187. =Three-ranked= arrangement,--that of all Sedges, and of White Hellebore. Here the second leaf is placed one third of the way round the stem, the third leaf two thirds of the way round, the fourth leaf accordingly directly over the first, the fifth over the second, and so on. That is, three leaves occur in each turn round the stem, and they are separated from each other by one third of the circumference. (Fig. 186, 187.) [Illustration: Fig. 186. Two-ranked arrangement, shown in a piece of the stalk of a Sedge, with the leaves cut off above their bases; the leaves are numbered in order, from 1 to 6. 187. Diagram or cross-section of the same, in one plane; the leaves similarly numbered; showing two cycles of three.] 188. =Five-ranked= is the next in the series, and the most common. It is seen in the Apple (Fig. 188), Cherry, Poplar, and the greater number of trees and shrubs. In this case the line traced from leaf to leaf will pass twice round the stem before it reaches a leaf situated directly over any below (Fig. 189). Here the sixth leaf is over the first; the leaves stand in five perpendicular ranks, with equal angular distance from each other; and this distance between any two successive leaves is just two fifths of the circumference of the stem. [Illustration: Fig. 188. Shoot with its leaves 5-ranked, the sixth leaf over the first; as in the Apple-tree.] [Illustration: 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.] [Illustration: 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.