← pacioli / De Divina Proportione

Part I: Compendio de Divina Proportione (Chapters I–LXXI)¶

Dedicated to Ludovico Maria Sforza, Duke of Milan (Milan, 1498 / Venice, 1509)

Dedicated to Ludovico Maria Sforza, Duke of Milan (completed at Milan, December 14, 1498; printed at Venice by Paganino de' Paganini, June 1, 1509). Each chapter is presented in four synchronized companion tracks: English, Italiano (1509), Deutsch (Winterberg 1889), and Critical Notes.

Conspectus Capitum (Part I)¶

Cap. I (Folio 0017–0018 (f. 1r–f. 1v)): Epistle on the Divine Proportion to the Most Excellent Prince Ludovico Maria Sforza Anglo, Duke of Milan, Ornament of Peace and War, by Brother Luca Pacioli of Borgo Sansepolcro of the Order of Minorites, Professor of Sacred Theology [English · Italiano · Deutsch · Notes]
Cap. II (Folio 0018–0021 (f. 1v–f. 3r)): Preface of the Reverend Father Magister Luca Pacioli of Borgo Sansepolcro, of the Order of Minorites and Professor of Sacred Theology, to the Compendium on the Divine Proportion drawn from the Mathematical Disciplines [English · Italiano · Deutsch · Notes]
Cap. III (Folio 0021–0022 (f. 3r–f. 3v)): The Preface Being Ended, There Follows an Explanation of What Is Meant by This Name ‘Mathematical’ [English · Italiano · Deutsch · Notes]
Cap. IV (Folio 0022 (f. 3v)): On Those Things Which the Reader Must Observe for the Understanding of This Treatise [English · Italiano · Deutsch · Notes]
Cap. V (Folio 0022–0023 (f. 3v–f. 4r)): On the Fitting Title of the Present Treatise [English · Italiano · Deutsch · Notes]
Cap. VI (Folio 0023 (f. 4r)): On Its Worthy Commendation [English · Italiano · Deutsch · Notes]
Cap. VII (Folio 0023–0025 (f. 4r–f. 5r)): On the First Effect of a Line Divided According to Our Proportion [English · Italiano · Deutsch · Notes]
Cap. VIII (Folio 0025 (f. 5r)): How a Quantity Divided According to the Proportion Having a Mean and Two Extremes Is to Be Understood [English · Italiano · Deutsch · Notes]
Cap. IX (Folio 0026 (f. 5v)): What the Root of a Number and of Any Other Quantity Is [English · Italiano · Deutsch · Notes]
Cap. X (Folio 0026 (f. 5v)): Sequel to the First Proposed Effect [English · Italiano · Deutsch · Notes]
Cap. XI (Folio 0026 (f. 5v)): On Its Second Essential Effect [English · Italiano · Deutsch · Notes]
Cap. XII (Folio 0027 (f. 6r)): On Its Third Singular Effect [English · Italiano · Deutsch · Notes]
Cap. XIII (Folio 0027 (f. 6r)): On Its Fourth Ineffable Effect [English · Italiano · Deutsch · Notes]
Cap. XIV (Folio 0027 (f. 6r)): On Its Fifth Marvelous Effect [English · Italiano · Deutsch · Notes]
Cap. XV (Folio 0027 (f. 6r)): On Its Sixth Inexpressible Effect [English · Italiano · Deutsch · Notes]
Cap. XVI (Folio 0028 (f. 6v)): On Its Seventh Inestimable Effect [English · Italiano · Deutsch · Notes]
Cap. XVII (Folio 0028 (f. 6v)): On the Eighth Effect, the Converse of the Preceding [English · Italiano · Deutsch · Notes]
Cap. XVIII (Folio 0028 (f. 6v)): On Its Ninth Effect, Surpassing All the Others [English · Italiano · Deutsch · Notes]
Cap. XIX (Folio 0029 (f. 7r)): On Its Tenth Supreme Effect [English · Italiano · Deutsch · Notes]
Cap. XX (Folio 0029 (f. 7r)): On Its Eleventh Most Excellent Effect [English · Italiano · Deutsch · Notes]
Cap. XXI (Folio 0029–0030 (f. 7r–f. 7v)): On Its Twelfth Almost Incomprehensible Effect [English · Italiano · Deutsch · Notes]
Cap. XXII (Folio 0030 (f. 7v)): On Its Thirteenth Most Worthy Effect [English · Italiano · Deutsch · Notes]
Cap. XXIII (Folio 0030–0031 (f. 7v–f. 8r)): How, Out of Reverence for Our Salvation, the Said Effects Conclude [English · Italiano · Deutsch · Notes]
Cap. XXIV (Folio 0031 (f. 8r)): On the Tenth Effect of Our Divine Proportion, Which Introduces the Formation of the Five Regular Bodies. [English · Italiano · Deutsch · Notes]
Cap. XXV (Folio 0031–0032 (f. 8r–f. 8v)): How It Is Not Possible for There to Be More Than Five Regular Bodies in Nature, and Why. [English · Italiano · Deutsch · Notes]
Cap. XXVI (Folio 0032 (f. 8v)): How the First Regular Body, Called the Tetracedron or Pyramid of Four Triangular Bases, Is Formed and Placed in the Sphere. [English · Italiano · Deutsch · Notes]
Cap. XXVII (Folio 0032–0033 (f. 8v–f. 9r)): How the Second Regular Body, Called the Exacedron or Cube, Is Formed and Placed in the Sphere. [English · Italiano · Deutsch · Notes]
Cap. XXVIII (Folio 0033–0034 (f. 9r–f. 9v)): How the Third Regular Body, Called the Octocedron of Eight Triangular Bases, Is Formed in the Sphere. [English · Italiano · Deutsch · Notes]
Cap. XXIX (Folio 0034–0036 (f. 9v–f. 10v)): How the Fourth Regular Body, Called the Ycocedron of Twenty Triangular Bases, Is Formed in the Sphere. [English · Italiano · Deutsch · Notes]
Cap. XXX (Folio 0036–0038 (f. 10v–f. 11v)): How the Fifth and Most Noble Regular Body, Called the Duodecedron of Twelve Pentagonal Bases, Is Formed in the Sphere. [English · Italiano · Deutsch · Notes]
Cap. XXXI (Folio 0038–0040 (f. 11v–f. 12v)): On the Rule and Way to Find the Sides of All Five Regular Bodies Placed in One and the Same Sphere, and Their Comparison with One Another. [English · Italiano · Deutsch · Notes]
Cap. XXXII (Folio 0040 (f. 12v)): On the Proportion of the Said Regular Bodies to One Another, Especially that of the Ycocedron to the Duodecedron. [English · Italiano · Deutsch · Notes]
Cap. XXXIII (Folio 0040–0041 (f. 12v–f. 13r)): On the Proportion of the Surfaces and Solid Volumes of All the Regular Bodies to One Another. [English · Italiano · Deutsch · Notes]
Cap. XXXIV (Folio 0041 (f. 13r)): On the Inscriptions of the Five Regular Bodies in One Another, How Many There Are in All, and Why. [English · Italiano · Deutsch · Notes]
Cap. XXXV (Folio 0041 (f. 13r)): How the Octocedron Is Formed and Inscribed in the Tetracedron. [English · Italiano · Deutsch · Notes]
Cap. XXXVI (Folio 0041 (f. 13r)): How the Tetracedron Is Placed in the Cube. [English · Italiano · Deutsch · Notes]
Cap. XXXVII (Folio 0041 (f. 13r)): How the Octocedron Is Placed in the Cube. [English · Italiano · Deutsch · Notes]
Cap. XXXVIII (Folio 0042 (f. 13v)): How the Exacedron or Cube Is Formed in the Octocedron. [English · Italiano · Deutsch · Notes]
Cap. XXXIX (Folio 0042 (f. 13v)): How the Tetracedron Is Formed in the Octocedron. [English · Italiano · Deutsch · Notes]
Cap. XL (Folio 0042 (f. 13v)): How the Duodecedron Is Formed in the Ycocedron. [English · Italiano · Deutsch · Notes]
Cap. XLI (Folio 0042 (f. 13v)): How the Exacedron or Cube Is Placed in the Ycocedron. [English · Italiano · Deutsch · Notes]
Cap. XLII (Folio 0042 (f. 13v)): How the Tetracedron Is Formed in the Ycocedron. [English · Italiano · Deutsch · Notes]
Cap. XLIII (Folio 0042 (f. 13v)): How the Ycocedron Is Formed in the Duodecedron. [English · Italiano · Deutsch · Notes]
Cap. XLIV (Folio 0042 (f. 13v)): How the Ycocedron Is Placed in the Cube. [English · Italiano · Deutsch · Notes]
Cap. XLV (Folio 0043 (f. 14r)): How the Octocedron Is Placed in the Duodecedron. [English · Italiano · Deutsch · Notes]
Cap. XLVI (Folio 0043 (f. 14r)): How the Exacedron and the Tetracedron Are Placed in the Duodecedron, and Why Only Twelve Inscriptions Are Possible. [English · Italiano · Deutsch · Notes]
Cap. XLVII (Folio 0043 (f. 14r)): How the Sphere Is Placed and Inscribed in Each of the Said Five Regular Bodies. [English · Italiano · Deutsch · Notes]
Cap. XLVIII (Folio 0043–0044 (f. 14r–f. 14v)): On the Form and Disposition of the Plane (Solid) and Hollow (Skeletal) Tetracedron, Regular and Truncated (Plates I–IV), and Elevated (Plates V–VI). [English · Italiano · Deutsch · Notes]
Cap. XLIX (Folio 0044 (f. 14v)): On the Disposition of the Exacedron or Cube: Regular (Plates VII–VIII), Truncated (Plates IX–X), and Elevated (Plates XI–XII). [English · Italiano · Deutsch · Notes]
Cap. L (Folio 0044–0045 (f. 14v–f. 15r)): On the Disposition of the Octocedron: Regular (Plates XIII–XIV), Truncated (Plates XV–XVI), and Elevated (Plates XVII–XVIII). [English · Italiano · Deutsch · Notes]
Cap. LI (Folio 0045 (f. 15r)): On the Disposition of the Ycocedron: Regular (Plates XIX–XX), Truncated (Plates XXI–XXII), and Elevated (Plates XXIII–XXIV). [English · Italiano · Deutsch · Notes]
Cap. LII (Folio 0045–0046 (f. 15r–f. 15v)): On the Disposition of the Duodecedron: Regular (Plates XXV–XXVI), Truncated (Plates XXVII–XXVIII), Elevated (Plates XXIX–XXX), and the Elevated Truncated Bodies (Plates XXXI–XXXIV). [English · Italiano · Deutsch · Notes]
Cap. LIII (Folio 0046–0047 (f. 15v–f. 16r)): On the Formation and Origin of the Body of Twenty-Six Bases: Eighteen Squares and Eight Equilateral Triangles (Plates XXXV–XXXVIII). [English · Italiano · Deutsch · Notes]
Cap. LIV (Folio 0047–0048 (f. 16r–f. 16v)): On the Body of Seventy-Two Bases (Septuaginta Duarum Basium), Its Origin, and Its Architectural Use in Cupolas and Vaults (Plates XXXIX–XL). [English · Italiano · Deutsch · Notes]
Cap. LV (Folio 0048–0049 (f. 16v–f. 17r)): On the Way to Know How to Form Still More Bodies Beyond Those Mentioned, How Their Forms Proceed to Infinity, and Why Plato Assigned the Five Regular Bodies to the Five Simple Bodies of the Universe. [English · Italiano · Deutsch · Notes]
Cap. LVI (Folio 0049–0050 (f. 17r–f. 17v)): On the Spherical Body and How It Is Formed (Plates XLI–XLII). [English · Italiano · Deutsch · Notes]
Cap. LVII (Folio 0050–0051 (f. 17v–f. 18r)): How All Five Regular Bodies Are Placed in the Sphere, How Stonemasons May Carve Them from a Spherical Stone, and the Anecdote of Count Girolamo Riario and Melozzo da Forlì in Rome. [English · Italiano · Deutsch · Notes]
Cap. LVIII (Folio 0051–0052 (f. 18r–f. 18v)): On Oblong Bodies, That Is, Bodies Longer or Higher Than They Are Wide: And First on the Round Column (Cylinder, Plates XLIII–XLIV). [English · Italiano · Deutsch · Notes]
Cap. LIX (Folio 0052 (f. 18v)): On Lateral (Faceted) Columns, and First on the Three-Sided (Triangular) Column, Called the Serratile Body (Plates XLV–XLVI). [English · Italiano · Deutsch · Notes]
Cap. LX (Folio 0052–0053 (f. 18v–f. 19r)): On Four-Sided (Quadrilateral) Lateral Columns and the Five Species of Quadrilaterals (Plates XLVII–XLVIII). [English · Italiano · Deutsch · Notes]
Cap. LXI (Folio 0053 (f. 19r)): On Five-Sided (Pentagonal) Lateral Columns and How the Species of Lateral Columns Proceed to Infinity (Plates XLIX–L). [English · Italiano · Deutsch · Notes]
Cap. LXII (Folio 0053–0054 (f. 19r–f. 19v)): On the Way to Measure All Kinds of Columns, and First the Round Columns (Cylinders). [English · Italiano · Deutsch · Notes]
Cap. LXIII (Folio 0054 (f. 19v)): On the Way to Know How to Measure All Lateral (Prismatic) Columns. [English · Italiano · Deutsch · Notes]
Cap. LXIV (Folio 0055 (f. 20r)): On Pyramids and All Their Differences, and First on the Round Pyramid (Cone, Plates XLIII–XLIV). [English · Italiano · Deutsch · Notes]
Cap. LXV (Folio 0055–0056 (f. 20r–f. 20v)): On Lateral (Polygonal) Pyramids and Their Varieties: Right, Oblique, and Truncated (Plates LI–LIX). [English · Italiano · Deutsch · Notes]
Cap. LXVI (Folio 0056 (f. 20v)): On the Way and Rule to Know How to Measure Every Complete Pyramid, Round or Lateral. [English · Italiano · Deutsch · Notes]
Cap. LXVII (Folio 0056–0057 (f. 20v–f. 21r)): How It Is Manifestly Demonstrated That Every Lateral Pyramid Is One-Third (Subtripla) of Its Column, and Why Every Polygon of N Sides Resolves into N − 2 Triangles. [English · Italiano · Deutsch · Notes]
Cap. LXVIII (Folio 0057–0058 (f. 21r–f. 21v)): How Short or Truncated Pyramids (Pyramidi Corte ouer Scapezze) Are Measured. [English · Italiano · Deutsch · Notes]
Cap. LXIX (Folio 0058–0059 (f. 21v–f. 22r)): On the Measure of All the Regular and Dependent Bodies, and the Panegyric to Duke Ludovico Maria Sforza and the 1498 Franciscan General Chapter in Milan. [English · Italiano · Deutsch · Notes]
Cap. LXX (Folio 0059 (f. 22r)): How to Locate in Order All the Bodies Drawn in Perspective in This Book, as Well as Their Physical Models According to Their Public Table. [English · Italiano · Deutsch · Notes]
Cap. LXXI (Folio 0059–0061 (f. 22r–f. 23r)): What Is Understood by the Mathematical Terms Used in This Compendium—Namely Hypothesis, Hypotenuse, Corausto, Pyramidal Cone, Pentagonal Chord, Perpendicular, Cathetus, Diameter, Parallelogram, Diagonal, Center, and Sagitta—and Conclusion of Part I (December 14, 1498). [English · Italiano · Deutsch · Notes]

Chapters I–LXXI (English)