before we find one vertically above the one from which we started, and if we measure the horizontal distance from any leaf to the next above or below it, it will be found to equal two-fifths of the total circumference, so that we have to go five times two-fifths way round the stem, or two complete revolutions, before completing the cycle. This is called a two-fifths phyllotaxis. In many other cases, the arrangement is immensely more complicated, and need not be entered on here. What is important for us to note at present is that by means of this orderly mathematical arrangement, the leaves are so distributed that each fulfils its functions to the best advantage. The shape of leaves offers an almost inexhaustible field for observation and scientific speculation. Mr. Ruskin has said: “The leaves of the herbage at our feet take all kinds of strange shapes, as if to invite us to examine them. Star-shaped, heart-shaped, spear-shaped, arrow-shaped, fretted, fringed, cleft, furrowed, serrated, sinuated, in whorls, in tufts, in spires, in wreaths, endlessly expressive, deceptive, fantastic, never the same from footstalk to blossom, they seem perpetually to tempt our watchfulness and take delight in outstripping our wonder.” The size of leaves will naturally vary inversely as their number. A plant of a certain size--say a tree--will require a certain total area of leaf for the manufacture of the requisite amount of plant-food. If we cut the branch of a horse chestnut and of a beech where each had exactly a diameter of one inch, or two, or six inches, and counted and measured the leaves on each, while the number of beech leaves would immensely exceed the number of chestnut leaves the total leaf-area would be about the same in each case. This area of green leaf, then, must be spread out to the best advantage. In this connection, a beautiful relation between the shape of leaves and their arrangement on the stem may frequently be remarked. Lay a twig of beech on a sheet of white paper, and note how small are the interstices between the leaves through which the paper may be seen. The shape of the leaves, and the intervals at which they are borne, are so related that an almost continuous expanse of green is offered to the sunlight. A more remarkable case may be seen in the lime, whose leaves are quite inequilateral, being contracted on one side at the base and expanded at the other, in order the more exactly to fill the space which is available. The elm likewise furnishes a beautiful example of close-fitting leaves. In most trees in which, like the beech, hazel, and elm, the leaves lie in close-ranked rows in the same plane as the twig which supports them, we find more or less oval leaves, their breadth varying with the space between the leaves, _i. e._, the length of the internode. In trees such as the horse chestnut or sycamore, on the other hand, the leaves grow in opposite pairs, and are typically arranged on upright twigs, the leaf-stems projecting at