142 



A TEXTBOOK OF BOTANY 



[Ch. IV, 5 



^ 



W 



mined. These fractions primarily express the angular diver- 

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

 secondarily the numerator shows the 

 number of turns made by the spiral 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 only ones 

 which ordinarily occur, the exceptions 

 r^^ being rare, and following an analogous 



^.J 1 plan. Furthermore, when a stem having 



one of these fractional systems becomes 

 twisted, the leaves are always brought 

 into the next system above or below. 

 When, now, the fractions are arranged 

 in sequence, — 



Fig. 98. — The al- 

 ternate, f spiral, ar- 

 rangement. 



5 

 T3 



A 



13 



T& 



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 of Houseleek, showing 

 diversely interpreted. Some botanists ^^q^ 9 "^ 

 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 



