Explorations and False Trails: The Innovative Techniques That Brought about Modern Algebra

ISBN-13: 9783031481574 / Angielski

Jens Hoyrup

Spis treści

Preface

 

Introduction

The changes that produced Modern algebra

Algebraic background(s)

Al-Khwārizmī; Fibonacci; Abbacus algebra

Chapter I. Geometric proofs

Dardi da Pisa

The Florentine encyclopedias

Pacioli

Cardano

Chapter II. Powers of the unknown

Abbacus algebra

Dardi da Pisa; Antonio de’ Mazzinghi; The Florentine Tratato sopra l’arte della arismetricha; The Florentine encyclopedias; An aside on the regula recta 

The end and aftermath of the abbacus tradition

A Modena manuscript; Nicolas Chuquet and Étienne de la Roche; Pacioli and Pacioli; del Sodo 

Algebra in German land

Eclectic beginnings; Andreas Alexander; Heinrich Schreyber; Christoph Rudolff and later coß; Jacques Peletier

Chapter III. Abbreviations, glyphs, symbols and symbolic calculation

The Maghreb symbolism

The Italian beginnings

Biagio “il vecchio”; Dardi and Alcibra amuchabile; The Liber restauracionis

Late 14th  and 15th-century algebra

Late-14th-century Florence; A collective work from 1429; The Florentine encyclopedias; The Modena manuscript; Chuquet; Giovanni del Sodo

Algebra in Italian print

Pacioli; Three 16th-century Italian writers

German notations

The German and Latin algebras of Dresden, C 80; Alexander; Schreyber, Rudolff and later coß

French algebraic writings

Chapter IV. Embedding and parenthesis function

Composite radicands

Dardi; Antonio – two levels; Chuquet and de la Roche; Mennher; Cardano, Tartaglia and Bombelli

The substitutes for and a dubious step toward a general parenthesis function

Viète

Chapter V. Several unknowns

Several unknowns in Arabic and post-Arabic algebra

Arabic use; Liber mahameleth; Fibonacci

Abbacus occurrences of several unknowns around 1400  

Antonio de’ Mazzinghi; The Florentine Tratato

The Florentine abbacus encyclopedias

The Ottoboniano Praticha; Benedetto

Pacioli, Chuquet and de la Roche

Coß

Stifel; Immediate impact in Germany; Mennher; French writers after de la Roche

Chapter VI. The transition to incipient modern algebra

Mathematical cultures

The raison-d’être and format of abbacus algebra; Scholastic mathematics; Humanist mathematics; Agonistic mathematics, once again

An apparently minor leap with immense consequences

Algebra as Descartes knew it; Viète’s “defiled and polluted” algebra; Receiving abstract coefficients as a gift not asked for; Geometry and algebra with one unknown; Viète’s abstract coefficients

Descartes’ other innovation

Newton and Wallis; The truly general parenthesis

Coda

Bibliography

Index

 

O autorze

Jens Egede Høyrup, born in 1943 in Copenhagen, is a Danish historian of mathematics, specializing in pre-modern and early modern mathematics, ancient Mesopotamian mathematics and the abbacus tradition in particular. He is especially known for his interpretation of what has often been referred to as Old Babylonian “algebra” as consisting of concrete, geometric manipulations.