The Method of Mechanical Theorems — Philological & Mathematical Notes¶
1. From Lost Work (1897) to Palimpsest Recovery (1906–1912)¶
Comparing heath:1897 with heath:1912 captures one of the greatest dramatic arcs in the history of scholarship:
In 1897 (heath:1897, p. xxxviii, verified from 300 DPI scan), Heath could list The Method (Ἐφόδιον) only under Lost Works, known solely from a single sentence in Suidas stating that Theodosius wrote a commentary upon it!
Just months later (1899/1903), Hermann Schöne published the newly discovered Constantinople codex of Heron's Metrica (heron:metrica), which quoted verbatim the two Method theorems on the cylindrical ungula and bicylinder!
In 1906, alerted by Papadopoulos-Kerameus’s catalogue of the Metochion of the Holy Sepulchre in Constantinople, Johan Ludvig Heiberg inspected Codex C—a thirteenth-century Euchologion prayer-book overwritten atop a tenth-century mathematical parchment—and deciphered The Method (heiberg:1907), prompting Heath’s 1912 Cambridge Supplement (heath:1912). (Leaf gaps at the very end of Prop. 15 were restored independently by Zeuthen and Heiberg from the Preface and Heron’s citation.)
2. Democritus’s Dilemma and Noetic Epistemology¶
Archimedes’ tribute to Democritus and Eudoxus in the Preface illuminates Plutarch’s report (De communibus notitiis 39) of Democritus’s atomic cone paradox: if a cone is sliced parallel to its base, are adjacent circular sections equal (making the cone a cylinder) or unequal (making the cone stepped)? By separating heuristic slice-weighing (ἔνδειξις) from finite-step Eudoxean exhaustion (ἀπόδειξις), Archimedes reaps the full creative power of Cavalieri indivisibles without sacrificing Aristotelian-Euclidean continuum rigor—resonating deeply with geometor/aristotle and geometor/hoenen on geometric intuition (φαντασία) versus demonstration.
3. GEOMETOR Cross-Corpus Concordance¶
Euclid (`geometor/euclid`) (geometor:euclid): Directly satisfies both
heath:method-of-archimedesandarchimedes:methoduscited in Heath’s Euclid Vol. III (Introduction to Book XII on Democritus and Eudoxus’s cone/pyramid proofs, Eucl. XII.7 & XII.10).Curved-to-Rectilinear Exact Cubatures: Notice that both Prop. 14 () and Prop. 15 () achieve rational cubatures of curved solids free of , prefiguring Hippocrates’ lunes in 3D and Steinmetz/Zu Chongzhi’s spherical-square cross-sections!