126 PHYLLOTAXY OR LEAF-ARRANGEMENT, 



241 , The successive grades of angular divergence of alternate 

 leaves, as expressed in degrees, are 



i = 180 f = 144 Vk = 138 27' 41.54" 



| = 120 f = 135 . = 137 8' 34.29" 



and so on ; and beyond, if not in the latter cases, the differences 

 become quite too small for determination by inspection. They 

 all fall within the and as to amount of divergence ; and they 

 form a series converging to a deduced typical angle of 137 30' 

 28", which, being irrational to the circumference, would place 

 no leaf exactly over any preceding one, but alternately and 

 more and more slightly on one and the other side of the vertical, 

 and so on, in an endless spiral. That is, according to Bravais, 

 the ranks in the higher grades tend to become curviserial, or 

 actually become so; while in the lower grades they are obviously 

 rectiserial. Unless, indeed, there is some torsion of the axis, 

 by which the vertical ranks are rendered oblique, as is often the 

 case in cones of the Norway Spruce. But, apart from this, the 

 difference between rectiserial in a high order and curviserial 

 soon becomes inappreciable. Any and all of the higher grades, 

 and practically one as low as the f , secures the utility of the 

 theoretical angle, viz., that " by which the leaves would be dis- 

 tributed most thoroughly and rapidly around the stem, exposed 

 most completely to light and air, and provided with the greatest 

 freedom for symmetrical expansion, together with a compact 

 arrangement in the bud." Even in the simpler grades of com- 

 monest occurrence, each leaf (according to Wright) is so placed 

 over the space between older leaves nearest in direction to it as 

 always to fall near the middle of the space, until the circuit is 

 completed, when the new leaf is placed over an old one. 1 



242. It is to be noted that the distichous or variety gives 

 the maximum divergence, viz. 180, and that the tristichous or 

 gives the least, or 120 ; that of the pentastichous or is nearly 

 the mean between the first two ; that of the f , nearly the mean 

 between the two preceding, &c. The disadvantage of the two- 

 ranked arrangement is that the leaves are soon superposed and 

 so overshadow each other. This is commonly obviated by the 

 length of the internodes, which is apt to be much greater in this 

 than in the more complex arrangements, therefore placing them 

 vertically farther apart ; or else, as in Elms, Beeches, and the 



1 This corresponds with Hofmeister's general rule, that "new lateral 

 members have their origin above the widest gaps between the insertions of 

 the nearest older members." Yet the fact that the character of the leaf- 

 arrangement is laid down at the beginning in the bud does not go far in the 

 way of the mechanical explanation which he invokes- 



