to look at all the
branches with alternate leaves that they have studied, and determine in
each case what leaves stand directly over each other. That is, beginning
with any leaf, count the number of leaves passed on the stem, till one is
reached that stands directly over the first.[1] In the Beech and the Elm
the leaves are on opposite sides of the stem, so that the third stands
directly over the first. This makes two vertical ranks, or rows, of
leaves, dividing the circle into halves. It is, therefore, called the
1/2 arrangement. Another way of expressing it is to say that the angular
divergence between the leaves is 180 deg., or one-half the circumference.

[Footnote 1: The pupils must be careful not to pass the bud-rings when
they are counting the leaves.]

The 1/3 arrangement, with the leaves in three vertical ranks, is not very
common. It may be seen in Sedges, in the Orange-tree, and in Black Alder
_(Ilex verticillata)_. In this arrangement, there are three ranks of
leaves, and each leaf diverges from the next at an angle of 120 deg., or
one-third of the circumference.

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