION. 



By DR. ARTHUR WEISSE. 



The older hypothesis of phyllotaxy occupied itself chiefly with the classi- 

 fication from an arithmetical standpoint of the relative positions of the lateral 

 organs as they were observed in their mature condition. The several arrange- 

 ments it framed have always a constant value which possess only mathematical 

 relationships one to the other. In sharp contrast therewith is Schwendener's 

 mechanical hypothesis * of the position of leaves which is based upon the 

 history of development. In it the lateral organs are not regarded as discrete 

 points but as geometric figures, which at definite stages of development mutually 

 touch, and must therefore influence one another mechanically. 



Of the factors which cause displacements of lateral organs in the course of 

 development of the shoots we must consider as of first importance inequalities 

 of growth in length and in thickness. If we suppose that a mother-organ grows 

 predominantly in thickness, whilst the lateral shoots retaining their form in cross- 

 section increase equally all round, it is evident that the resistances will reach their 

 maximum in the longitudinal direction, their minimum in the transverse direction. 

 The displacements brought about by this will be the same as they would be were 

 the axis subjected to parallel pressure. If, conversely, the growth in length pre- 

 dominates, the displacements that occur will be of a kind such as would be produced 

 by a longitudinal pull. 



In order to state the problem as simply as possible we may start, like Schwen- 

 dener, with the assumption that the form and size of the lateral organs during the 

 displacement remain constant and that their cross-section is circular. Let us 

 consider a concrete case, such as is represented in Fig. 33, which shows a spiral 



I 3 



arrangement of the chief series with a divergence of , upon a cylindric axis 



34 



which has been unrolled and spread out. If longitudinal pressure acts upon 

 this arrangement it is evident that it can only be propagated in the direction 

 of those parastichies of which the lateral organs are in contact. We obtain 

 then two components of which the one operates in the direction of the third 

 row (that is to say from organ 27 in the series 27, 24, 21 ... .), the other 

 in the direction of the fifth row (that is to say from organ 27 in the series 27, 22, 

 1 7 ....). The problem then is the well-known mechanical one of the movement 

 of a span-roof with unequal length of rafters. In our example organ 27 is the 

 apex, the two contact lines 27, 24, 21 .... and 27, 22, 17 .... are the rafters 

 of the span-roof. Without following out the mathematical solution of the problem 



1 Schwendener, Mechanische Theorie der Blattstellungen. Leipzig, 1878. 



