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My attention was afterwards called to "What may be learned from a Tree," by Prof. Harlan Couitas. D. Appleton & Co., New York, 1863. The leaf-arrangement of the Beech is alternate, on the one-half plan. The small twigs turn upwards, so that all the spray is on the upper side, giving a flat appearance to the branch. This gives the leaves a better exposure to the light.

The flower-cluster leaves a concave, semicircular scar, in the leaf-axil. Balm-of-Gilead. 1. Branch in winter state: a, leaf-scar; b, bud-scar. 2. Branch, with leaf-buds expanded. 3. The terminal buds are the strongest and not very many axillary buds develop, so that the tree has not fine spray. The leaf-arrangement is alternate, on the 2/5 plan.

The outer four or six pairs are so minute that the arrangement is not very clear, but as we proceed we perceive that the scales are in alternate pairs, as in Horsechestnut; that is, that two scales are exactly on the same plane. But we have learned in the Lilac that the scales are modified leaves, and follow the leaf-arrangement of the species.

These arrangements can be found in nature in cones, the scales of which are modified leaves and follow the laws of leaf-arrangement. By Chauncey Wright. Memoirs Amer. Acad., IX, p. 389. This essay is an abstruse mathematical treatise on the theory of phyllotaxy.

Looking along the shoot, I find that every six buds stand in the same line: the sixth bud is over the first, seventh over the second, eighth over the third. This is the leaf-arrangement or phyllotaxy of the apple-tree, expressed by the fraction 2/5.

The 2/5 leaf-arrangement obtains on cherry, peach, apricot, pear, raspberry and many others; but a very different order is that of the linden, grape, currant, lilies, elm, maple. A year ago, in the spring of 1920, a terminal bud that had formed in 1919 expanded and gave rise to this rapidly growing shoot.

The leaf-arrangement is alternate on the 2/5 plan. The pupils may compare the branching with that of their other specimens. This is a good specimen for the study of accessory buds. There is usually a bud in the axil of each lower scale of the axillary buds, making three side by side. We have already noticed this as occurring sometimes in Lilac. It is habitually the case with the Red Maple.

The buds that spring from the inner angle of the leaf with the stem are axillary buds; those that crown the stems are terminal. Since a bud is an undeveloped branch, terminal buds carry, on the axis which they crown, axillary buds give rise to side-shoots. The leaf-scars show the leaf-arrangement and the number of leaves each year.

The bud may also be studied in cross-section. This shows the whole arrangement. The plan is not so simple as in Horsechestnut, where the leaves are opposite. The subject of leaf-arrangement should be passed over until phyllotaxy is taken up. The scars on the stem differ from Horsechestnut in having no distinct bands of rings.

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.