By far the commonest arrangement is with the leaves in five vertical ranks.  The Cherry, the Poplar, the Larch, the Oak, and many other trees exhibit this.  In this arrangement there are five leaves necessary to complete the circle.  We might expect, then, that each leaf would occupy one-fifth of the circle.  This would be the case were it not for the fact that we have to pass twice around the stem in counting them, so that each leaf has twice as much room, or two-fifths of the circle, to itself.  This is, therefore, the 2/5 arrangement.  This can be shown by winding a thread around the stem, passing it over each leaf-scar.  In the Beech we make one turn of the stem before reaching the third leaf which stands over the first.  In the Apple the thread will wind twice about the stem, before coming to the sixth leaf, which is over the first.

Another arrangement, not very common, is found in the Magnolia, the Holly, and the radical leaves of the common Plantain and Tobacco.  The thread makes three turns of the stem before reaching the eighth leaf which stands over the first.  This is the 3/8 arrangement.  It is well seen in the Marguerite, a greenhouse plant which is very easily grown in the house.

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 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.”