XIV] OR PHYLLOTAXIS 915 



which Moseley gave to the spirals of his moUuscan shells*. But in 

 the sunflower, to judge by the eye, the spirals remain self-similar 

 as they grow; each fresh increment forms, or seems to form, a 

 gnofnon to what went before ; each new floret falls into hne as part 

 of a continuous and self-§imilar curve : and this goes a long way to 

 justify our use of the famihar term logarithmic, or equiangular spiral. 

 But the leaf-arrangement or the inflorescence are far less simple 

 than the shell. The shell grew as one continuous and indivisible 

 whole; its tip is the oldest part, it remains the smallest part, and 

 the spiral tube expands continuously as it goes on. But each floret 

 of the sunflower has its own separate and individual growth; the 

 oldest is also the largest, and the youngest is the least; and as 

 younger and younger florets are added on, the spiral advances in 

 the direction of its own focus, or its own httle end. And the con- 

 ditions may be less simple still in other cases, as in the fir-cone 

 itself. 



The spiral tesselation of th-e fir-cone was carefully studied in the 

 middle of the eighteenth century by the celebrated Bonnet, with the 

 help of Calandrini the mathematician. Memoirs pubHshed about 1 835, 

 by Schimper and Braun, greatly amphfied Bonnet's investigations, 

 and introduced a nomenclature which still holds its own in botanical 

 textbooks. Naumann and the brothers Bravais are among those 

 who continued the investigation in the years immediately following, 

 and Hofmeister, in 1868, gave an admirable account and summary 

 of the work of these and many other writers |. 



* Thus Dr A. H. Church, in his Interpretation of Phyllotaxis Phenomena, 1920, 

 p. 3, begins by saying that "angular measurements on actual plant-specimens. . . 

 can never hope to come within a range of accuracy admitting of an error of less 

 than half a degree, while precise mathematical theory soon begins to tabulate 

 minutes and seconds." 



f Besides papers referred to below, and many others quoted in Sachs's Botany 

 and elsewhere, the following are important: Alex. Braun, Vergl. Untersuchung 

 liber die Ordnung der Schuppen an den Tannenzapfen, etc., Nova Acta Acad. Car. 

 Leop. XV, pp. 199-401, 1831; C. F. Schimper's Vortrage iiber die Moglichkeit 

 eines wissenschaftlichen Verstandnisses der Blattstellung, etc., Flora, xvui, 

 pp. 145-191, 737-756, 1835; C. F. Schimper, Geometrische Anordnung der um 

 eine Achse peripherischen Blattgebilde, Verhandl. Schweiz. Ges. 1836, pp. 113-117; 

 L. and A. Bravais, Essai sur la disposition des feuiUes curviseriees, Ann. Sci. Nat. 

 (2), VII, pp. 42-110, 1837; Sur la disposition symetrique des inflorescences, 

 ibid. pp. 193-221, 291-348, vni, pp. 11^2, 1838; Sur la disposition generale des 

 feuilles rectiseriees, ibid, xn, pp. 5-41, 65-77, 1839; Memoire sur la disposition 

 geometrique des feuilles et des inflorescences, Paris, 1838; Zeising, Normalverhdltnisa 



