Summary — Metrica Book III (Preface & Props. 1–23)

Structural Overview of Proportional Divisions of Planar Estates & Three-Dimensional Solids

1. Deductive Architecture of Metrica Book III

Whereas Books I and II compute a scalar measure (area or volume) from given dimensions, Book III solves the inverse constructive problem: given a figure and a target ratio \(m : n\), construct the dividing line or plane with zero geometric error (“not exceeding or falling short by a single millet-seed”):

  1. Philosophical Prooemium: Grounds cadastral division in classical Greek ethics—superposition (ἐφαρμογή) establishes arithmetic equality, geometric proportion realizes distributive justice (δικαιοσύνη), and Euclidean proof renders the boundary indisputable.

  2. Triangle Divisions (Props. III.1–III.3): Divides any triangle in ratio \(m : n\) from a vertex (III.1, linear base division), parallel to a base (III.2, duplicate ratio / mean-proportional quadrance \(AD^2 = \frac{m}{m+n} AB^2\)), or from an arbitrary side point (III.3, reciprocal side product).

  3. Quadrilaterals, Polygons, Annuli & Exterior Points (Props. III.4–III.19): Solves parallel trapezium cuts via root-mean-square quadrance (III.5), concentric circle cuts (III.9), and straight-line cuts constrained to pass through an exterior landmark (III.10–III.13).

  4. Stereometric Divisions & Cube-Root Extraction (Props. III.20–III.23): Divides pyramids, cones, and frusta by parallel planes via triplicate ratios (III.20–III.22), introduces Hero’s cube-root algorithm (\(\sqrt[3]{100} \approx 4\frac{9}{14}\)), and concludes with Dionysodorus's conic solution for dividing a sphere (III.23).

2. Canonical Division Laws across Dimensions

Scaling Laws of Proportional Division (Ratio \(m : n\), Fraction \(\lambda = \frac{m}{m+n}\))

Dimension / Configuration

Exact Constructive Law

Metrica III Exemplar

1D / Common-Altitude (III.1)

\(BV = \lambda\,BC\)

\(\frac{5}{8} \times 14 = 8\frac{3}{4}\)

2D Similar Triangle (III.2)

\(AD^2 = \lambda\,AB^2\) (quadrance linear!)

\(\frac{3}{4} \times 169 = \frac{507}{4} = 126\frac{3}{4}\)

2D Parallel Trapezium (III.5)

\(x^2 = (1 - \lambda)a^2 + \lambda b^2\)

Weighted mean of base quadrances

3D Pyramid / Cone (III.20)

\(x^3 = \lambda\,L^3\)

\(\frac{4}{5} \times 125 = 100 \implies x \approx 4\frac{9}{14}\)

3D Frustum (III.21–22)

\(x^3 = (1 - \lambda)a^3 + \lambda b^3\)

Weighted mean of base cubes