160 



towards these points, the farther the original point (m, n) was chosen 

 distant from O, that original point being determined, for instance, 

 by coordinates like m = 34, n = 89, etc. The angular points men- 

 tioned have the coordinate-fractions characteristic for the consecutive 

 terms of the Fibonaccian series; and the straight line considered 

 will, in infinity, pass through the point, having the ^rational coordi- 

 nates: Af[(3-l/5), 2] However it is clear, that there is in the whole 

 net-plane infinitely extended, no such point really present; and 

 it cannot maintained that a parallelism between the space-lattice 

 character of crystals, with their rational ratios, and living organisms 

 is really established by this way of reasoning, suggestive as for the 

 remaining it may be. For the generatrix of the cylindrical stem would 

 in this case be a line of the supposed space-lattice, which does not 

 pass through any real angular point of the net-plane considered; 

 this straight line would therefore in a crystalline medium have no 

 significance at all. Instead of supporting such a parallelism between 

 the two groups of phenomena, the fundamental contrast between 

 living and inanimate nature with respect to form-symmetry, is again 

 more strikingly brought to the fore by these considerations, in so 

 far as what is impossible in the one domain of phenomena, should 

 even be the most perfect state of things in the other. 



31. However it must be borne in mind that we have no right 

 to consider these remarkable views on phyllotaxis, as pictured in 

 the above, to be of real ontogenetic significance, as long as we have 

 no indications about the mechanical or physiological causes of such 

 mathematically determinable arrangements of similar organs. 



In this respect now it is of importance to remark that the cor- 

 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 ob- 

 servation teaches that a leaf perhaps never stands vertically above any 



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



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

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



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



