Sourcebook in the Mathematics of Medieval Europe and North Africa

"Medieval Europe was a meeting place for the Christian, Jewish, and Islamic civilizations, and the fertile intellectual exchange of these cultures can be seen in the mathematical developments of the time. This sourcebook presents original Latin, Hebrew, and Arabic sources of medieval mathematics, and shows their cross-cultural influences. Most of the Hebrew and Arabic sources appear here in translation for the first time. Readers will discover key mathematical revelations, foundational texts, and sophisticated writings by Latin, Hebrew, and Arabic-speaking mathematicians, including Abner of Burgos's elegant arguments proving results on the conchoid--a curve previously unknown in medieval Europe; Levi ben Gershon's use of mathematical induction in combinatorial proofs; Al-Mu'taman Ibn Hūd's extensive survey of mathematics, which included proofs of Heron's Theorem and Ceva's Theorem; and Muhyī al-Dīn al-Maghribī's interesting proof of Euclid's parallel postulate. The book includes a general introduction, section introductions, footnotes, and references. The Sourcebook in the Mathematics of Medieval Europe and North Africa will be indispensable to anyone seeking out the important historical sources of premodern mathematics."-- Publisher's website Contents Preface Permissions General Introduction Chapter 1. The Latin Mathematics of Medieval Europe Introduction I Latin Schools, 800–1140 I-1. Brief Selections 1. The Quadrivium of Martianus Capella 2. The Quadrivium of Cassiodorus 3. The Quadrivium of Isidore of Seville 4. A New Creation: Rules for Addition of Signed Numbers I-2. Numbering 1. Roman numerals 2. Finger reckoning 3. Isidore of Seville, Liber numerorum (Book of Numbers) 4. Hindu-Arabic numerals I-3. Arithmetic 1. Boethius, De institutione arithmetica (Introduction to Arithmetic) 2. Computus 3. “Gerbert‘s jump” 4. Pandulf of Capua, Liber de calculatione I-4. Geometry 1. Gerbert, Geometry 2. Gerbert: Area of an equilateral triangle 3. Units of measurement 4. Franco of Liège, De quadratura circuli 5. Hugh of S? Victor, Practica geometriae I-5. Recreational Mathematics 1. Number puzzles 2. Thought problems 3. The Josephus Problem 4. The most important medieval number game: the Rithmimachia II. A School Becomes a University: 1140–1480 II-1. Translations 1. Translators and translations 2. Translations: A practical illustration II-2. Arithmetic 1. Al-Khwarizmi, Arithmetic 2. Leonardo of Pisa (Fibonacci), Liber abbaci (Book on Calculation) 3. John of Sacrobosco, Algorismus vulgaris 4. Johannes de Lineriis, Algorismus de minuciis 5. Jordanus de Nemore (Nemorarius), De elementis arithmetice artis 6. Combinatorics and probability, De Vetula II-3. Algebra 1. Al-Khwarizmi, Algebra 2. Leonardo of Pisa, Liber abbaci (Book on Calculation) 3. Leonardo of Pisa, Book of Squares 4. Jordanus de Nemore, De numeris datis (On Given Numbers) 5. Nicole Oresme, Algorismus proportionum (Algorithm of Ratios) 6. Nicole Oresme, De proportionibus proportionum (On the Ratio of Ratios) II-4. Geometry 1. Banu Musa ibn Shakir, The Book of the Measurement of Plane and Spherical Figures 2. Abu Bakr, Liber mensurationum (On Measurement) 3. Leonardo of Pisa, De practica geometrie (Practical Geometry) 4. John of Murs, De arte mensurandi 5. Jordanus de Nemore, Liber philotegni 6. Dominicus de Clavasio, Practica geometriae II-5. Trigonometry 1. Ptolemy, On the Size of Chords in a Circle 2. Leonardo of Pisa, De practica geometrie (Practical Geometry) 3. Johannes de Lineriis, Canones 4. Richard of Wallingford, Quadripartitum 5. Geoffrey Chaucer, A Treatise on the Astrolabe 6. Regiomontanus, On Triangles II-6. Mathematics of the infinite 1. Angle of contingence 2. Thomas Bradwardine, Tractatus de continuo (On the Continuum) 3. John Duns Scotus, Indivisibles and Theology 4. Does light travel instantaneously or over time? 5. Nicole Oresme, Questiones super geometriam Euclidis (Questions on the Geometry of Euclid) II-7. Statics, Dynamics, and Kinematics 1. Robert Grosseteste, De lineis, angulis et figuris (On lines, angles and figures) 2. Jordanus de Nemore, De ratione ponderis (On the Theory of Weights) 3. Thomas Bradwardine, Tractatus de proportionibus 4. William Heytesbury, Regule solvendi sophismata (Rules for Solving Sophisms) 5. Giovanni di Casali, De velocitate motus alterationis (On the Velocity of Motion of Alteration) 6. Nicole Oresme, De configurationibus qualitatum et motuum (On the Configurations of Qualities and Motions) III. Abbacist Schools: 1300–1480 III-1. Foreign Exchange III-2. Geometry III-3. Algebra 1. Gilio da Siena, A Lecture in Introductory Algebra 2. Paolo Girardi, Libro di Ragioni 3. Jacobo da Firenze, Tractatus algorismi 4. Master Dardi, New equations solved Sources References Chapter 2. Mathematics in Hebrew in Medieval Europe Introduction I. Practical and Scholarly Arithmetic 1. Abraham ibn Ezra, Sefer Hamispar (The Book of Number) 2. Aaron ben Isaac, Arithmetic 3. Immanuel ben Jacob Bonfils, On decimal numbers and fractions 4. Jacob Canpan?on, Bar Noten ?a‘am 5. Elijah MizrA?i, Sefer Hamispar (The Book of Number) 6. Levi ben Gershon, Ma‘ase ?oshev (The Art of the Calculator) II. Numerology, Combinatorics, and Number Theory 1. Abraham ibn Ezra, Sefer Ha’e ?ad (The Book of One) 2. Abraham ibn Ezra, Sefer Ha‘olam (Book of the World) 3. Levi ben Gershon, Ma‘ase ?oshev (The Art of the Calculator) 4. Levi ben Gershon, On Harmonic Numbers 5. Qalonymos ben Qalonymos, Sefer Melakhim (Book of Kings) 6. Don Benveniste ben Lavi, Encyclopedia 7. Aaron ben Isaac, Arithmetic III. Measurement and Practical Geometry 1. Abraham ibn Ezra (?), Sefer Hamidot (The Book of Measure) 2. Abraham bar ?iyya, ?ibur Hameshi?a Vehatishboret (The Treatise on Measuring Areas and Volumes) 3. Rabbi Shlomo I??aqi (Rashi), On the Measurements of the Tabernacle Court 4. Simon ben ?ema?, Responsa 165 concerning Solomon‘s Sea 5. Levi ben Gershon, Astronomy IV. Scholarly Geometry 1. Levi ben Gershon, Commentary on Euclid‘s Elements 2. Levi ben Gershon, Treatise on Geometry 3. Qalonymos ben Qalonymos, On Polyhedra 4. Immanuel ben Jacob Bonfils, Measurement of the Circle 5. Solomon ben Isaac, On the Hyperbola and Its Asymptote 6. Abner of Burgos (Alfonso di Valladolid), Sefer Meyasher ‘Aqov (Book of the Rectifying of the Curved) V. Algebra 1. Quadratic word problems 2. Simon Mo?o?, Algebra 3. Ibn al-A?dab, Igeret Hamispar (The Epistle of the Number) Sources References Chapter 3. Mathematics in the Islamic World in Medieval Spain and North Africa Introduction I. Arithmetic 1. Ibn al-Banna’, Arithmetic 2. ‘Ali b. Mu?ammad al-Qala?adi, Removing the Veil from the Science of Calculation 3. Mu?ammad ibn Mu?ammad al-Fullani al-Kishnawi, On magic squares II. Algebra 1. A?mad ibn al-Banna’, Algebra 2. Mu?ammad ibn Badr, An Abridgement of Algebra III. Combinatorics 1. A?mad ibn Mun‘im, Fiqh al-?isab (On the Science of Calculation) 2. Ibn al-Banna’ on Combinatorics, Raising the Veil 3. Shihab al-Din ibn al-Majdi, On enumerating polynomial equations IV. Geometry 1. Abu ‘Abd Allah Mu?ammad ibn ‘Abdun, On Measurement 2. Abu al-Qasim ibn al-Sam?, The Plane Sections of a Cylinder and the Determination of Their Areas 3. Abu ‘Abd Allah Mu?ammad ibn Mu‘adh al-Jayyani, On ratios 4. Al-Mu’taman ibn Hud, Kitab al-Istikmal (Book of Perfection) 5. Mu?yi al-Din ibn Abi al-Shukr al-Maghribi, Recension of Euclid‘s Elements V. Trigonometry 1. Abu ‘Abd Allah Mu?ammad ibn Mu‘adh al-Jayyani, Book of Unknowns of Arcs of the Sphere 2. Abu ‘Abd Allah Mu?ammad ibn Mu‘adh al-Jayyani, On Twilight and the Rising of Clouds 3. Abu ‘Abd Allah Mu?ammad ibn Mu‘adh al-Jayyani, On the qibla 4. Ibrahim ibn al-Zarqalluh, On a universal astrolabe 5. Abu Mu?ammad Jabir ibn Afla?, Correction of the Almagest Sources References Appendices Appendix 1. Byzantine Mathematics 1. Maximus Planudes, The Great Calculation According to the Indians 2. Manuel Moschopoulos, On Magic Squares 3. Isaac Argyros, On Square Roots 4. Anonymous fifteenth-century manuscript on arithmetic Sources Appendix 2. Diophantus Arithmetica, Book I, #24 Appendix 3. From the Ganitasarasangraha of Mahavira Appendix 4. Time Line Editors and Contributors Index Medieval Europe was a meeting place for the Christian, Jewish, and Islamic civilizations, and the fertile intellectual exchange of these cultures can be seen in the mathematical developments of the time. This book presents original Latin, Hebrew, and Arabic sources of medieval mathematics, and shows their cross-cultural influences. Most of the Hebrew and Arabic sources appear here in translation for the first time. Readers will discover key mathematical revelations, foundational texts, and sophisticated writings by Latin, Hebrew, and Arabic-speaking mathematicians, including Abner of Burgos's elegant arguments proving results on the conchoid—a curve previously unknown in medieval Europe; Levi ben Gershon's use of mathematical induction in combinatorial proofs; Al-Muʾtaman Ibn Hūd's extensive survey of mathematics, which included proofs of Heron's Theorem and Ceva's Theorem; and Muhyī al-Dīn al-Maghribī's interesting proof of Euclid's parallel postulate. The book includes a general introduction, section introductions, footnotes, and references. Cover -- Title -- Copyright -- Contents -- Preface -- Permissions -- General Introduction -- Chapter 1. The Latin Mathematics of Medieval Europe -- Introduction -- I Latin Schools, 800-1140 -- I-1. Brief Selections -- 1. The Quadrivium of Martianus Capella -- 2. The Quadrivium of Cassiodorus -- 3. The Quadrivium of Isidore of Seville -- 4. A New Creation: Rules for Addition of Signed Numbers -- I-2. Numbering -- 1. Roman numerals -- 2. Finger reckoning -- 3. Isidore of Seville, Liber numerorum (Book of Numbers) -- 4. Hindu-Arabic numerals -- I-3. Arithmetic II. A School Becomes a University: 1140-1480 -- II-1. Translations -- 1. Translators and translations -- 2. Translations: A practical illustration -- II-2. Arithmetic -- 1. Al-Khwārizmī, Arithmetic -- 2. Leonardo of Pisa (Fibonacci), Liber abbaci (Book on Calculation) -- 3. John of Sacrobosco, Algorismus vulgaris -- 4. Johannes de Lineriis, Algorismus de minuciis -- 5. Jordanus de Nemore (Nemorarius), De elementis arithmetice artis -- 6. Combinatorics and probability, De Vetula -- II-3. Algebra -- 1. Al-Khwārizmī, Algebra 2. Leonardo of Pisa, Liber abbaci (Book on Calculation) -- 3. Leonardo of Pisa, Book of Squares -- 4. Jordanus de Nemore, De numeris datis (On Given Numbers) -- 5. Nicole Oresme, Algorismus proportionum (Algorithm of Ratios) -- 6. Nicole Oresme, De proportionibus proportionum (On the Ratio of Ratios) -- II-4. Geometry -- 1. Banū Mūsā ibn Shākir, The Book of the Measurement of Plane and Spherical Figures -- 2. Abū Bakr, Liber mensurationum (On Measurement) -- 3. Leonardo of Pisa, De practica geometrie (Practical Geometry) 1. Boethius, De institutione arithmetica (Introduction to Arithmetic) -- 2. Computus -- 3. "Gerbert's jump"--4. Pandulf of Capua, Liber de calculatione -- I-4. Geometry -- 1. Gerbert, Geometry -- 2. Gerbert: Area of an equilateral triangle -- 3. Units of measurement -- 4. Franco of Liège, De quadratura circuli -- 5. Hugh of St. Victor, Practica geometriae -- I-5. Recreational Mathematics -- 1. Number puzzles -- 2. Thought problems -- 3. The Josephus Problem -- 4. The most important medieval number game: the Rithmimachia Medieval Europe was a meeting place for the Christian, Jewish, and Islamic civilizations, and the fertile intellectual exchange of these cultures can be seen in the mathematical developments of the time. This work presents original Latin, Hebrew, and Arabic sources of medieval mathematics, and shows their cross-cultural influences