The World of the Abbaco: Abbacus Mathematics Analyzed and Situated Historically Between Fibonacci and Stifel (Frontiers in the History of Science)

The abbacus was a thorough and complete system of arithmetical calculations, which saw its dawn in the Indian and Arabic tradition of the Middle Ages, but which was developed in its fully fledged aspects especially in Italy, between Genoa, Milan, Venice and the region of Umbria. In this monograph, Høyrup explores how the abbacus tradition has developed in Europe, with a focus on Italy and Germany between the 14th and the 17th Century. With the analysis of texts from Fibonacci, Pacioli, and Stifel to name but a few, this book offers a critical historical analysis of the development and diffusion of a tradition of calculus that has deeply influenced the way in which mathematics has developed in the West. The primary purpose of the book is to present a fairly detailed portrait of the abbacus tradition as it developed historically; as will be argued, Fibonacci was much less important for the emergence of this tradition than mostly assumed – but since his importance isbroadly taken for granted, that argument needs to be made, for which reason Fibonacci’s Liber Abbaci is also described and analyzed in some depth. A secondary purpose is to show how the adoption of abbacus mathematics in German lands gave rise to the creation of a different tradition. The end of the book investigates the interplay of abbacus algebra with other intellectual currents which turned the whole mathematical undertaking upside-down in the 17th Century.​ Preface Table of contents Introduction I. The Home of Abbacus Mathematics: the Abbacus School II. An Example: Jacopo da Firenze’s Tractatus Algorismi, the Short Version The Introduction About the Numerals and the Place-Value System Multiplication, Division, Fractions The Rule of Three Basic Commercial Techniques Mixed Problems Practical Geometry with Approximate Determination of Square Roots The Coin List Alloying Problems III. Fibonacci and the Fibonacci Story The Liber Abbaci, the Autobiography, and the Meaning of the Title Some General Characteristics of Fibonacci’s Project Chapter 1 – Introducing the Hindu-Arabic Numerals Chapter 2 – Multiplication of Integers Chapters 3–4 – Addition and Subtraction Chapter 5 – Division Chapter 6 – Multiplication of Mixed Numbers Chapter 7 – Addition, Subtraction and Division of Mixed Numbers Chapter 8 – the Rule of Three Chapter 9 – Barter Chapter 10 – Partnership Chapter 11 – Alloying Chapter 12 – Mixed “Abbacus Questions” 12.1, Summation of Series 12.2, “Proportions of Numbers” 12.3, “Questions of Trees ...” 12.4, Finding a Purse 12.5, Buying a Horse 12.6, Repeated Travels with Gain and Expenses 12.7, “Rambling Problems” 12.8, Divinations 12.9, Chess-Board and Other Geometric Series Chapter 13 – Elchatayn Rule 13.1, Problems Already Dealt With 13.2, New Problems Chapter 14 – Square and Cube Roots A Puzzling Preamble 14.1, Extraction of Square Roots 14.2a, the Multiplication of Roots and Binomials 14.2b, Multiplying Roots by Roots 14.3, Addition and Subtraction of Monomials, Binomials and Apotomes 14.4a, Mutual Division of Monomials 14.4b, Division of Binomials and Apotomes 14.5, Cube Roots Chapter 15 – Theory of Means, Rules of Geometry, and Algebra 15.1, An Investigation of Means 15.2, Geometric and Other Questions 15.3, Introduction to Algebra 99 Questions with Interspersed Theory IV. The Real Story in Select Detail “Generation 1”: “Generation 1”: Livero de l’Abbecho, Pisan Libro di Ragioni, “Columbia Algorism” and Liber Habaci The Livero de l’Abecho The Pisa Libro di Ragioni The “Columbia Algorism” The Liber Habaci An Ibero-Provençal aside The “Second Generation”: Crystallization of a Tradition, and the Arrival of Algebra Jacopo’s Tractatus Paolo Gherardi Tutta l'Arte The Lucca Libro d’Abaco Giovanni di Davizzo Biagio il Vecchio The Second Generation – Summary Observations Further into the 14th Century Dardi da Pisa and the Aliabraa Argibra Alcibra Amuchabile Antonio de’ Mazzinghi The Florentine Tratato sopra l’Arte della Arismetricha The Abbacus Encyclopedias The Ottoboniano and Palatino Benedetto Summary Observations Concerning the Florentine Encyclopedias V. Abbacus Goes into Print and Abroad Early Prints in Italy The Basic to Intermediate Levels The Higher Level Luca Pacioli The German-Speaking Area Wolack’s Lecture The Regensburg Practica, Friedrich Amann – and Regiomontanus Two Manuscripts, a University Lecture, and the First Prints Rechenmeister Culture Attains Maturity Coß Stifel’s Arithmetica Integr VI. A Double Conclusion A Mathematical Practice Sui Generis The Zilsel Thesis and the Transformation of Mathematics 12th-13th-Century Reception of Algebra The 14th Century – Early Abbacus Algebra, and First Interaction The 15th Century – the Beginnings of a Ménage à Trois 1500–1575: a Changing Scenery The Take-Off of modern Algebra Sigla Bibliography Source Index Sources referred to by author, title or modern designation Manuscripts referred to as such Name and Subject Index The abbacus was a thorough and complete system of arithmetical calculations, which saw its dawn in the Indian and Arabic tradition of the Middle Ages, but which was developed in its fully fledged aspects especially in Italy, between Genoa, Milan, Venice and the region of Umbria. In this monograph, Høyrup explores how the abbacus tradition has developed in Europe, with a focus on Italy and Germany between the 14th and the 17th Century. With the analysis of texts from Fibonacci, Pacioli, and Stifel to name but a few, this book offers a critical historical analysis of the development and diffusion of a tradition of calculus that has deeply influenced the way in which mathematics has developed in the West. The primary purpose of the book is to present a fairly detailed portrait of the abbacus tradition as it developed historically; as will be argued, Fibonacci was much less important for the emergence of this tradition than mostly assumed – but since his importance is broadly taken for granted, that argument needs to be made, for which reason Fibonacci’s Liber Abbaci is also described and analyzed in some depth. A secondary purpose is to show how the adoption of abbacus mathematics in German lands gave rise to the creation of a different tradition. The end of the book investigates the interplay of abbacus algebra with other intellectual currents which turned the whole mathematical undertaking upside-down in the 17th Century.