ARRANGEMENT OF LEAVES. The distribution of leaves on the stem has given rise to a subject called phyllotaxy, which undertakes to study the laws which govern the distribution. Tht general conclusion reached is that leaves are distributed so as to economize space and to ob tain a light exposure, hut this is to be regarded as the result of the arrangement rather than its cause. The most fundamental classification of leaves on the basis of arrangement is into the cyclic and spiral arrangements. In the former, two or more leaves stand together at the same joint (node) of the stein, dividing the circum ference between them. If the cycle consists of two leaves they are called opposite. while if it consists of three or more they are called whorled or vertieillate. In the spiral arrange ments the leaves stand singly one after an other—that is. each point of the stem bears but a single leaf, and they are commonly spoken of as alternate. It is the spiral arrangement which has developed the largest discussion in reference to the laws of phyllotaxy, for the cyclic arrangement represents merely two or more spirals ascending the stem. In the simplest alter nate arrangement the second leaf stands upon the opposite side of the stem from the first, and the third leaf stands directly over the first. This results in two vertical rows of leaves, one on each side of the stem, an arrangement indicated by the fraction one-half. The whole fraction signi
fies the angular divergence between two successive leaves, the denominator the number of vertical rows. The next higher arrangement is one in which the angular divergence between two suc cessive leaves is one-third of the circumference, and as a consequence the leaves occur in three vertical rows, and the fractional expression is one-third. The next higher arrangement is indi valet] by the fraction two-fifths, which means that the angular divergence is two-fifths of the circum ference of the stem, that there are five vertical rows, and that the spiral line makes two turns around the stem before it reaches the same vertical row with which it started. The curious feature of the system appears at this point. Succeeding fractions may be obtained by adding the numerators and denominators of the two pre ceding fractions. For example, the fraction which follows the one-half and one-third arrange ments is two-fifths, and the next would be three eighths, and so on. The higher numbers, such as five-thirteenths, eight twenty-firsts, etc., oc cur in certain pine-cones, but in ordinary foliage leaves the lower numbers of the series are the common ones. It is often difficult to determine the normal arrangement, since the stem axis is not always perfectly straight in its growth.