and this distance between any two successive leaves is
just two fifths of the circumference of the stem.

[Illustration: Fig. 188. Shoot with its leaves 5-ranked, the sixth leaf
over the first; as in the Apple-tree.]

[Illustration: Fig. 189. Diagram of this arrangement, with a spiral line
drawn from the attachment of one leaf to the next, and so on; the parts
on the side turned from the eye are fainter.]

[Illustration: Fig. 190. A ground-plan of the same; the section of the
leaves similarly numbered; a dotted line drawn from the edge of one leaf
to that of the next marks out the spiral.]

189. The five-ranked arrangement is expressed by the fraction 2/5. This
fraction denotes the divergence of the successive leaves, i. e. the
angle they form with each other: the numerator also expresses the number
of turns made round the stem by the spiral line in completing one cycle
or set of leaves, namely, two; and the denominator gives the number of
leaves in each cycle, or the number of perpendicular ranks, namely,
five. In the same way the fraction 1/2 stands for the two-ranked mode,
and 1/3 for the three-ranked: and so these different sorts are expressed
by the series of fractions 1/2, 1/3, 2/5. Other cases follow in the same
numerical progression, the next being the

190. =Eight-ranked= arrangement. In this the ninth leaf stands over the
first, and three turns are made around the stem to reach it; so it is
expressed by the fraction 3/8. This is seen in the Holly, and in the
common Plantain. Then comes the

191. =Thirteen-ranked= arrangement, in which the fourteenth leaf is over
the first, after five turns around the stem. The common Houseleek (Fig.
191) is a good example.

192. The series so far, then, is 1/2, 1/3, 2/5, 3/8, 5/13; the numerator
and the denominator of each fraction being those of the two next
preceding ones added together. At this rate the next higher should be
8/21, then 13/34, and so on; and in fact just such cases are met with,
and (commonly) no others. These higher sorts are found in the Pine
Family, both in the leaves and the cones and in many other plants with
small and crowded leaves. But in those the number of the ranks, or of
leaves in each cycle, can only rarely be made out by direct inspection.
They may be indirectly ascertained, however, by studying the _secondary_
spirals, as they are called, which usually become conspicuous, at least
two series of them, one turning to the right and one to