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 understand the relations of the leaves. The divergences may also be represented diagrammatically on a horizontal projec tion of the vertical axis, as in fig. 21. 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 be seen at once that the leaves are ar ranged in orthostichies marked I–V, and that these divide the cir cumference into five equal portions. But the divergence between leaf i and leaf 2 is equal to of the circumference, and the same is the case between 2 and 3, 3 and 4, etc. The divergence, then, is 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 spiral. In cases where the internodes are very short and the leaves are closely applied to each other, as in the house-leek, it is difficult to trace 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 development of parts or by a torsion of the axis.
In dicotyledonous plants the first leaves produced (the coty ledons) are opposite. This arrangement often continues during the life of the plant, but at other times it changes, passing into distichous and spiral forms. Some tribes of plants are distin guished by the opposite or verticillate, others by their alternate, leaves. Labiate plants have decussate leaves, while Boraginaceae have alternate leaves, and Tiliaceae usually have distichous leaves; Rubiaceae have opposite leaves. Such arrangements as -I, A and are common in Dicotyledons. In monocotyledonous plants there is only one seed-leaf or cotyledon, and hence the arrange ment is at first alternate ; and it generally continues so more or less, rarely being verticillate. Such arrangements as 1, and 3 are common in Monocotyledons, as in grasses, sedges and lilies. It has been found in general that, while the number 5 occurs in the phyllotaxis of Dicotyledons, 3 is common in that of Monocoty ledons.
nature, serving a temporary purpose, and then falling off when the leaf is expanded. They are frequently covered with a resinous matter, as in balsam-poplar and horse-chestnut, or by a thick downy covering as in the willow. In plants of warm climates the buds have often no protective apparatus, and are then said to be naked.
The arrangement of the leaves in the bud is termed vernation or prefoliation. In considering vernation we must take into account both the manner in which each individual leaf is folded and also the arrangement of the leaves in relation to each other. These vary in different plants, but in each species they follow a regular law. The leaves in the bud are either placed simply in apposition, or they are folded or rolled up longitudinally or laterally, giving rise to different kinds of vernation, as delineated in figs. 22 to 31, where the folded or curved lines represent the leaves, the thickened part being the midrib. The leaf taken individually is either folded longitudinally from apex to base, as in the tulip-tree, and •called reclinate or replicate; or rolled up in a circular manner from apex to base, as in ferns (fig. 22), and called circinate ; or folded lat erally, conduplicate (fig. 23), as in oak; or it has several folds like a fan, plicate or plaited (fig. 24), as in vine and sycamore, and in leaves with radiating vernation, where the ribs mark the foldings; or it is rolled upon itself, convolute (fig. 25), as in banana and apricot ; or its edges are rolled inwards, involute (fig. 26), as in violet ; or outwards, revolute (fig. 27), as in rosemary. The differ ent divisions of a cut leaf may be folded or rolled up separately, as in ferns, while the entire leaf may have either the same or a different kind of vernation. The leaves have a definite relation to each other in the bud, being either opposite, alternate or verticil late ; and thus different kinds of vernation are produced. Some times they are nearly in a circle at the same level, remaining flat or only slightly convex externally, and placed so as to touch each other by their edges, thus giving rise to valuate vernation. At other times they are at different levels, and are applied over each other, so as to be imbricated, as in lilac, and in the outer scales of sycamore ; and occasionally the margin of one leaf overlaps that of another, while it in its turn is overlapped by a third, so as to be twisted, spiral or contortive. When leaves are applied to each other face to face, without being folded or rolled together, they are appressed. When the leaves are more completely folded they either touch at their extremities and are accumbent or opposite (fig. 28), or are folded inwards by their margin and become induplicate; or a conduplicate leaf covers another similarly folded, which in turn covers a third, and thus the vernation is equitant (fig. 29), as in privet; or conduplicate leaves are placed so that the half of the one covers the half of another, and thus they be come half-equitant (fig. 3o), as in sage. When in the case of convolute leaves one leaf is rolled up within the other, it is super volute (fig. 31). The scales of a bud sometimes exhibit one kind of vernation and the leaves another. The same modes of ar rangement occur in the flower-buds.