916 ON LEAF-ARRANGEMENT [ch. 



The surface of a pine-cone shews a crowded assemblage of woody 

 scales, close-packed and pressed together in such a way that each 

 has a quadrangular, rhomboidal form*. Each scale forms part of, 

 and marks the intersection of, two hnear series; these run upwards 

 in a spiral course, one in one direction and one in the other, and 

 are called accordingly diadromous si^iials. In the little cones of the 

 Scotch Fir (Pinus silvestris), the whose assemblage of scales may be 

 looked on as forming five linear series, or spiral bands, running side 

 by side the one way, or as eight such series running the other. But 

 these two sets are far from being all the spirals which we can trace 

 upon the cone. Sometimes the packing is closer still, especially if 

 the cone be long and slender. Then each scale tends to come in 

 contact with six others, and so to become roughly hexagonal; we 

 recognise a third spiral series besides the other two, and this new 

 series is found to consist of thirteen rows. But let us disregard for 

 the moment this perplexing phenomenon of a cone composed of so 

 many series of scales, five, eight or thirteen in number as we happen 

 to look at them ; and try to find a single series in which every scale 

 takes part. We are in no way limited to the fir-cone, which is a 

 somewhat special case; but may consider, in a very general way, 

 the case of any leafy stem. 



Starting from some given level and proceeding upwards, let us 

 mark the position of some one leaf (A) upon the cylindrical stem. 



der chemischen und morphologischen Proportionen, Leipzig, 1856; C. F. Naumann, 

 Ueber den Quincunx als Gesetz der Blattstellung bei Sigiilaria, etc., Neues Jahrb. 

 f. Miner. 1842, pp. 410-417; T. Lestiboudois, Phyllotaxie anatomique, Paris, 1848; 

 G. Henslow, Phyllotaxis, or the arrangement of leaves according to mathematical 

 laws, Jl. Victoria Inst, vi, pp.- 129-140, 1873; On the origin of the prevailing 

 systems of Phyllotaxis, Tr. Linn. Soc. (Bot.), i, pp. 37-45, 1880. J. Wiesner, 

 Bemerkungen xiber rationale und irrationale Divergenzen, Flora, Lvin, pp. 113-115, 

 139-143, 1875; H. Airy, On leaf arrangement, Proc. B.S. xxi, p. 176, 1873; 

 S. Schwendener, Mechanische Theorie der Blattstellungen, Leipzig, 1878; F. Delpino, 

 Causa mecca'nica della filotasse quincunciale, Genova, 1880; Teoria generate di 

 Filotasse, ibid. 1883; S. Giinther, Das mathematische Grundgesetz im Bau des 

 Pflanzenkorpers, Kosmos, iv, pp. 270-284, 1879; F. Ludwig, Wichtige Abschnitte 

 aus der mathematischen Botanik, Zeitschr. f. truithem. u. naturw. Unterricht, xiv, 

 p. 161, 1883; Weiteres iiber Fibonacci-Kurven und die numerische Variation der 

 gesammten Bluthenstande der Kompositen, Botan. Chit, lxviii, p. 1, 1896; Alex. 

 Dickson, Phyllotaxis of Lepidodendron and Knossia, Jl. Bot. ix, p. 166, 1871. For 

 a historical account of the earlier literature, see Casimir de Candolle's Considerations 

 generates sur Vetude de la phyllotaxie, Geneve, 1881. 

 * Cf. supra, p. 515. 



