220. As the position of the leaf upon the stem marks the position on the exillary bud, it follows that the order of the leaf-arrangement will be the order of the branches also. The careful investigation of this subject has developed a science of unexpected exactness and beauty, called phyllotaxy (Ývλλov, a leaf, τύξĮς, order.

85, Ladies' slippcr (leaves alternate); 86

85, Ladies'-slippcr (leaves alternate); 86. Synandra grrandiflora (leaves opposite); 88, Medeela Virginica (leaves verticillate); 87, Larix Americana (leaves fasciculate).

221. Position upon the stem. Leaves are radical when they grow out of the stem at or beneath the surface of the ground, so as to appear to grow from the roots; cauline when they grow from the stem, and ramial (ramus, a branch), when from the branches.

222. Insertion upon the axis. The arrangement of the scales and young leaves in the bud appears to be in close, contiguous circles. By the development of the axis the leaves are separated, and their order variously modified, according to the following general modes:-

Alternate, one above another on opposite sides, as in the elm.

Scattered, irregularly spiral, as in the potato vine.

Rosulate, clustered regularly, like the petals of a rose, as in the plantain and shepherd's-purse.

Fasciculate, tufted, clustered many together in the axil, as seen in the pine, larch, berberry.

Opposite, two, against each other, at the same node. Ex. maple.

Verticillate, or whorled, more than two in a circle at each node, as in the meadow-lily, trumpet-weed. We may reduce all these modes to

223. Two general types, - the alternate, including all cases with one leaf at each node, - the opposite, including cases with two or more leaves at each node.

224. The true character of the alternate type may be learned by an experiment. Take a straight leafy shoot or stem of the elm or flax, or any other plant with seemingly scattered leaves, and beginning with the lowest leaf, pass a thread to the next above, thence to the next in the same direction, and so on by all the leaves to the top; the thread will form a regular spiral.

225. Fasciculate leaves are the members of an undeveloped branch, and in

Phyllotaxy

Phyllotaxy. 89, leafy branch of elm, - cycle 1/2 90, leafy branch of alder, - cycle 1/2; 91, leafy branch of cherry, - cycle 2/5• case of the subsequent development of the branch, as often occurs in the Berb-eris and larch, their spiral arrangement becomes manifest In the pines the fascicles have fewer leaves, their number being definite and characteristic of the species. Thus P. strobus, the white pine, has 5 leaves in each fascicle, P. palustris, the long-leaved pine, has 3, P. inops, 2.

226. The opposite leaved type is also spiral. The leaves in each circle, whether two or more, are equidistant, dividing the circumference of the stem into equal arcs. The members of the second circle are not placed directly above those of the first, but are turned, as it were, to the right or left, so as to stand over the intervening spaces. Hence there may be traced as many spirals as there are leaves in each whorl.

227. Decussate leaves result from this law, as in the motherwort, and all the mint tribe, where each pair of opposite leaves crosses in direction the next pair, forming four vertical rows of leaves. Therefore, it is

228. An established law that the course of development in the growing plant is universally spiral. But this, the formative cycle as it is called, has several variations.

92, 93, 94, showing the course of the spiral thread and the order of the leaf succession in the axes of elm, alder, and cherry

92, 93, 94, showing the course of the spiral thread and the order of the leaf-succession in the axes of elm, alder, and cherry. 95, axis of Osage-orange with a section of the bark peeled, displaying the order of the leaf-scars (cycle §).

229. The elm cycle. In the strictly alternate arrangement (elm, linden, grasses) the spiral thread makes one complete circuit and commences a new one at the third leaf. The third leaf stands over the first, the fourth over the second, and so on, forming two vertical rows of leaves. Here (calling each complete circuit a cycle) we observe

230. First, That this cycle is composed of two leaves; second, that the angular distance between its leaves is 1/2 a cycle (180°); third, if we express this cycle mathematically by 1/2, the numerator (1) will denote the turns or revolutions, the denominator (2) its leaves, and the fraction itself the angular distance between the leaves (1/2 of 3600).

231. The alder cycle. In the alder, birch, sedges, etc., the cycle is not complete until the fourth leaf is reached. The fourth leaf stands over the first, the fifth over the second, etc, forming three vertical rows. Here call the cycle 1/3; 1 denotes the turns, 3 the leaves, and this fraction itself the angular distance (1/3 of 360°).

232. The cherry cycle. In the cherry, apple, peach, oak, willow, etc., neither the third nor the fourth leaf, but the sixth, stands over the first; and in order to reach it the thread makes two turns around the stem. The sixth leaf is over the first, the seventh over the second, etc, forming five vertical rows. Call this the 2/5 cycle; 2 denotes the turns, 5 the leaves in the cycle, and the fraction itself the angular distance (§ of 360°).

233. The Osage-orange cycle. In the common hedge plant, Osage-orange, the holly, evening primrose, flax, etc, we find no leaf exactly over the first until we come to the 9th, and in reaching it the spiral makes three turns. Here the leaves form eight vertical rows. It is a $• cycle; 3 the number of turns, 8 the number of leaves, and the fraction the angular distance between the leaves (3/8 of 360°).

234. The cycles compared. These several fractions which represent the above cycles form a series as follows: 1/2, 1/3, 2/5, 3/8, in which each term is the sum of the two preceding. The fifth terms in order will, therefore, be 5/13; and this arrangement is actually realized in

96, Phyllotaxy of the cone (cycle 8 / 21 ) of Pinus serotina

96, Phyllotaxy of the cone (cycle 8 / 21 ) of Pinus serotina. 97, cherry cycle (2 /5), as seen from above, forming necessarily that kind of aestivation called quincuntial.

235. The white pine cycle. In the young shoots of the white pine, in cones of most pines, in flea-bane (Erigeron Canadense), etc., the fourteenth leaf stands over the first, the fifteenth over the second, etc. The spiral thread makes five revolutions to complete the cycle, which is, therefore, truly expressed by 5/13 .

236. The houseleek cycle is next in order, expressed by the fraction (3+5 / 8+13) A. having eight turns and twenty-one leaves. Examples are found in the Scotch pine, houseleek, etc.

237. How to determine the HIGHER cycles. To trace the course of the for-mative spiral in these higher cycles becomes difficult on account of the close proximity of the leaves. In the pine cone (Fig. 96, Pinus serotina) several sets of secondary spirals are seen; one set of five parallel spirals turning right (1 - 6 - 11 - 16, etc., the common difference being also five); two sets (one of three, the other of eight) turning left; and still another set, of thirteen, steepest of all, turning right (1 - 14 - 27, etc.). Now the sum of the spirals contained in the two steepest sets gives the denominator of the fraction expressing the true formative spiral sought. Thus, 8+13=21. The numerator corresponding is already known, and the fraction is 8 / 21. See also the white pine cone, whose cycle is 5/13.

238. Diagram 97 represents the leaves of a cherry cycle as seen from above, and verified in the aestivation of the flowers in the rose-family.