Fig. 189. Diagram of this arrangement, with a spiral line drawn from the attachment of one leaf to the next, and so on; the parts on the side turned from the eye are fainter.
Fig. 190. A ground-plan of the same; the section of the leaves similarly numbered; a dotted line drawn from the edge of one leaf to that of the next marks out the spiral.
189. The five-ranked arrangement is expressed by the fraction 2/5. This fraction denotes the divergence of the successive leaves, i. e. the angle they form with each other: the numerator also expresses the number of turns made round the stem by the spiral line in completing one cycle or set of leaves, namely, two; and the denominator gives the number of leaves in each cycle, or the number of perpendicular ranks, namely, five. In the same way the fraction 1/2 stands for the two-ranked mode, and 1/3 for the three-ranked: and so these different sorts are expressed by the series of fractions 1/2, 1/3, 2/5. Other cases follow in the same numerical progression, the next being the
[190.] Eight-ranked arrangement. In this the ninth leaf stands over the first, and three turns are made around the stem to reach it; so it is expressed by the fraction 3/8. This is seen in the Holly, and in the common Plantain. Then comes the
[191.] Thirteen-ranked arrangement, in which the fourteenth leaf is over the first, after five turns around the stem. The common Houseleek (Fig. [191]) is a good example.
192. The series so far, then, is 1/2, 1/3, 2/5, 3/8, 5/13; the numerator and the denominator of each fraction being those of the two next preceding ones added together. At this rate the next higher should be 8/21, then 13/34, and so on; and in fact just such cases are met with, and (commonly) no others. These higher sorts are found in the Pine Family, both in the leaves and the cones and in many other plants with small and crowded leaves. But in those the number of the ranks, or of leaves in each cycle, can only rarely be made out by direct inspection. They may be indirectly ascertained, however, by studying the secondary spirals, as they are called, which usually become conspicuous, at least two series of them, one turning to the right and one to the left, as shown in Fig. [191]. For an account of the way in which the character of the phyllotaxy may be deduced from the secondary spirals, see Structural Botany, Chapter IV.
Fig. 191. A young plant of the Houseleek, with the leaves (not yet expanded) numbered, and exhibiting the 13-ranked arrangement; and showing secondary spirals.