DISTRIBUTION OF THE GREEN LEAVES ON THE STEM. 



401 



another ^j of the circumference; in the otlier case by J^; 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 



1 2. S _S 8 13 



'S' 6> 8> l7r> TT> TTT-' 



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

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



which can be placed together in the series ^, i, f, ^, -/jj , and also in the series 



T> f > TT> Ts Iii iill these series this very remarkable peculiarity occurs, that 



Fig. ino. — Plau 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 fractious. 



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 ^J- arrangements, especially in green hei'baceous stems, is very difiicult and 



uncertain. 



Vol,. I. 36 



