64 The Living Plant 



as in the Elm and Grasses, in which case one must pass once 

 round the stem and cover two spaces to reach a leaf over the 

 first (figure 16, C). In others, (e. g., the Sedges), the fraction is 

 1 /3, and a spiral drawn through the bases of the leaves passes 

 once round the stem and across three spaces to reach a leaf over the 

 first (figure 16, Z>). In others, (e. g., the Apple) it is 2 / 5 , when the 

 spiral must pass twice around the stem and cross five spaces to 

 come to a leaf over the first (figure 16, E), an arrangement which 

 is, perhaps, the commonest of all. In others the fraction is 3 / 8 

 (in Holly and Plantain figure 16, F), or / 13 , as in cones of White 

 Pine, while 8 / 21 , 13 / 34 , and even some higher fractions are said to 

 have been traced in special places where the leaves are greatly 

 condensed together in rosettes. And a curious thing is this, that 

 while these fractions occur, the various possible intermediate 

 ones do not. In these fractions, which primarily express the 

 amount of circumference between two successive leaves, the 

 numerator also expresses the number of turns that must be made 

 around the stem to reach a leaf over the first, while the denomina- 

 tor expresses the number of spaces that must be passed over for 

 this purpose, and also the number of vertical ranks into which the 

 leaves fall. Moreover, these fractions bear to one another a very 

 curious relationship, for when they are arranged in a series, viz., 



*/2, Va, 2 /5, 3 /8, 5 /13, 8 /21, 13 /34 



it is found that each numerator is the sum of the two numerators 

 preceding, and each denominator likewise the sum of its two pre- 

 decessors, and moreover each numerator is the same as the de- 

 nominator next before the preceding. This curious series, known 

 in mathematics as the Fibonacci series, is said to find expression 

 in other phenomena of nature, including the arrangement of the 

 planets, and is therefore not peculiar to the phyllotaxy of plants. 

 The question of present importance, however, is this, what is its 

 meaning in connection with leaf-arrangement? Of course one's 

 first natural thought is, adaptation, which appears reasonable 

 enough with the opposite system and the whorls, and even with 



