DISTRIBUTION' OF THE GUEEN LEAVES ON THE STEM. 



401 



another ^ of the circumference; in the other case by l^; 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 



T> ■?> fi t> TJi TT' 3T 



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

 exhausted bj' a long way. Although but seldom, still cases have been observed 



which Cixn be placed together in the series |, \, f, ^, ^^^ , and also in the series 



}> t> A> iV I" 'il' these series this very remarkable peculiarity occurs, that 



Fig. 100 —Plan of Five-thiiteenths Phyllotaas. 



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 

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

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



uncertain. > 



Vol. L 26 



