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 the left, as shown in Fig. 191. For an account of the way in which the character of the phyllotaxy may be deduced from the secondary spirals, see Structural Botany, Chapter IV. [Illustration: Fig. 191. A young plant of the Houseleek, with the leaves (not yet expanded) numbered, and exhibiting the 13-ranked arrangement; and showing secondary spirals.] 193. =Phyllotaxy of Opposite and whorled Leaves.= This is simple and comparatively uniform. The leaves of each pair or whorl are placed over the intervals between those of the preceding, and therefore under the intervals of the pair or whorl next above. The whorls or pairs alternate or cross each other, usually at right angles, that is, they _decussate_. Opposite leaves, that is, whorls of two leaves only, are far commoner than whorls of three or four or more members. This arrangement in successive decussating pairs gives an advantageous distribution on the stem in four vertical ranks. Whorls of three give six vertical ranks, and so on. Note that in descriptive botany leaves in whorls of two are simply called _opposite_ leaves; and that the term _verticillate_ or _whorled_, is employed only for cases of more than two, unless the latter number is specified. [Illustration: Fig. 192. Opposite leaves of Euonymus, or Spindle-tree, showing the successive pairs crossing each other at right angles.] 194. =Vernation or Præfoliation=, the disposition of the leaf-blades in the bud, comprises two things; 1st, the way in which each separate leaf is folded, coiled, or packed up in the bud; and 2d, the arrangement of the leaves in the bud with respect to one another. The latter of course depends very much upon the phyllotaxy, i. e. the position and order of the leaves upon the stem. The same terms are used for it as for the arrangement of the leaves of the flower in the flower-bud. See, therefore, "Æstivation, or Præfloration."