DISTRIBUTION OF THE GREEN LEAVES ON THE STEM. 



401 



another ^ of the circumference; in the other case by ^; and from this it follows 



that in the one instance there are twenty-one, and in the other thirty-four orthostichies. 



If we place these actually-observed instances together, we have the series 



^. S> 5> 



TS> ^T» s^- 



But the variety of the conditions on which the leaves are arranged is not 

 I exhausted by a long way. Although but seldom, still cases have been observed 



' which can be placed together in the series ^, ^, f, ^, -^^ , and also in the series 



hh-Ti'-TS I^ ^11 these series this very remarkable peculiarity occurs, that 



Fig. 100.— Plan of Five-thirteenths Phyllotaxis. 



in each individual fraction the denominator is equal to the sum of the denom- 

 inators, and the numerator is equal to the sum of the numerators of the two 

 preceding fractions. 



Moreover it must be here particularly mentioned that the divergence, by which 

 the leaves following one another in age are separated in a horizontal direction, is 

 the more difficult to establish the smaller it becomes. The one-third, two-fifths, and 

 three-eighths arrangements are the most easily demonstrable on the full-grown 

 shoots, although occasionally doubt arises as to whether the three, five, and eight 

 orthostichies represent completely straight lines. But the demonstration of ^ and 

 the ^ arrangements, especially in green herbaceous stems, is very difficult and 

 uncertain. 



