Ibn al-Haytham and Analytical Mathematics

Roshdi Rashed is one of the most eminent authorities on Arabic mathematics and the exact sciences. A historian and philosopher of mathematics and science and a highly celebrated epistemologist, he is currently Emeritus Research Director (distinguished class) at the Centre National de la Recherche Scientifique (CNRS) in Paris, and is the former Director of the Centre for History of Arabic and Medieval Science and Philosophy at the University of Paris (Denis Diderot, Paris VII). He also holds an Honorary Professorship at the University of Tokyo and an Emeritus Professorship at the University of Mansourah in Egypt.

CONTENTS

Preface ..................................................................................... xi

Note ....................................................................................... xiii

INTRODUCTION: IBN AL-HAYTHAM AND HIS WORK ON INFINITESIMAL

MATHEMATICS

1. Ibn al-Haytham: from Basra to Cairo .............................................. 1

2. Al-Îasan ibn al-Îasan and MuÌammad ibn al-Îasan:

mathematician and philosopher ......................................................... 11

3. The works of al-Îasan ibn al-Haytham on infinitesimal mathematics ......... 25

CHAPTER I: THE QUADRATURE OF LUNES AND CIRCLES

1.1. INTRODUCTION ....................................................................... 39

1.2. MATHEMATICAL COMMENTARY ................................................ 42

1.2.1. Treatise on lunes ................................................................. 42

1.2.2. Treatise on the quadrature of the circle ........................................ 46

1.2.3. Exhaustive treatise on the figures of lunes .................................... 49

1.3. TRANSLATED TEXTS

1.3.1. Treatise on Lunes ............................................................... 93

1.3.2. Treatise on the Quadrature of the Circle .................................... 99

1.3.3. Exhaustive Treatise on the Figures of Lunes .............................. 107

CHAPTER II: CALCULATION OF VOLUMES OF PARABOLOIDS AND SPHERES

AND THE EXHAUSTION METHOD

2.1. INTRODUCTION ....................................................................... 143

2.2. MATHEMATICAL COMMENTARY ................................................ 144

2.2.1. Calculation of volumes of paraboloids ........................................ 144

2.2.2.1. Arithmetical lemmas ................................................... 144

2.2.2.2. Volume of a paraboloid of revolution ................................ 151

2.2.2.3. The volume of the second species of paraboloid ................... 160

2.2.2.4. Study of surrounding solids .......................................... 164

2.2.3. Calculation of the volume of a sphere ......................................... 168

2.3. TRANSLATED TEXTS:

2.3.1. On the Measurement of the Paraboloid ...................................... 177

2.3.2. On the Measurement of the Sphere .......................................... 221

2.3.3. On the Division of Two Different Magnitudes as Mentioned

in the First Proposition of the Tenth Book of Euclid’s Elements .............. 235

CHAPTER III: THE PROBLEMS OF ISOPERIMETRIC AND ISEPIPHANIC

FIGURES AND THE STUDY OF THE SOLID ANGLE

3.1. INTRODUCTION ....................................................................... 239

3.2. MATHEMATICAL COMMENTARY ................................................ 242

x CONTENTS

3.3. TRANSLATED TEXT: On the Sphere which is the Largest of all the Solid

Figures having Equal Perimeters and On the Circle which is the Largest

of all the Plane Figures having Equal Perimeters ...................................... 305

APPENDIX: THE APPROXIMATION OF ROOTS

4.1. MATHEMATICAL COMMENTARY ................................................ 343

4.2. TRANSLATED TEXTS

4.3.1. On the Cause of the Square Root, its Doubling and its Displacement ... 351

4.3.2. On the Extraction of the Side of a Cube ........................................ 357

SUPPLEMENTARY NOTES

1. On the Arithmetic of Transactions ................................................ 361

2. The Configuration of the Universe: a Book by al-Îasan ibn al-Haytham ? . 362

3. Ibn Sinæn and Ibn al-Haytham on the subject of ‘shadow lines’ ................ 377

4. Commentary in the Resolution of Doubts by Ibn al-Haytham on

Proposition X.1 of the Elements ...................................................... 381

5. List of Ibn al-Haytham’s works .................................................... 391

BIBLIOGRAPHY ............................................................................. 429

INDEXES

Index of names ........................................................................... 439

Subject index .............................................................................

Index of works .........................................................