Unit I: Mathematical Morphology, Magnitude, and Rate of Growth — Notes & Concordance¶
Critical Apparatus & GEOMETOR Cross-References — I. Magnitude & Growth
Scholarly Commentary & GEOMETOR Concordance (Unit I)¶
1917 Edition Provenance & Epigraph Architecture¶
1917 First Edition Structure: Verified directly against 300 DPI page scans of the 1917 Cambridge University Press imprint (ongrowthform1917thom), the First Edition opens with Stephen Hales's Vegetable Staticks epigraph preceding the Prefatory Note (dated University College, Dundee, December, 1916) and places the celebrated triad of Aquinas, Newton, and J. H. Jellett on p. xvi immediately facing Chapter I (p. 1). (In the 1942 Second Edition—strictly excluded from this codex—Chapter I was heavily rewritten and split.)
Link to `geometor/aquinas`: Thompson’s quotation from aquinas:depot (“Cum formarum naturalium et corporalium esse non consistat nisi in unione ad materiam…”, which he had earlier introduced in his 1911 Portsmouth British Association address, thompson:1911) anchors mathematical morphology in Aristotelian–Thomistic natural philosophy: physical agents transmuting matter suffice to account for natural configuration without appealing to vitalistic entelechies.
Dimensional Scaling & Euclidean Similitude¶
Link to `geometor/euclid` & `geometor/elements`: Thompson’s dimensional equations $S \propto L^2$ and $V \propto L^3$ rest directly on Euclid's Book VI (Props. 19–20, duplicate ratio of similar polygons) and Book XI (Prop. 33, triplicate ratio of similar parallelepipeds; euclid:elements, heath:1908).
Froude’s Law & Locomotion: Thompson’s remarkable proof (pp. 29–35) that maximal jumping height and stride mechanics remain nearly scale-invariant across flea, kangaroo, and horse anticipates modern biomechanical scaling and dimensional analysis.
Differential Growth as Precursor to Transformations: Chapter III’s demonstration that different anatomical axes possess distinct growth rates ($dx/dt \neq dy/dt$) supplies the exact ontogenetic justification for both Chapter XI’s gnomonic shell spirals (where ratios of marginal growth remain constant) and Chapter XVII’s method of coordinates (where growth gradients vary smoothly across space).
Structured Concordance Loci in This Unit¶
- Mathematical Morphology, Efficient Cause, and Physical Law (Prefatory Note & Ch. I, pp. v–vi, 1–15)
Establishes morphology as a mathematical science of physical forces acting in the present and immediate past of the growing organism. Algebraic Invariants:
F = m a·Dimensional homogeneitygeometor/aquinas (De Potentia Dei, Q. 3, a. 11): Thompson’s opening Latin epigraph affirms that corporeal forms require no extrinsic agent beyond the physical causes transmuting matter.
geometor/aristotle (Physics II & Historia Animalium): Balances Aristotelian formal/teleological morphology with Newtonian and Galilean efficient mechanics.
Dynamic Symmetry Parallel: Dynamic Symmetry: The Greek Vase (1920), Preface & Introduction — recovering mathematical law behind organic and classical artistic form
- Principle of Similitude, Galileo’s Square-Cube Law, and Scale Regimes (Ch. II (On Magnitude) & Bridge Chs. III–IV, pp. 16–49 (bridge pp. 50–200))
Demonstrates why gravity governs terrestrial vertebrates (L³), surface tension and viscosity govern protists and cells (L² and L¹), and Brownian agitation rules bacteria. Algebraic Invariants:
S / S₀ = (L / L₀)²·V / V₀ = (L / L₀)³·S / V ∝ L⁻¹geometor/euclid (Book VI, Def. 1 & Props. 19–20; Book XI, Prop. 33): Duplicate ratio of areas (L²) and triplicate ratio of similar solids (L³) supply the exact geometric basis for Galileo’s scaling bounds.
geometor/elements (Book VI Similitude DAG): Formalizes proportional scaling of similar rectilinear and solid figures.
Dynamic Symmetry Parallel: The Diagonal (1919–1920), Lesson I — invariance of proportional shape versus absolute linear scale