Look now at these fractions, 1/2, 1/3, 2/5, and 3/8. The numerator of
the third is the sum of the numerators of the first and second, its
denominator, the sum of 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°, 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