142 



A TEXTBOOK OF BOTANY 



[Ch. IV, 5 



mined. The^;c fractions primarily express the angular diver- 

 gence of the leaves from one another around the stem, but 

 secondarilj^ the numerator shows the 

 number of turns made by the s]iiral in 

 reaching a leaf directly over any given 

 one, while the denominator expresses the 

 number of leaves in such a complete turn. 

 It is not only true that these fractions 

 are actually found in phyllotaxy, but 

 also a fact that they are the onlj' ones 

 which ordinarily occur, the exceptions 

 l)eing rare, and following an analogous 

 plan. Furthermore, when a stem having 

 one of these fractional sj^stems becomes 

 twisted, the leaves are always lirought 

 into th(> next system above or below. 

 When, now, the fractions are arranged 

 in seciuence, — 



I 3- JL. _8_ 13 IX 



5 8 13 J 1 3 4 .5 .5 



some remarkable relations among them 

 come out, — viz. in all cases after the first and second, the 

 numerators and denominators are each the sum of the two 

 preceding, while each numerator is the 

 same as the denominator next before the 

 preceding. This curiously related series, 

 which as a mathematical abstraction is 

 known from its discoverer as the Fibo- 

 nacci series, finds actual physical expres- 

 sion not only in phyllotaxy, but also in 

 some other phenomena of nature. 



The significance of phyllotaxy has been 

 diversely interpreted. Some liotanists 

 have explained it as adaptive, thinking 

 it must give to clusters of leaves the best aggregate ex- 

 posure to light. But such reasonableness as this theory may 



Rosetlo 



•;howiiiy; 



irraiisoment. 



(After eiray.) 



