[572] Cf. also the immense range of variation in elks’ horns, as described by Lönnberg, P.Z.S. II, pp. 352–360, 1902.
[573] Besides papers referred to below, and many others quoted in Sach’s Botany and elsewhere, the following are important: Braun, Alex., Vergl. Untersuchung über die Ordnung der Schuppen an den Tannenzapfen, etc., Verh. Car. Leop. Akad. XV, pp. 199–401, 1831; Dr C. Schimper’s Vorträge über die Möglichkeit eines wissenschaftlichen Verständnisses der Blattstellung, etc., Flora, XVIII, pp. 145–191, 737–756, 1835; Schimper, C. F., Geometrische Anordnung der um eine Axe peripherische Blattgebilde, Verhandl. Schweiz. Ges., pp. 113–117, 1836; Bravais, L. and A., Essai sur la disposition des feuilles curvisériées, Ann. Sci. Nat. (2), VII, pp. 42–110, 1837; Sur la disposition symmétrique des inflorescences, ibid., pp. 193–221, 291–348, VIII, pp. 11–42, 1838; Sur la disposition générale des feuilles rectisériées, ibid. XII, pp. 5–41, 65–77, 1839; Zeising, Normalverhältniss der chemischen und morphologischen Proportionen, Leipzig, 1856; Naumann, C. F., Ueber den Quincunx als Gesetz der Blattstellung bei Sigillaria, etc., Neues Jahrb. f. Miner. 1842, pp. 410–417; Lestiboudois, T., Phyllotaxie anatomique, Paris, 1848; Henslow, G., Phyllotaxis, London, 1871; Wiesner, Bemerkungen über rationale und irrationale Divergenzen, Flora, LVIII, pp. 113–115, 139–143, 1875; Airy, H., On Leaf Arrangement, Proc. R. S. XXI, p. 176, 1873; Schwendener, S., Mechanische Theorie der Blattstellungen, Leipzig, 1878; Delpino, F., Causa meccanica della filotassi quincunciale, Genova, 1880; de Candolle, C., Étude de Phyllotaxie, Genève, 1881.
[574] Allgemeine Morphologie der Gewächse, p. 442, etc. 1868.
[575] Relation of Phyllotaxis to Mechanical Laws, Oxford, 1901–1903; cf. Ann. of Botany, XV, p. 481, 1901.
[576] “The proposition is that the genetic spiral is a logarithmic spiral, homologous with the line of current-flow in a spiral vortex; and that in such a system the action of orthogonal forces will be mapped out by other orthogonally intersecting logarithmic spirals—the ‘parastichies’ ”; Church, op. cit. I, p. 42.
[577] Mr Church’s whole theory, if it be not based upon, is interwoven with, Sachs’s theory of the orthogonal intersection of cell-walls, and the elaborate theories of the symmetry of a growing point or apical cell which are connected therewith. According to Mr Church, “the law of the orthogonal intersection of cell-walls at a growing apex may be taken as generally accepted” (p. 32); but I have taken a very different view of Sachs’s law, in the eighth chapter of the present book. With regard to his own and Sachs’s hypotheses, Mr Church makes the following curious remark (p. 42): “Nor are the hypotheses here put forward more imaginative than that of the paraboloid apex of Sachs which remains incapable of proof, or his construction for the apical cell of Pteris which does not satisfy the evidence of his own drawings.”
[578] Amer. Naturalist, VII, p. 449, 1873.
[579] This celebrated series, which appears in the continued fraction
etc. and is closely connected with the Sectio aurea or Golden Mean, is commonly called the Fibonacci series, after a very learned twelfth century arithmetician (known also as Leonardo of Pisa), who has some claims to be considered the introducer of Arabic numerals into christian Europe. It is called Lami’s series by some, after Father Bernard Lami, a contemporary of Newton’s, and one of the co-discoverers of the parallelogram of forces. It was well-known to Kepler, who, in his paper De nive sexangula (cf. supra, p. 480), discussed it in connection with the form of the dodecahedron and icosahedron, and with the ternary or quinary symmetry of the flower. (Cf. Ludwig, F., Kepler über das Vorkommen der Fibonaccireihe im Pflanzenreich, Bot. Centralbl. LXVIII, p. 7, 1896). Professor William Allman, Professor of Botany in Dublin (father of the historian of Greek geometry), speculating on the same facts, put forward the curious suggestion that the cellular tissue of the dicotyledons, or exogens, would be found to consist of dodecahedra. and that of the monocotyledons or endogens of icosahedra (On the mathematical connexion between the parts of Vegetables: abstract of a Memoir read before the Royal Society in the year 1811 (privately printed, n.d.). Cf. De Candolle, Organogénie végétale, I, p. 534).