LEAF-ARRANGEMENT 75 



individuals comprising it are always separated from each other 

 by regular angular intervals. Thus, if two leaves are present 

 they are half the circumference of the stem apart or exactly 

 opposite each other, and not both on the same side ; if three 

 arise at the same node, they are separated from each other 

 by regular intervals of 120 degrees, or one-third of the circum- 

 ference, and so on for any number of leaves. 



On many stems the leaves are not in whorls but scattered 

 singly along it, only one leaf arising at each node : such an 

 arrangement is spoken of as alternate or spiral. A line drawn 

 from the bottom to the top of a shoot in such a manner that 

 it touches the base of each successive leaf is a spiral. The 

 distances between the leaves measured along the stem are 

 variable, some being an inch apart, others two or more; their 

 angular intervals apart are, however, as definite and regular as 

 in plants with the whorled arrangement. 



The divergence or angular distance is usually expressed in 

 fractions of the circumference. In elm, Spanish chestnut and 

 grasses, it is ^, that is, the spiral in passing from one leaf to the 

 next winds half round the stem. In birch it is J, while in pear 

 and plum the angular distance is f of the circumference. 



The divergences most frequently met with are, ^, \, -f, |, and 

 Y°^. On inspection these spirally arranged leaves are seen to be 

 in straight longitudinal rows along the stems ; plants with a 

 divergence of ^ having two rows, those with \ three rows, and 

 those with f five rows, and so on, the denominator or lower 

 figure of the above fractions indicating the number of rows 

 present. 



If any particular leaf in a row is selected and the spiral traced 

 round the stem touching each successive leaf until another leaf 

 is reached on the same row, the number of leaves touched, not 

 counting the one at which we start, is equal to the number 

 of the denominator of the fractions expressing the angular 

 divergence, and the numerator indicates the number of com- 



