166 



rectness of the theory of phyllotaxis mentioned, as developed by 

 Bravais, Schimper and Braun, has been partially contested in 

 later times by several workers in this field of research, for instance 

 by Hofmeister 1 ), Sachs 2 ), Church 3 ), and others. 



The latter has demonstrated in a convincing way, that the determi- 

 nation of a member exactly vertically superposed to one taken as a 

 point of reference, is practically impossible, either by direct observa- 

 tion or by angular measurements as proposed by Bravais. Direct 

 observation teaches us that a leaf never stands vertically above 

 any other given one, a fact already stated in some exceptional cases 

 by Bravais. All so-called "orthostichies" seem to be really curvi- 

 serial lines, especially in the higher divergencies. But then they 

 cannot be distinguished from parastichies, and therewith one of 

 the premises of the Schimper- Braun-theory has lost its value. 



Church concludes that only the number of intersecting para- 

 stichies, dextro-, orlaevogyratory, determines the numerical character 

 of the arrangement, as already suggested by Braun. Moreover, 

 he points to the fact already stated by Hofmeister, De Candolle 

 and Sachs, that the phyllo taxis-fraction, whatever numerical 

 value is given to it, must appear greater or smaller in the same rate, 

 as the axis about which the leaves are arranged, is shortened or 

 lengthened, the phenomena of varying phyllotaxis, therefore, being 

 partially caused by the varying rates of growth 4 ). De Candolle 4 ) 

 has drawn attention to the same fact; according to this author the 

 character of the phyllotaxis, even if a constant angular divergence 

 between consecutive members be supposed, - - must vary when 



length 



the ratio : -rr-^ of the stem during the process of growth changes 



diameter 



appreciably. In condensed and multiple fruits and inflorescences 



1 ) W. Hofmeister, Allgemeine Morphologic der Gewdchse, (1868). 



2 ) J. Sachs, Vorlesungen tiber Pflanzenphysiologie, Leipzig, (1882). p. 603, 

 606; S. Schwendener, Mechanische Theovie der Blattstellungen, (1878). 



3 ) A. H. Church, On the Relation of Phyllotaxis to Mechanical Laws, (1904). 



4 ) Cf.: G. v. Iterson Jr., Mathematische und Mikvoskopisch-anatomische 

 Studien tiber Blattstellungen, Jena, (1907), p. 222; C. de Candolle, Considerations 

 sur I'Etude de la Phyllotaxie, Geneve, (1881), p. 29, 47, 52. Certainly, however, there 

 are a number of other important causes acting during the first design of the lateral 

 organs, which govern this phenomenon of varying phyllotaxis ; and it is highly 

 probable that an exact and exhaustive knowledge of the specific nature of 

 these causes would simultaneously bring the final solution of the problem 

 concerning the true meaning of the remarkable numerical relations exposed 

 in the above. 



