Unit III: The Logarithmic Spiral, Hero’s Gnomon, Foraminifera, and Horns — Summary

Analytical Synopsis — 1917 Cambridge First Edition — Chapters XI–XIII (pp. 493–634)

Overview of Unit III (Chapters XI–XIII)

Unit III presents verbatim the geometric centerpiece of On Growth and Form—Chapter XI (The Logarithmic Spiral, pp. 493–586)—together with its companion chapters on Foraminifera (Chapter XII, pp. 587–611) and Horns, Teeth, and Torsion (Chapter XIII, pp. 612–634):

  1. Chapter XI — The Logarithmic Spiral (pp. 493–586):

    • Contrasts the uniform arithmetic spiral of Archimedes ($r = a\theta$, constant whorl width, changing shape; archimedes:spirals) with the geometric or equiangular spiral of Descartes and Jakob Bernoulli ($r = a e^{\theta \cot\alpha}$, constant tangent angle $\alpha$, geometrically increasing whorl widths; bernoulli:1692).

    • Theory of the Gnomon (pp. 508–514): Recovers Aristotle's (Categories 15a30, aristotle:cat) and Hero of Alexandria's (Definitiones 58, hero:def) doctrine of the gnomon: whenever dead material (shell, horn, tooth, claw) grows by terminal accretion without remodeling older parts, preservation of geometric similarity requires each increment to be a gnomon to the existing form, generating an equiangular spiral.

    • Illustrates gnomonic subdivision on the $\sqrt{2}$ rectangle (Fig. 251), the whirling-square golden rectangle ($1 : \phi$, Fig. 252), and Euclid's IV.10 golden triangle ($36^\circ\text{–}72^\circ\text{–}72^\circ$, Fig. 254).

    • Formulates the 3D helicospiral geometry of turbinated gastropods, bivalves, ammonite sutures, and Nautilus septa in three parameters $(\alpha, \beta, \gamma)$ building upon Canon Henry Moseley (moseley:1838).

  2. Chapter XII — The Spiral Shells of the Foraminifera (pp. 587–611): Shows how discrete pulsatile protoplasmic extrusions harden by surface-tension equilibrium into linear (Nodosaria), textularian, and logarithmic (Rotalia, Globigerina, Nummulites) chamber sequences.

  3. Chapter XIII — The Shapes of Horns, and of Teeth or Tusks: with a Note on Torsion (pp. 612–634): Extends unbalanced axial growth gradients and spatial torsion ($\tau$) to sheep and antelope horns, narwhal tusks, rodent incisors, nails, and beaks.