Chttrch. — Note on Phyllotaxis. 485 



phenomena, it is clearly impossible to found any modern 

 scientific generalizations on angles which cannot be measured, 

 and lines which cannot be proved to be straight : it thus 

 follows that all speculations based on the assumption of the 

 Schimper-Braun series must rest on a purely hypothetical 

 foundation which may at any time be overturned. Such 

 expressions, as Sachs constantly pointed out, attempt to 

 imitate the phenomena observed without giving any reason 

 for such geometrical construction. 



Again, taking the mathematical interpretation of the 

 Schimper-Braun system, that the genetic spiral and the 

 parastichies are represented by spirals of Archimedes, while 

 the orthostichies are radii vectores, a simple geometrical con- 

 struction in terms of these spirals should bring out either the 

 truth or error of this hypothetical relationship of the lateral 

 members. 



Thus, from the equation to the Archimedean spiral (r — aQ), 

 it is easy to construct a pair of spirals whose variable a shall 

 have the ratio of the parastichies observed on any given speci- 

 men. Take for example the t^^j system, the primary contact 

 parastichies of which are 8 and 13 ; Fig. 3 shows such a system 

 geometrically planned for a left-hand genetic spiral: the 

 members along the twenty-one orthostichy lines differ by 

 twenty-one, and fall on the mathematically straight radii 

 vectores of the system. The intersections of these parastichy 

 spirals mark the points at which the lateral members are 

 inserted, and the views of Schimper and Braun included only 

 the consideration of such points. It is clear, however, that if 

 the spaces between the spiral planes are regarded as contain- 

 ing the members pressed into close lateral contact, as seen in 

 the transverse section of a foliage bud, the appearance of the 

 progressive dorsiventrality of such lateral members is very 

 fairly imitated. The construction, in fact, becomes more and 

 more like the appearances seen in the plant as the periphery 

 of the system is reached, but the central part which includes 

 the actual seat of development is very inadequately repre- 

 sented: thus, the areas become so relatively elongated in the 



