The Transformation of Mathematics in the Early Mediterranean World: From Problems to Equations

Reviel Netz 

Cambridge University Press | 2007 | páginas | pdf | 731 kb

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The transformation of mathematics from its ancient Greek practice to its development in the medieval Arab-speaking world is approached by focusing on a single problem proposed by Archimedes and the many solutions offered. From a practice of mathematics based on the localized solution (originating in the polemical practices of early Greek science), we see a transition to a practice of mathematics based on the systematic approach (grounded in the deuteronomic practices of Late Antiquity and the Middle Ages). A radically new interpretation is accordingly offered of the historical trajectory of pre-modern mathematics.

CONTENTS
Acknowledgments page viii
Introduction 1
1 The problem in the world of Archimedes 11
1.1 The problem obtained 11
1.2 The problem solved by Archimedes 16
1.3 The geometrical nature of Archimedes’ problem 19
1.4 The problem solved by Dionysodorus 29
1.5 The problem solved by Diocles 39
1.6 The world of geometrical problems 54
2 From Archimedes to Eutocius 64
2.1 The limits of solubility: Archimedes’ text 66
2.2 The limits of solubility: distinguishing Archimedes from Eutocius 71
2.3 The limits of solubility: the geometrical character of Archimedes’ approach 85
2.4 The limits of solubility: Eutocius’ transformation 91
2.5 The multiplication of areas by lines 97
2.6 The problem in the world of Eutocius 121
3 From Archimedes to Khayyam 128
3.1 Archimedes’ problem in the Arab world 129
3.2 A note on Al-Khwarizmi’s algebra 137
3.3 Khayyam’s solution within Khayyam’s algebra 144
3.4 The problem solved by Khayyam 155
3.5 Khayyam’s equation and Archimedes’ problem 160
3.6 Khayyam’s polemic: the world of Khayyam and the world of Archimedes 171
3.7 How did the problem become an equation? 181
Conclusion 187
References 193
Index 196

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