636 ON LEAF-ARKANGEMENT [en. 



elongated fir-cone ; or, on the other hand, we may be more 

 attracted by, and regard as of greater importance, the logarithmic 

 spirals which we trace in the curving rows of florets in the discoidal 

 mflorescence of a sunflower. Whether one way or the other be 

 the better, or even whether one be not positively correct and the 

 other radically wrong, has been vehemently debated. In my 

 judgment they are, both mathematically and biologically, to be 

 regarded as inseparable and correlative phenomena. 



The helical arrangement (as in the fir-cone) was carefully 

 studied in the middle of the eighteenth century by the celebrated 

 Bonnet, with the help of Calandrini, the mathematician. Memoirs 

 published about 1835, by Schimper and Braun, greatly amplified 

 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*. 



Starting from some given level and proceeding upwards, let 

 us mark the position of some one leaf {A) upon a cylindrical stem. 

 Another, and a younger leaf (B) will be found standing at a certain 

 distance around the stem, and a certain distance (dong the stem, 



* Besides papers referred to below, and many others quoted in Sach's Botany 

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

 Tiber die Ordnung der Schuppen an den Tannenzapfen, etc., Verh. Car. Leop. 

 Akad. XV, pp. 199-401, 1831; Dr C. Schimper's Vortrage iiber die Moghchkeit 

 eines wissenschafthchen Verstandnisses der BlattsteUung, etc., Flora, xvm, 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 curviseriees, A^ui. Sci. Nat. (2), 

 vn, pp. 42-110, 1837; Sur la disposition symmetrique des inflorescences, ibid., 

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

 rectiseriees, ibid, xii, pp. 5-41, 65-77, 1839; Zeising, Normalverlidltniss der 

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

 Ueber den Quincunx als Gesetz der BlattsteUung bei Sigillaria, etc., Neue-s Jahrb. 

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

 Henslow, G., Phyllotaxis, London, 1871; Wiesner, Bemerkungen iiber rationale 

 nnd irrationale Divergenzen, Flora, Lvni, pp. 113-115, 139-143, 1875; Airy, H., 

 On Leaf Arrangement, Proc. R. S. xxi, p. 176, 1873; Schwendener, S., Mechanische 

 Theorie der BlattsteUung en, Leipzig, 1878; Delpino, F., Causa meccanica della 

 filotassi quincunciale, Genova, 1880; de Candolle, C., Etude de Phyllotaxie, Geneve, 

 1881. 



