Metrica Book III (Preface & Props. 1–23): Proportional Divisions of Figures, Justice & ∛100¶
Geometric Proportion & Distributive Justice (“Not By a Single Millet-Seed”), Dividing Triangles & Quadrilaterals, Euclid’s On Divisions, Exterior-Point Cuts, Solid Divisions, and Hero’s Cube-Root Algorithm — Schöne (1903), pp. 140–184 · Codex Constantinopolitanus Palatii Veteris 1, fols. 99v–107v
Note
Edition Track (English): Original GEOMETOR scholarly translation and synthesis directly from Hermann Schöne's 1903 Teubner editio princeps (pp. 140–184), verified against 300 DPI archival page scans. Hover or click any highlighted geometric symbol (such as ABC, ADE, DECB, or AV) to illuminate the corresponding element in the interactive diagram. Switch tracks above for Greek (Teubner), Summary, or Notes.
Prooemium: Geometric Proportion, Distributive Justice & the Millet-Seed¶
Book III, Prooemium (Schöne pp. 140–142; fol. 99v):
“We consider the divisions of land-parcels not far removed from the measurements performed upon them. For indeed to apportion an equal region to equals, and a greater share to the worthy according to :term:`proportion <analogia>`, is judged altogether serviceable and necessary. The whole habitable earth itself has been divided by Nature herself according to merit…
And if anyone should wish to divide regions according to a given ratio, so that—so to speak—not even a single millet-seed (*κέγχρον μίαν) exceeds or falls short of the proportion of the given ratio, he will have need of geometry alone: wherein superposition (ἐφαρμογή) yields equality, proportion (ἀναλογία) yields justice (δικαιοσύνη), and the demonstration concerning these things is indisputable—something that none of the other arts or sciences promises.”*
Echoing Plato (Gorgias 508a, Laws VI.757b) and Aristotle (Nicomachean Ethics V.3, 1131a–b), Hero elevates land-division from cadastral craft to philosophical exemplar: arithmetic equality superposes congruent magnitudes, geometric proportion distributes shares in exact ratio, and apodictic demonstration silences dispute.
Props. III.1–III.3: Dividing Triangles from a Vertex, Parallel to a Base & From a Side Point¶
(Diagram Mapping: Both **Proposition III.1* and Proposition III.2 act upon Hero’s canonical (13, 14, 15) triangle* ABC of area \(84\) with A = \((5, 12)\), B = \((0, 0)\), C = \((14, 0)\) ).
Proposition III.1 (Division from a Vertex in Ratio 5 : 3, Schöne p. 142; fol. 100r): To divide a given triangular region ABC (:math:`AB = 13, BGamma = 14, AGamma = 15`) into two triangles in a given ratio (:math:`5 : 3`) having the same vertex A.
Let the dividing cevian be AV (Hero’s \(A\Delta\)). Because triangles \(ABV\) and \(AV\Gamma\) share the same altitude from A, their areas have the same ratio as their bases ([VI.1]):
\[\frac{\text{Area}(\triangle ABV)}{\text{Area}(\triangle AV\Gamma)} = \frac{BV}{V\Gamma} = \frac{5}{3}.\]Componendo ([V.18]), \(B\Gamma : V\Gamma = (5 + 3) : 3 = 8 : 3\). Since \(B\Gamma = 14\) is given, \(V\Gamma = \frac{3}{8} \times 14 = \frac{21}{4} = 5\frac{1}{4}\) and \(BV = \frac{5}{8} \times 14 = \frac{35}{4} = 8\frac{3}{4}\), partitioning the total area \(84\) into exact rational shares \(52\frac{1}{2}\) and \(31\frac{1}{2}\).
Proposition III.2 (Division by a Line Parallel to the Base in Ratio 3 : 1, Schöne pp. 142–144; fol. 100v): To divide the given triangle ABC into two regions—an upper triangle ADE and a lower trapezium DECB —in the given ratio :math:`m : n = 3 : 1` by a straight line DE parallel to base BC.
Analysis: Since DE is parallel to BC, triangle ADE is equiangular and similar to ABC ([I.29], [VI.4]). When the ratio of upper triangle to lower trapezium is \(3 : 1\), componendo the ratio of the whole triangle ABC to ADE is \((3 + 1) : 3 = 4 : 3\).
By Euclid’s *Elements* VI.19, similar triangles are to one another in the duplicate ratio of their homologous sides:
\[\frac{\text{Area}(\triangle AB\Gamma)}{\text{Area}(\triangle A\Delta E)} = \frac{AB^2}{A\Delta^2} = \frac{4}{3}.\]Geometric Mean-Proportional Construction: Mark P on AB so that \(AP = \frac{3}{4} AB\), describe the semicircle on diameter AB, erect the perpendicular PR to meet the semicircle at R, and transfer radius \(AR\) via circle (AR) back onto AB at D ([VI.13]), drawing DE parallel to BC.
Synthesis on (13, 14, 15): Since \(AB^2 = 13^2 = 169\), the quadrance of \(A\Delta\) is exact over \(\mathbb{Q}\):
\[A\Delta^2 = \frac{3}{4} \times 169 = \frac{507}{4} = 126\frac{3}{4}, \qquad A\Delta = \frac{13\sqrt{3}}{2} \approx 11\frac{1}{4},\]partitioning the area \(84\) into \(\text{Area}(\triangle ADE) = 63\) and \(\text{Area}(DECB) = 21\) (\(63 : 21 = 3 : 1\) with zero error!).
Proposition III.3 (Division from an Arbitrary Point on a Side, Schöne pp. 144–146): Given a point \(\Delta\) anywhere on side \(AB\), to cut off a fraction \(\frac{m}{m+n}\) of \(\triangle AB\Gamma\) by a line \(\Delta E\), Hero constructs \(AE\) from the reciprocal proportion \(\Delta A \cdot AE = \frac{m}{m+n} BA \cdot A\Gamma\) ([VI.15]), solving every boundary-point triangle cut in two compass strokes.
Props. III.4–III.19: Trapezia, Quadrilaterals, Annuli & Exterior-Point Cuts¶
Extending Euclid's treatise On Divisions of Figures (Περὶ διαιρέσεων) to survey engineering, Hero solves sixteen increasingly intricate planar division problems:
Propositions III.4–III.8 (Dividing Trapezia & General Quadrilaterals, Schöne pp. 146–160):
Prop. III.4: Dividing a trapezium in ratio \(m : n\) by a line through a vertex or given point on a parallel base.
Prop. III.5 (Parallel Cut of a Trapezium): Given parallel bases \(a < b\), a cut \(x\) parallel to the bases divides the trapezium in ratio \(m : n\) when \(x\) satisfies the weighted root-mean-square law:
\[\frac{x^2 - a^2}{b^2 - x^2} = \frac{m}{n} \qquad \Longrightarrow \qquad x^2 = \frac{n a^2 + m b^2}{m + n}.\]Hero proves this by extending the non-parallel legs to meet at an auxiliary apex pyramid-style, reducing the trapezium to nested similar triangles ([VI.19])!
Props. III.6–III.8: Dividing arbitrary quadrilaterals from a vertex, parallel to a side, or from a given point on any side by reducing the quadrilateral to an equivalent triangle ([I.37]).
Proposition III.9 (Dividing a Circle & Concentric Ring, Schöne pp. 160–162): To divide a circle of diameter \(d\) into an inner disk and an outer annulus in ratio \(m : n\), Hero applies [XII.2] (circles are to one another as the squares on their diameters): \(d_{\text{inner}}^2 = \frac{m}{m+n} d^2\).
Propositions III.10–III.19 (Divisions by Lines Through Exterior Points & Polygons, Schöne pp. 162–172): Suppose a well, boundary stone, or road junction \(\Delta\) lies outside a triangular or polygonal estate, and the surveyor must draw a straight boundary line passing through \(\Delta\) that cuts off a prescribed fraction of the estate! In Props. III.10–III.13, Hero reduces this problem to application of areas with hyperbolic/elliptic excess ([VI.28]–[VI.29], equivalent to solving a quadratic equation geometrically) and extends the method to arbitrary polygons in Props. III.14–III.19.
Props. III.20–III.23: Stereometric Divisions & Hero’s Cube-Root Algorithm (∛100 ≈ 4 9/14)¶
In the final four propositions of the Metrica, Hero ascends from two-dimensional area divisions (governed by duplicate ratios and square roots) to three-dimensional solid divisions (governed by triplicate ratios [XII.12] and cube roots):
Proposition III.20 — Dividing a Pyramid or Cone Parallel to Its Base & Extracting :math:`sqrt[3]{100}` (Schöne pp. 172–178; fols. 105v–106v): To cut a pyramid (or cone) of slant edge \(5\) into ratio \(4 : 1\) from the apex (\(\frac{4}{5}\) upper pyramid to \(\frac{1}{5}\) lower frustum), similar pyramids stand in the triplicate ratio of their homologous edges:
\[\frac{x^3}{5^3} = \frac{4}{5} \qquad \Longrightarrow \qquad x^3 = \frac{4}{5} \times 125 = 100.\]How can one extract the cube root of 100 (\(\sqrt[3]{100}\)), the sole surviving cube-root algorithm in all of Greek mathematical literature? Hero gives his celebrated weighted secant/error-interpolation rule (Schöne p. 178):
Take the nearest integer cubes above and below \(N = 100\): \(5^3 = 125\) (root \(u = 5\)) and \(4^3 = 64\) (root \(\ell = 4\)).
Compute the upper excess \(e_u = 125 - 100 = 25\) and the lower deficit \(e_\ell = 100 - 64 = 36\).
Multiply the upper root by the lower deficit (\(u \cdot e_\ell = 5 \times 36 = 180\)), and the lower root by the upper excess (or in Schöne’s text \(\sqrt{100 \dots}\) / \(4 \times 25\)—specifically \(90 + \dots\) or \(180 / (180 + 100)\)): adding \(180\) and \(100\) (\(= 280\)) and dividing \(180\) by \(280\) gives \(\frac{9}{14}\)!
Add \(\frac{9}{14}\) to the lower root \(4\):
\[x \approx 4 + \frac{9}{14} = \frac{65}{14} \approx 4.642857\dots \qquad \text{(exact } \sqrt[3]{100} \approx 4.6415888\dots\text{)},\]achieving relative accuracy of :math:`0.027%` (\((\frac{65}{14})^3 = \frac{274{,}625}{2744} \approx 100.082\)) in a single rational step!
Propositions III.21–III.22 (Dividing a Conical or Pyramidal Frustum Parallel to Its Bases, Schöne pp. 178–182): Extending the frustum to its virtual apex reduces the frustum cut to \(x^3 = \frac{n a^3 + m b^3}{m + n}\), solved immediately via the cube-root method of Prop. III.20.
Proposition III.23 (Dividing a Sphere in a Given Ratio by a Plane, Schöne pp. 182–184; fol. 107v): Concluding the entire three-book treatise, Hero poses the famous cubic problem of Archimedes' Sphere and Cylinder II.4—to cut a sphere by a plane so that its segments have a given ratio \(m : n\)—and cites Dionysodorus's conic-section solution via the intersection of a parabola and a rectangular hyperbola!