The Method of Mechanical Theorems — Analytical Summary¶
Scope: Heiberg (1907 / Vol. II², pp. 426–507) · Heath Supplement (1912), pp. 12–51 · Epistle to Eratosthenes, Lemmas 1–11 & Propositions 1–15
Unveiling the Workshop of Antiquity’s Greatest Geometer¶
For over two millennia, Archimedes' published treatises inspired awe and perplexity: how did he guess exact ratios like , , , or before setting up his intricate double-reductio exhaustion proofs? Deciphered by J. L. Heiberg in Constantinople in 1906 from the tenth-century Palimpsest (Codex C), The Method (archimedes:methodus) reveals Archimedes’ private analytic engine:
Cavalieri Cross-Sections + Virtual Lever (:term:`ἔφοδος <ephodion>`): Decompose an unknown area or solid X and a known comparison figure Y into one-parameter families of lower-dimensional slices along axis . Using conic/quadric symptomata, construct a virtual lever arm H beyond the fulcrum A (HA = a or 2r) so that . Moving every slice to H concentrates X at a single point, balancing Y at its known centroid :
\[a \cdot \text{Vol}(X) = \bar{x}_Y \cdot \text{Vol}(Y).\]Methodological Separation of Discovery and Proof: Archimedes draws a razor-sharp philosophical boundary between indication (ἔνδειξις, summing unextended planar/linear indivisibles on a mechanical lever) and synthetic demonstration (ἀπόδειξις, rigorous Eudoxean exhaustion by finite magnitudes).
Register of Propositions 1–15 in The Method¶
Prop. |
Subject |
Mechanical / Geometric Result |
|---|---|---|
1 |
Parabolic Segment Area |
Chords balanced against triangleCFAimpliestextSegmentfrac43triangleABC |
2 |
Sphere Volume & Surface Analogy |
Sphere + Cone slices balance Cylinder |
3 |
Spheroid Cubature |
Spheroid + Cone slices balance Cylinder cylinder |
4 |
Paraboloid Segment Cubature |
Circular slices balance Cylinder cylinder cone |
5 |
Paraboloid Centroid |
Divides axis in ratio from vertex () |
6 |
Hemisphere Centroid |
Divides radius in ratio from pole ( from base centre) |
7–10 |
Spherical & Spheroid Segments |
Volumes and exact axial centroids of arbitrary segments |
11 |
Hyperboloid Segment & Centroid |
Cubature and centroid of obtuse-angled conoid |
12–14 |
Cylindrical Ungula |
Triple proof (mechanical lever + parabolic prisms + exhaustion): prism () |
15 |
Archimedean Bicylinder |
Intersection of two orthogonal cylinders in cube: cube () |