986 A TEXTBOOK OF THEORETICAL BOTANY 



tnat every leaf is separated from the next by a fixed proportion of the circum- 

 ference of the stem. This is called the leaf divergence. By comparing a 

 number of different plants we find that the characteristic divergences are not 

 all the same, but that the number of different divergences is nevertheless 

 limited, and that certain figures occur again and again. Expressed as fractions 

 of the circumference, the divergences found experimentally can be arranged 

 in order thus : |, -|, f , |, 3^3, etc., a numerical series called, from its discoverer, 

 the Fibonacci series. This series has the property that the numerator and 

 denominator of each fraction are equal to the respective sums of those of 

 the two preceding fractions. It has the further property that the first term, 

 h, expresses the maximum divergence, and the second term, ^, expresses 

 the minimum divergence in the series. 



llie maximum divergence, |, is not very common. It represents leaves 

 which are arranged, singly, in two rows, on opposite sides of the stem and is 

 specially distinguished as the alternate or distichous arrangement. We have 

 already mentioned that it is commoner among Monocotyledons than among 

 Dicotyledons, and is especially characteristic of Gramineae, Orchidaceae, 

 and Iridaceae. The second or minimum divergence is likewise restricted 

 in occurence, but it is noteworthy that it predominates in the Cyperaceae, 

 a family closely related to the Gramineae. It is also frequent in the flowers 

 of Monocotyledons but is rare among Dicotyledons. A closer approximation 

 than I is very rare among higher plants, but cases of ^ or ^ divergence are 

 known, and they form the first term of another Fibonacci series. 



The divergences of | and | are the commonest among Dicotyledons. 

 Higher fractions are characteristic of shoots with either very narrow or very 

 closely set leaves or branches, like some inflorescences. As the higher fractions 

 of the series are approached, it becomes increasingly difiicult to say which 

 leaves are directly above each other, as the number of intervening turns of 

 the spiral increases, approaching the limiting case in which no two leaves 

 are directly superposed, which occurs in some very condensed shoots such 

 as gymnospermic cones in which the fractional denominator approaches 

 infinity. 



In describing these spiral arrangements it is customary to refer to the 

 vertical rows of superposed leaves as orthostichies. The number of ortho- 

 stichies present will obviously equal the denominator of the divergence 

 fraction. The mature stem has normally only one spiral of leaves, traced 

 conventionally in the anticlockwise direction round the stem, which represents 

 the sequence in which the initials were produced at the apex and is therefore 

 referred to as the genetic spiral (Fig. 974). The existence of this spiral does 

 not, however, imply any twisting of the axis during growth, nor does it imply 

 any spirality or " spiral tendency " as it used to be called, at the apex. It is 

 simply a geometrical figure which is " described " as the mathematicians 

 would say, by the equiangular spacing of the primordia on an elongating 

 axis. As the genetic spiral approaches the apex it becomes more and more 

 condensed towards a flat helix, and the young leaves or primordia approach 

 closer and closer together. The number of orthostichies represents, in each 



