The Transformation of Mathematics in the Early Mediterranean World

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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