[Footnote 1: See the uses and origin of the arrangement of leaves in plants. By Chauncey Wright. Memoirs Amer. Acad., IX, p. 389. This essay is an abstruse mathematical treatise on the theory of phyllotaxy. The fractions are treated as successive approximations to a theoretical angle, which represents the best possible exposure to air and light. Modern authors, however, do not generally accept this mathematical view of leaf-arrangement.] [1]"It is to be noted that the distichous or 1/2 variety gives the maximum divergence, namely 180°, and that the tristichous, or 1/3, gives the least, or 120°; that the pentastichous, or 2/5, is nearly the mean between the first two; that of the 3/8, nearly the mean between the two preceding, etc. 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 further apart; or else, as in Elms, Beeches, and the like, the branchlets take a horizontal position and the petioles a quarter twist, which gives full exposure of the upper face of all the leaves to the light. The 1/3 and 2/5, with diminished divergence, increase the number of ranks; the 3/8 and all beyond, with mean divergence of successive leaves, effect a more thorough distribution, but with less and less angular distance between the vertical ranks." [Footnote 1: Gray's Structural Botany, Chap, iv, p. 126.] For directions for finding the arrangement of cones, see Gray's Structural Botany, Chap. IV, Sect. 1. The subject appears easy when stated in a text-book, but, practically, it is often exceedingly difficult to determine the arrangement. Stems often twist so as to alter entirely the apparent disposition of the leaves. The general principle, however, that the leaves are disposed so as to get the best exposure to air and light is clear. This cannot be shown by the study of the naked branches merely, because these do not show the beautiful result of the distribution.[1] Many house plants can be found, which will afford excellent illustrations (Fig. 21). The Marguerite and Tobacco, both easily grown in the house, are on the 3/8 plan. The latter shows the eight ranks most plainly in the rosette of its lower leaves. The distribution is often brought about by differences in the lengths of the petioles, as in a Horsechestnut branch (Fig. 22) where the lower, larger leaves stand out further from the branch than the upper ones; or by a twist in the petioles, so that the upper faces of the leaves are turned up to the light, as in Beech (Fig. 23). If it is springtime when the lessons are given, endless adaptations can be found. [Footnote 1: Reader in Botany. IX. Leaf-Arrangement.] [Illustration: FIG. 21. Branch of Geranium, viewed from above.] [Illustration: FIG. 22.] [Illustration: FIG. 23.]