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

Reviel Netz
Cambridge University Press
Publication Date: 
Number of Pages: 
Cambridge Classical Studies
[Reviewed by
Fernando Q. Gouvêa
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The current consensus among historians of Ancient mathematics is that one should make a sharp distinction between the Greek "tradition of geometric problems" and our modern way of thinking about such problems, namely, algebra. Tempting as it is to simply translate Greek problems into quadratic and cubic equations, such translation, scholars tell us, fundamentally mischaracterizes what the Greeks were actually doing. The exact nature of the differences are still being discussed, but most have come to agree that the fact of the difference is undeniable.

(Of course, not all scholars are convinced, and the new consensus has certainly not been noticed by a great many authors of popular books on the history of mathematics. But the description given above is nevertheless accurate: the vast majority of scholars would agree.)

Given that consensus, a crucial question poses itself: how did we get from there to here? Scholars have not always answered that question very well. For example, readers of Jacob Klein's Greek Mathematical Thought and the Origin of Algebra might well get the impression that the new way of thinking sprung, fully grown and ready for battle, from the mind of François Viète.

As Reviel Netz points out in the introduction to The Transformation of Mathematics in the Early Mediterranean World, without an answer to this question we do not yet have a history; rather, we simply have two ahistorical monolyths -- Greek and modern -- that remain unrelated. In order to begin to construct such a history, Netz examines one particular problem from Archimedes' "The Sphere and the Cylinder," looking first at the solutions given by Archimedes, Diocles, and Dionysodorus, then at the writing of Eutocius in Late Antiquity, and finally at Omar Khayyam's 10th century analysis of the problem. The result is engaging, provocative, and definitely worth reading and thinking about. Anyone who wants to know what current scholarship on Greek mathematics could do much worse than to begin with this book.

Fernando Q. Gouvêa is Professor of Mathematics at Colby College in Waterville, ME.

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