DISTRIBUTION OF LEAVES ON THE STEM. 121 



straight equidistant vertical ranks are produced. In ternate or 

 trimerous whorls there are six vertical ranks ; in quaternate or 

 tetramerous whorls, eight vertical ranks, and so on. 1 



236. The cases in which successive 

 pairs of leaves do not decussate at right 

 angles, or the members of whorls are not 

 exactly superposed to intervals, but as it 

 were wind spirally (as in Dipsacus, man}- 

 Caryophyllacese, &c.) , may some of them 

 be explained by torsion of the stem, 

 such as is very manifest in numerous in- 

 stances ; and others may be resolved into 



instances of alternate leaves simulating or passing into whorls 

 by the non-development of internodes. 2 



237. Alternate or Spiral Arrangement. Here the leaves are 

 distributed singly at different heights of the stem, and at equal 

 intervals as respects angular divergence. (Fig. 236.) This angu- 

 lar divergence (i. e. the angular distance of any two successive 

 leaves) differs in the various kinds of this system of phyllotaxy, 

 but is alwa}*s large enough to place the leaves which immediately 



1 These vertical ranks have, by some German botanists, been named 

 Orthostichies ; but this technical Greek is no clearer and no shorter than the 

 equivalent English, which answers every purpose. 



2 In Lilium Canadense, superbum, &c., with whorls of variable number 

 of leaves and vague relation to each other (when of the same number some- 

 times the members superposed), and above and below passing into the alter- 

 nate arrangement normal to the family, these whorls are evidently formed 

 of alternate leaves brought together by non-development of internodes. 



Here may also be mentioned the not uncommon anomaly in Fir-cones, 

 notably those of Norway Spruce, the normal phyllotaxy of which is simply 

 spiral, but in occasional instances the cone is composed of pairs of opposite 

 scales, spirally arranged, i. e. the pairs not decussating at right angles, thus 

 forming double spirals. In the abnormal spruce-cones, the fractions usually 

 observed are f$ or ^, or, as expressed by Braun, (i)-j^ and ()^. 



Braun's mode of notation for the ordinary succession (t. e. the decussation ) 

 of opposite leaves is (|)i, the ^ meaning that the two leaves of the pair are 

 half the circumference of the circle apart, the denoting that each leaf of 

 the succeeding pair diverges one fourth of the circumference from the pre- 

 ceding. Braun finds cases in which pairs (and equally whorls) are super- 

 posed (e.g. certain species of Mesembryanthemum and Euphorbia), these 

 are expressed in this notation by the formula (), that is, the corresponding 

 leaves of the succeeding pair diverge 180 from their predecessors. He 

 recognizes also some cases of intermediate divergence; such as ()$ in the 

 upper leaves of Mercurialis perennis, (-J-)-^ on certain stems of Linaria vul- 

 garis, (i)^ exceptionally in the leaves of Epilobium angustifolium and the 

 scales of Norway Spruce, (i)-j 8 ^ exceptionally in the scales of Norway 

 Spruce. See Ordnung des Schuppen an der Tannenzapfen, 376, &c. 



FIG. 239. Ground-plan diagram of six trimerous whorls, showing their alternation. 



