On the Equilibrium of Planes — Analytical Summary¶
Scope: Heiberg Vol. II, pp. 142–241 · Heath (1897), pp. 189–220 · Book I (7 Postulates, Props. 1–15) & Book II (Props. 1–10)
Deductive Architecture of Rational Mechanics¶
On the Equilibrium of Planes (Ἐπιπέδων ἰσορροπιῶν) is the founding treatise of theoretical mechanics and barycentric geometry. Without appealing to dynamic virtual velocities (as in the Peripatetic Mechanica), Archimedes axiomatizes static equilibrium from pure symmetry, monotonicity, convexity, and Eudoxean proportion:
Symmetry to Commensurable Lever (I.Post.1–7 → I.1–6): By splitting two commensurable weights A, B into and equal unit weights distributed uniformly along a beam of total length , Archimedes reduces unequal reciprocal leverage directly to the midpoint symmetry of a single homogeneous rod!
Eudoxean Extension to Irrationals (I.7): Using Euclid V. Def. 4 (euclid:elements), Archimedes extends the Law of the Lever to incommensurable ratios (, , etc.) via rational insertion.
Rectilinear Barycenters (I.8–15): Derives in strict sequence the remainder theorem (I.8), the parallelogram diagonal intersection (I.9–10), the triangle centroid
X(2)as concurrency of the medians (I.13–14), and the trapezium centroid formula (I.15).Self-Similar Parabolic Centroids (II.1–10): Exploiting the affine self-similarity of parabolic segments (II.7), Book II solves for the parabolic centroid algebraically without needing a fresh infinite summation of moments—a breathtaking anticipation of functional equations and renormalization!
Canonical Centroid Ledger¶
Plane Figure |
Archimedes Locus |
Exact Barycentric Locus (κέντρον τοῦ βάρεος) |
|---|---|---|
Two Equal Masses |
Book I, Prop. 4 |
Midpoint of joining segment () |
General Lever (A, B) |
Book I, Props. 6–7 |
Divides DE inversely: |
Parallelogram ABCD |
Book I, Props. 9–10 |
Intersection of diagonals |
Triangle ABC |
Book I, Props. 13–14 |
Concurrency of medians , , barycentric |
Trapezium () |
Book I, Prop. 15 |
On midpoint join EF with |
Parabolic Segment |
Book II, Prop. 8 |
On diameter AO with () |
Parabolic Frustum |
Book II, Prop. 10 |
Exact rational expression in the two parallel diameter intercepts |