_When leaves are_ =opposite= _the pairs usually alternate._ That is, if one pair stands north and south, the next pair stands east and west. See the box elder shoot, on the left in Fig. 110. _One pair does not shade the pair beneath._ The leaves are in four vertical ranks. [Illustration: FIG. 110.—PHYLLOTAXY OF BOX ELDER, ELM, APPLE.] _There are several kinds of_ =alternate= _arrangement._ In the elm shoot, in Fig. 110, the third bud is vertically above the first. This is true no matter which bud is taken as the starting point. Draw a thread around the stem until the two buds are joined. Set a pin at each bud. Observe that two buds are passed (not counting the last) and that the thread makes one circuit of the stem. Representing the number of buds by a denominator, and the number of circuits by a numerator, we have the fraction ½, _which expresses the part of the circle that lies between any two buds_. That is, the buds are one half of 360 degrees apart, or 180 degrees. Looking endwise at the stem, the leaves are seen to be 2-ranked. Note that in the apple shoot (Fig. 110, right) the thread makes two circuits and five buds are passed: _two fifths represents the divergence between the buds_. The leaves are 5-ranked. [Illustration: FIG. 111.—PHYLLOTAXY OF THE POTATO TUBER. Work it out on a fresh long tuber.] _Every plant has its own arrangement of leaves._ For opposite leaves, see maple, box elder, ash, lilac, honeysuckle, mint, fuchsia. For 2-ranked arrangement, see all grasses, Indian corn, basswood, elm. For 3-ranked arrangement, see all sedges. For 5-ranked (which is one of the commonest), see apple, cherry, pear, peach, plum, poplar, willow. For 8-ranked, see holly, osage orange, some willows. More complicated arrangements occur in bulbs, house leeks, and other condensed plants. The buds or “eyes” on a potato tuber, which is an underground stem (why?), show a spiral arrangement (Fig. 111). _The arrangement of leaves on the stem is known as_ =phyllotaxy= (literally, “leaf arrangement”). Make out the phyllotaxy on six different plants nearest the schoolhouse door. In some plants, several leaves occur at one level, being arranged in a circle around the stem. Such leaves are said to be =verticillate=, or =whorled=. Leaves arranged in this way are usually narrow: why? Although a definite arrangement of leaves is the rule in most plants, _it is subject to modification_. On shoots that receive the light only from one side or that grow in difficult positions, the arrangement may not be definite. Examine shoots that grow on the under side of dense tree tops or in other partially lighted positions.