Unit VI: On the Theory of Transformations and Epilogue — Summary

Analytical Synopsis — 1917 Cambridge First Edition — Chapter XVII & Epilogue (pp. 719–779, Complete Verbatim Text)

Overview of Unit VI (Chapter XVII & Epilogue — Complete Verbatim Text)

Unit VI presents the crowning achievement of On Growth and Form—the complete verbatim 1917 text of Chapter XVII (*On the Theory of Transformations, or the Comparison of Related Forms*, pp. 719–777) and the Epilogue (pp. 778–779):

  1. The Method of Coordinates in Morphology (pp. 719–735):

    • Elevates Descartes's coordinate geometry and Albrecht Dürer's 1528 human-proportion grids (durer:1528) from descriptive measurement to comparative transformation: inscribing a type species in a rectangular net $(x, y)$ and deforming the coordinate lines into $(x', y') = (f(x, y), g(x, y))$ reveals harmonious global laws of differential growth.

    • Classifies canonical deformation families (verified on 300 DPI scans pp. 729–733):

      1. Uniform axial extension $(x, cy)$ turning circles into ellipses (cannon-bones of ox, sheep, giraffe, Fig. 354);

      2. Logarithmic/monotonic gradient spacing (dicotyledon leaf growth around a basal node, Fig. 355–358);

      3. Oblique affine shear $(x + y\cot\omega, y\sin\omega)$ (Fig. 357);

      4. Radial/polar fan coordinates (palmate and pedate leaves, Helleborus, Fig. 360–362);

      5. Coaxial-hyperbolic curvilinear nets.

  2. Comparative Zoological & Palaeontological Transformations (pp. 736–777):

    • Crustacea: Geryon, Corystes, Scyramathia, Paralomis, Lupa, Chorinus carapaces (Figs. 366–371);

    • Teleost Fishes: Argyropelecus Olfersi $\to$ Sternoptyx diaphana via $70^\circ$ oblique shear (Figs. 373–374); Scarus $\to$ Pomacanthus via coaxial circular arcs (Figs. 375–376); Polyprion $\to$ Pseudopriacanthus $\to$ Scorpaena $\to$ Antigonia; and Diodon $\to$ Orthagoriscus mola via coaxial-hyperbolic expansion (Figs. 381–382);

    • Reptiles, Birds & Mammals: Crocodilian skulls, Archaeopteryx $\to$ Apatornis pelves (with Gerhard Heilmann’s interpolated intermediates, Figs. 388–393), Hyrachyus $\to$ Aceratherium rhinoceros skulls, Eohippus $\to$ Mesohippus $\to$ Equus equine crania (Figs. 401–402), and human, chimpanzee, and baboon cranial coordinates (Figs. 404–408).

  3. Epilogue (pp. 778–779): Closes with Thompson’s unforgettable lyric meditation on Form and Number—linking Colin Maclaurin's honeybee, Philolaus the Pythagorean (πάντα γα μὰν τὰ γιγνωσκόμενα), Milton’s golden compasses, James Joseph Sylvester, and Jean-Henri Fabre.