UNIT I SUMMARY — PROLOGUE, PISAN METROLOGY & DISTINCTIO I–II

Structural Synopsis of Boncompagni Vol. II, pp. 1–30 (fol. 1r–18v)

English Edition · Latin (1220 / Boncompagni 1862) · GEOMETOR Notes

Overview

Composed in Pisa in $1220$ for Master Dominicus, Unit I establishes the dual epistemological charter of Practica Geometriae: uniting rigorous Euclidean synthetic demonstration (secundum geometricam demonstrationem) with the concrete computational algorithms of Mediterranean surveyors (secundum vulgarem consuetudinem).

Key Structural Movements

  1. Architectonic Plan of the Eight Distinctions (fol. 1r): Leonardo Pisano organizes the entire science of practical geometry into a progressive ladder:

    • Dist. I–II: Rectangular mensuration, Euclidean geometric algebra, and square roots;

    • Dist. III–IV: General planar mensuration (triangles, polygons, circles) and consortium land division;

    • Dist. V–VI: Cube roots, solid stereometry, and the five regular polyhedra in a sphere;

    • Dist. VII–VIII: Optical surveying with the quadrans and higher algebraic-geometric subtilitates.

  2. Euclidean Foundations (fol. 1v–2v): Definitions of point, straight line, plane surface, right angle / cathetus, and solid body from euclid:elements Books I and XI, paired with Common Notions $1$–$5$.

  3. The Pisan Agrarian Lattice (fol. 2v–3v): Linear and superficial conversion hierarchy anchored on the $6$-foot Pisan pertica:

    Pisan Superficial & Linear Hierarchy

    Unit

    Definition

    Exact Equivalent in Perticae

    Linear Pertica

    $6$ linear pedes $= 108$ linear unciae

    $1$ linear pertica

    Superficial Pertica

    $1\text{ pertica} \times 1\text{ pertica}$ ($6 \times 6\text{ pedes}^2$)

    $1$ ($= 36$ denarii $= 108$ sup. unciae)

    Superficial Pes

    Strip $1\text{ pertica} \times 1\text{ linear pes}$

    $\frac{1}{6}$ pertica ($= 6$ denarii)

    Superficial Denarius

    Unit square $1\text{ linear pes} \times 1\text{ linear pes}$

    $\frac{1}{36}$ pertica

    Panorum

    Seed-basket measure of land

    $5\frac{1}{2} = \frac{11}{2}$ perticae ($198$ denarii)

    Stariorum

    $12$ panora ($12 \times \frac{11}{2}$)

    $66$ perticae ($2{,}376$ denarii)

  4. Distinctio I — Rectangular Mensuration & Euclid Books II–III (fol. 4r–11v): Area (embadum) of squares and rectangles, cross-multiplication table for mixed subunits, the $1$-stariorum ($66$ perticae) frontage division ($66 \div 7 = 9\text{p } 2\text{ft } 10\frac{2}{7}\text{in}$), Euclid II.1–10 identities ($(a+b)^2$, $ad \cdot db + gd^2 = gb^2$), and intersecting chords (Euclid III.35).

  5. Distinctio II — Positional & Fractional Square Roots (fol. 11v–18v): Hindu-Arabic digit-pair extraction, two-step rational Heron/Babylonian refinement $a_2 = a_1 - \frac{a_1^2 - N}{2a_1}$, and reduction of fractional areas ($\sqrt{\frac{2}{3}\text{ pertica}} = \sqrt{24\text{ denarii}} \approx 4\text{ pedes } 16\frac{2}{11}\text{ unciae}$).