384 RELATION TO ENVIRONMENT. 



these forms are found in certain green stems which do not bear 

 leaves. An example of the first is found in asparagus with its 

 numerous crowded slender branches. But such forms in our 

 climate are rare, since foliage leaves are more efficient. The 

 second and third forms are found among cacti, which usually 

 grow in dry regions under conditions which would be fatal to 

 ordinary thin foliage leaves. 



747. Relation of foliage leaves to the stem. In the study of 

 the position of the leaves on the stem we observe two important 

 modes of distribution: (i) the distribution along the individual 

 stem . or branch which bears them, usually classed under the 

 head of Phyllotaxy; (2) the distribution of the leaves with refer- 

 ence to the plant as a whole. 



748. Phyllotaxy, or arrangement of leaves. In examining buds on the 

 winter shoots of woody plants, we cannot fail to be impressed with some 

 peculiarities in the arrangement of these members on the stem of the plant. 



In the horse-chestnut, as we have already observed, the leaves are in 

 pairs, each one of the pair standing opposite its partner, while the pair 

 just below or above stand across the stem at right angles to the position of 

 the former pair. In other cases (the common bed-straw) the leaves are 

 in whorls, that is, several stand at the same level on the axis, distributed 

 around the stem. By far the larger number of plants have their' leaves 

 arranged alternately. A simple example of alternate leaves is presented 

 by the elm, where the leaves stand successively on alternate sides of the 

 stem, so that the distance from one leaf to the next, as one would measure 

 around the stem, is exactly one half the distance around the stem. This 

 arrangement is one half, or the angle of divergence of one leaf from the 

 next is one half. In the case of the sedges the angle of divergence is less, 

 that is one third. 



By far the larger number of those plants which have the alternate arrange- 

 ment have the leaves set at an angle of divergence represented by the frac- 

 tion two fifths. Other angles of divergence have been discovered, and 

 much stress has been laid on what is termed a law in the growth of the 

 stem with reference to the position which the leaves occupy. Singularly 

 by adding together the numerators and denominators of the last two fractions 

 gives the next higher angle of divergence. Example: - + - = ^; 2 + !=J-- 



_, 3+5 8 5+8 13 



and so on. I here are, however, numerous exceptions to this regular 



arrangement, which have caused some to question the importance of any 

 theory like that of the "spiral theory" of growth propounded by Goethe 

 and others of his time. 



