the two denominators. The same is true of the
fourth fraction and the two immediately preceding it. Continuing the
series, we get the fractions 5/13, 8/21, 13/34. These arrangements can
be found in nature in cones, the scales of which are modified leaves and
follow the laws of leaf-arrangement.[1]

[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 deg., and that the tristichous, or 1/3, gives the
least, or 120 deg.; 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