[Footnote 1:  Reader in Botany.  VIII.  Young and Old Leaves.]

3. Phyllotaxy.  The subject of leaf-arrangement is an extremely difficult one, and it is best, even with the older pupils, to touch it lightly.  The point to be especially brought out is the disposition of the leaves so that each can get the benefit of the light.  This can be seen in any plant and there are many ways in which the desired result is brought about.  The chief way is the distribution of the leaves about the stem, and this is well studied from the leaf-scars.

The scholars should keep the branches they have studied.  It is well to have them marked with the respective names, that the teacher may examine and return them without fear of mistakes.

In the various branches that the pupils have studied, they have seen that the arrangement of the leaves differs greatly.  The arrangement of leaves is usually classed under three modes:  the alternate, the opposite, and the whorled; but the opposite is the simplest form of the whorled arrangement, the leaves being in circles of two.  In this arrangement, the leaves of each whorl stand over the spaces of the whorl just below.  The pupils have observed and noted this in Horsechestnut and Lilac.  In these there are four vertical rows or ranks of leaves.  In whorls of three leaves there would be six ranks, in whorls of four, eight, and so on.

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 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.