leaves sometimes occurs as in Bryophyllum, and many plants of the order Gesneraceae. The leaf of Venus's liy-trap (Dionaea muscipula) when cut off and placed in damp moss, with a pan of water underneath and a bell-glass for a cover, has produced buds from which young plants were obtained. Some species of saxifrage and of ferns also produce buds on their leaves and fronds. In Nyrrtphaea micrantha buds appear at the upper part of the petiole. Leaves occupy various positions on the stem and branches, and have received different names according to their situation. Thus leaves arising from the crown of the root, as in Phyllo- . . h h ax, s the primrose, are called radical, t ose on the stem are two leaves; the third leaf being placed vertically over the first, and the divergence between the first and second leaf being one-half the circumference of a circle, 36O°>< é = I8O°. Again, in a tristichous arrangement the number is § , 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 in many cases such a mode of expression is of excellent service in enabling us readily to understandcauline; on flower-stalks, floral leaves (see FLOWER). The first leaves developed are known as seed leaves or cotyledons. The arrangement of the leaves on the axis and its appendages is called phyllotaxis. In their arrangement leaves follow a definite order. The points on the stern at which leaves appear are called nodes; the part of the stem between the nodes is the inter node. Vi/hen two leaves are produced at the same node, one on each side of the stem or axis, and at the same level, they are opposite (fig. 29); when more than two are produced they are verticillate, and the circle of leaves is then called a verticil or whorl. When leaves are opposite, each successive pair may be placed at right angles to the pair immediately preceding. They are then said to decussate, following thus a law of alternation (fig. 29). The same occurs in the verticillate arrangement, the leaves of each whorl rarely being superposed on those of the whorl next it, but usually alternating so that each leaf in a the relations of the leavesi Tlge divergence, so erepresente, 51 1, = diagrammatically on a 3 "('&' > horizontal projection of ' /, -=' the vertical axis, as in 3 i " fig. 33. Here the outer- //, ;, f Q ° 2 most circle represents a “i <, ' ', 3 SH., f section of that portion » . ° i, of the axis bearing the ' ; l /if f' 2 lowest leaf, the inner- L ' ii most represents the X Q, g1'.-Q ¢ UA highest. The brofad xl ' 5 dark lines represent the / ' . ”~ f leaves, and they are -it Im numbered according to ', ;, n their age and position. '~ .g“' l It will be seen at once G that the leaves are arranged in orthostichies marked I.-V., and that these divide the circumference into live equal portions. But the divergence between leaf I and leaf 2 is equal to § ths of the circumference, and the same

FIG. 31.-Portion of a branch of a Lime tree, with four leaves arranged in a distichous manner, or in two rows. a, The branch with the leaves numbered in their order, n being the node and rn the inter node; b is a magnified representation of the branch, show-1ng the points of insertion of the leaves and their spiral arrangement, which is expressed by the fraction § , or one turn of the spiral for two inter nodes. a FIG. 29.-A stem with opposite leaves. The pairs are placed at right angles alternately, 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 are placed laterally, and so on. a

FIG. 30.-A stem with alternate leaves, arranged in a pen- tastichous or quincuncial manner. The sixth leaf is directly above the first, and commences the second cycle. The fraction of the circumference of the stem expressing the di- vergence of the 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. 30). A plane passing through the point of insertion of the leaf in the node, dividing the leaf into 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. is the case between 2 and 3, 3 and 4, &c. The divergence, then, is 2, and from this we learn that, starting from any leaf on the axis, we must pass twice round the stem in a spiral through five leaves before reaching one directly over that with which we started. The line which, winding round an axis either to the right or to the left, passes through the points of insertion of all the leaves on the axis is termed the genetic or generating spiral; a nd that margin of each leaf which is towards the direction from which the spiral proceeds is the kathodic side, the other margin facing the point whither the spiral passes being the anodie side., In cases where the inter nodes leaves is two-On every axis there are usually fifths. two or more orthostichies. In fig. 31, leaf 1 arises from a node n; leaf 2 is separated from it by an inter node ni, and is placed to the right or left; while leaf 3 is situated directly above leaf I. 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 latter case the 7th leaf is above the 4th, the 10th above the 7th; also the 5th above the 2nd, the 6th above the 3rd and so on. The size of the angle between the median planes of two consecutive leaves in an alternate arrangement is their diverge me; and it is expressed in fractions of the circumference of the axis which is supposed to be a circle. In a regularly formed straight branch covered with leaves, if a thread is passed from one to the other, turning always in the same direction, a spiral is described, and a certain number of leaves and of complete turns occur before reaching 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 inter nodes in the spiral cycle, the fraction will be found to represent the angle of divergence of the consecutive leaves on the axis. Thus, in fig. 32, ~a, b, the cycle consists of five leaves, the 6th leaf being placed vertically over the 1st, are very short and the leaves are 6 ll 5 6, v closely applied to each other, as W 3 ~ > 5 5 in the house-leek, it is difficult ». '} to trace the generating spiral. 'T .5 Thus, in fig. 34 there are thirteen 'E leaves which are numbered in Q 3 f their order, and five turns of the V47 i * ' i 4 spiral marked by circles in the, I i, ig# centre (358 indicating the arrange- if ment); but this could not be », ' Q detected at once. So also in hr ', ;§ cones (fig. 35), which are com- 1 1? ' 2 posed of scales or modified leaves, 1 q, the generating spiral cannot be : determined easily. But in such 1 'l cases a series of secondary spirals a 1 , , or paras tic hies are seen running parallel with each other both 5 right and left, which to a certain extent conceal the genetic spiral. The spiral is not always constant throughout the whole length of an axis. The angle of divergence may alter either abruptly or gradually, and the phyllotaxis thus becomes very complicated. This change may be brought about by arrest of development, by increased de-FIG. 32.-Part of a branch of a Cherry with six leaves, the sixth being placed vertically over the first, after two turns of the spiral. This is expressed by two-fifths. a, The branch, with the leaves numbered in order; b, a magnified representation of .the branch, showing the points of insertion of the leaves and their spiral arrangement.

the the the the 7th over the 2nd and so on; while the number of turns between 1st and 6th leaf is two; hence this arrangement is indicated by fraction § . In other words, the distance or divergence between first and second leaf, expressed in parts of a circle, is § of a circle or 360°><§ = 144° In fig. 31, a, b, the spiral is § , i.e. one turn and velopment of parts or by a torsion of the axis. The former are exemplified in many Crassulaceae and aloes. The latter is seen well in the screw-pine (Pandanus). In the bud of the screw-pine the leaves are arranged in three orthostichies with the phyllotaxis é, but by torsion the developed leaves become arranged in three strong spiral rows running round the stem. These causes of change in phyllotaxis are also well exemplified in the alteration of an opposite or verticillate arrangement to an alternate, and vice versa; thus the effect of interruption of growth, in causing alternate leaves to become opposite and verticillate, can be distinctly shown in

Rhododendron ponticum. The primitive or generating spiral may