Unit IV: On Leaf-Arrangement, or Phyllotaxis — Summary¶
Analytical Synopsis — 1917 Cambridge First Edition — Chapter XIV (pp. 635–651, Complete Verbatim Text)
Overview of Unit IV (Chapter XIV — Complete Verbatim Text)¶
Unit IV presents the complete, unabridged 1917 text of Chapter XIV: On Leaf-Arrangement, or Phyllotaxis (pp. 635–651), the direct historical companion to geometor/phyllotaxis:
Cylindrical vs. Discoidal Formulations (pp. 635–638): Unifies the cylindrical helical view of Schimper, Braun, and the brothers Bravais (bravais:1837, fir-cones and palm stems) with the planar logarithmic-spiral capitulum view of A. H. Church (church:1904, sunflower heads).
The Schimper–Braun Series & Continued Fractions (pp. 638–642): Analyzes the principal fractional divergences:
$$\frac{1}{2},\; \frac{1}{3},\; \frac{2}{5},\; \frac{3}{8},\; \frac{5}{13},\; \frac{8}{21},\; \frac{13}{34}, \dots$$
generated by the unit continued fraction $\frac{1}{2 + \frac{1}{1 + \frac{1}{1 + \dots}}}$, converging to $\phi^{-2} = \frac{3 - \sqrt{5}}{2} \approx 0.381966$ (the golden divergence angle $137^\circ 30' 28''$), alongside accessory Lucas/secondary series ($\frac{1}{4}, \frac{1}{5}, \frac{2}{9}\dots$).
P. G. Tait’s Cylindrical Parastichy Lattice (pp. 643–646, Fig. 325): Reproduces Peter Guthrie Tait's 1872 geometric proof (tait:1872) that whenever a cylindrical point-lattice exhibits two intersecting contact-spiral families (parastichies) of orders $m$ and $n$ winding in opposite directions, the next steeper intermediate family necessarily has order $m + n$—automatically generating the Fibonacci recurrence without assuming a single physical “genetic spiral.”
Critique of Teleology & Mechanical Close-Packing (pp. 646–651): Rejects Darwinian sunlight-optimization myths (noting deep-sea hydroids, subterranean bulbs, and tightly closed pine cones show identical Fibonacci numbers) in favor of Hofmeister's mechanical gap-rule (hofmeister:1868), Schwendener’s mutual contact pressure, and Church’s orthogonal log-spiral growth trajectories.