When the leaves are alternate, or single at each node of the stem, they are arranged in many different ways. Ask the pupils 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°, 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°, 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 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]