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SUMMA DE ARITHMETICA, GEOMETRIA, PROPORTIONI ET PROPORTIONALITA¶

Fra Luca Pacioli (Venice, 1494) — Critical Multi-Track Digital Codex & Symbolic Edition

A folio-verified digital edition of Fra Luca Pacioli's encyclopedic Renaissance masterpiece, Summa de arithmetica, geometria, proportioni et proportionalita (printed in Venice by Paganino de' Paganini, November 10–20, 1494, under Doge Agostino Barbarigo and dedicated to Guidobaldo da Montefeltro, Duke of Urbino, and the Venetian patrician Marco Sanudo).

As the first printed Renaissance synthesis of speculative arithmetic, Hindu–Arabic algorism, Cossic algebra (Regola dela Cosa), Euclidean and Boethian proportion theory, Venetian double-entry bookkeeping (Particularis de computis et scripturis), early probability (Problem of Points), practical mensuration, and stereometry of the regular and Archimedean solids, the 1494 Summa forms the direct mathematical foundation for Pacioli’s 1509 *De Divina Proportione*. Each canonical unit is presented across four synchronized companion tracks:

English (GEOMETOR Edition): Original scholarly English translation with live interactive geometric and algebraic markup (:g:, :eq:, :just:, .. folio::, .. enunciation::, .. assertion::, .. qed::) linked to nine exact .euclid-graphic vector diagrams.
Italiano (1494): Diplomatic Venetian volgare and Latin text transcribed directly by visual inspection of the 1494 Paganini editio princeps folios.
Summary: Executive mathematical and architectural digest of definitions, algorithms, conservation laws, and exact SymPy / geometor.model invariants.
Notes: Historical, philological, and deductive commentary tracing Pacioli’s sources in Euclid, Boethius, Al-Khwārizmī, Leonardo Pisano, Jordanus Nemorarius, Campanus of Novara, and Piero della Francesca, forward to Scipione del Ferro, Gerolamo Cardano, Blaise Pascal, and Pierre de Fermat.

Conspectus Totius Operis¶

Unit I — Preliminaries & Speculative Arithmetic (Prelim. fols. 1r–8v; Part I, Dist. I, fols. 1r–18v) - Continentia de tutta lopera, Latin Epistle to Marco Sanudo, Epigrams of Paolo Pompilio and Giorgio Sommaripa, Italian and Latin Dedications to Duke Guidobaldo da Montefeltro, the Quadrivium Hierarchy, Explicit Debt to Leonardo Pisano, Polygonal Figurate Numbers, and [IX.36] Generation of Perfect Numbers (fol. 8r).
Unit II — Algorismus, Surds & Cossic Algebra: Arte Maggiore / Regola dela Cosa (Part I, Dist. II–V & VIII, fols. 19r–67v, 112r–150r) - Eight Modes of Multiplication (scacchieri, gelosia), Galea Division, Regola del Tre, El Cataym (Double False Position), Syncopated Cossic Powers (.co., .ce., .cu., .ce.ce.), Operational Signs (.p̄. / .m̄., .℞.u.), Euclid Book X Binomials and Apotomes ([X.36], [X.73]), the Six Canonical Quadratic Equations (fols. 145r–146r, [II.4]–[II.6]), and the Historic Verdict on General Cubics (fol. 150r).
Unit III — Tractatus de Proportionibus et Proportionalitatibus (Part I, Dist. VI, fols. 68r–98r) - Universal Praise of Proportion across Philosophy, Perspective (Piero della Francesca), and Architecture (Leon Battista Alberti, Vitruvius, fol. 68v), the Five Boethian Genera (multiplex, superparticularis, superpartiens), Semicircle Construction of the Arithmetic, Geometric, and Harmonic Means (fol. 73v, [VI.13]), the Six Euclidean Modi Arguendi (fol. 74r, [V.16]–[V.22]), and Division of $10$ in Extreme and Mean Ratio ($\sqrt{125}-5$ and $15-\sqrt{125}$, [II.11], [XIII.1]).
Unit IV — Mercantile Arithmetic, Problem of Points & Particularis de Computis et Scripturis (Part I, Dist. IX, fols. 150v–224v) - Partnerships (compagnie), Barter, Bills of Exchange, the Problem of Points (fol. 197r–197v: Two-Player Balla & Three-Player Balestrare vs. Pascal–Fermat), and the 36-Chapter Treatise on Double-Entry Bookkeeping al modo de Vinegia (fols. 198v–210v): Inventario, Memoriale, Giornale (Per // A), Quaderno Grande in Venetian l. s. g. p., Cassa Non-Negativity, Skew-Symmetric Conservation ($F^T = -F$), Summa Summarum, and Bilancio del Libro.
Unit V — Tractatus Geometriae: Practical Mensuration, Surface Division & Inscribed Polyhedra (Part II, Dist. I–VIII, fols. 1r–76r) - Heron's $(13, 14, 15)$ Triangle & Cathetus (Part II, fol. 9v, Area $= \sqrt{7056} = 84$), Archimedes's $96$-Gon Circle Bounds ($3\frac{10}{71} < \pi < 3\frac{1}{7}$, fol. 31r), De divisione superficierum (Dist. IV, [VI.19]), Wine-Barrel Stereometry (botte), Malfatti-Type Three-Circle Algebraic Packing (fol. 57r), Regular & Archimedean Solids Inscribed in the Sphere of Diameter $6$ (Dist. VIII, fols. 68v–70r, Truncated Tetrahedron $\sqrt{6\frac{6}{11}}$, Cuboctahedron $3$, Dodecahedron $\sqrt{20}-2$ in Sphere of Possanza $48$), and the November 10, 1494 Venetian Colophon (fol. 76r).
Scholarly Apparatus: Bibliography & Cited Works · Index of Persons · Technical Glossary

Contents

Scholarly Apparatus