116 BOTANY [NUTRITIVE ORGANS. formed together, and a simultaneous whorl results ; or they may be formed one after the other, as in Characese, when a successive whorl is formed. When leaves are opposite, each successive pair may be placed at right angles to the pair immediately preceding. They are then said to decus sate, following thus a law of alternation. The same occurs V Fig. 126. Fig. 127. Fig. 128. FIG 126 A stem with opposite leaves. The pairs are placed at right angles Rlter- nately! or in what is called a decussate manner. In the lowest pair one leaf is In front, and the other at the back; in the second pair the leaves arc placed Fio* 127. Vertidllate or whorled leaves of a species of Madder. There are five leaves in the verticil or whorl. Fio 128 A stem with alternate leaves, arranged in a pentastjchous or quincun- cial manner The sixth leaf is directly above the first, and commences the second cycle. The fraction of the circumference of the stem expressing the divergence of the leaves is two-fifths. in the verticillate arrangement, the leaves of each whorl rarely being superposed on those of the whorl next it, as on the branches of Chara, but usually alternating so that each leaf in a whorl occupies the space between two leaves of the whorl next to it. There are considerable irregularities, however, in this respect, and the number of leaves in different whorls is not always uniform, as may be seen in Lysimachia vulgaris. When a single leaf is produced at a node, and the nodes are separated so that each leaf is placed at a different height on the stem, the leaves are alternate (fig. 128). A plane passing through the point of insertion of the leaf in the node, dividing the leaf into two similar halves, is the median plane of the leaf ; and when the leaves are arranged alternately on an axis so that their median planes coincide they form a straight row or orthostichy. On every axis there are usually two or more orthostichies. The leaves in such a case are said to be rectiserial. In fig. 129, leaf 1 arises from a node n ; leaf 2 is separated by an internode m, and is placed to the right or left ; while leaf 3 is situated directly above leaf 1. In this case, then, there are two orthostichies, and the arrangement is said to be distichous. When the fourth leaf is directly above the first, the arrangement is tristichous. The same arrangement continues throughout the branch, so that in the lat ter case the 7th leaf is above the 4th, the 10th above the 7th ; also the 5ti above the 2d, the 6th above the 3d, and so on. The size of the angle between the median planes of two consecutive leaves in an alternate arrange ment is their divergence ; and it is expressed in frac tions of the circumference of the axis which is sup posed to be a circle. In a regularly-formed straight branch covered with leaves, if a thread is passed from Fig. 130. One to the Other, turning P art a branch of a Cherry with six leaves, . n . the sixth being placed vertically over the always 111 the Same direc- first, after two turns of the spiral. This is expressed by two-fifths, or the quincunx. a, the branch, with the leaves numbered in order; 6, a magnified representation of the branch, showing the cicatrices of the leaves or their points of insertion, and their spiral arrangement. tion, a spiral is described, and a certain number of leaves and of complete turns occur before reach ing the leaf directly above that from which the enumeration commenced. If this arrangement is expressed by a fraction, the numerator of which indicates the number of turns, and the denominator the number of internodes in the spiral cycle, the fraction will be found to represent the angle of diverg ence of the consecutive leaves on the axis. Thus, in fig. 130, a, 6, the cycle consists of five leaves, the 6th leaf being placed vertically over the 1st, the 7th over the 2d, and so on ; while the number of turns between the 1st and 6th leaf is two ; hence this arrangement is indicated by the fraction ^. In other words, the distance or divergence between the first and second leaf, expressed in parts of a circle, is of a circle, or 360-=--| = 144. In fig. 129, a, b, the spiral is -|, i.e., one turn and two leaves ; the third leaf being placed vertically over the first, and the diver gence between the first and second leaf being one-half the circumference of a circle, 360 -j- ^ = 180. Again, in a tristichous arrangement the number is -j, or one turn and three leaves, the angular divergence being 120. By this means we have a convenient mode of expressing on paper the exact position of the leaves upon an axis. And I in many cases such a mode of expression is of excellent ser vice in enabling us readily to understand the relations of the leaves. The diver gences may also be represented dia- grammatically on a horizontal projection Fig. 131. of the vertical axis, Diagram of a phyllotaxis represented by the fraction . _ , - -- 2-6ths. (Sachs.) asm fig. 131. Here the outermost circle represents a section of that portion of the axis bearing the lowest leaf, the innermost represents the highest. The broad dark lines represent the leaves, and they
are numbered according to their age and position. It will