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.

• CONTENTSPreface ..................................................................................... xiNote ....................................................................................... xiiiINTRODUCTION: IBN AL-HAYTHAM AND HIS WORK ON INFINITESIMALMATHEMATICS1. Ibn al-Haytham: from Basra to Cairo .............................................. 12. Al-Îasan ibn al-Îasan and MuÌammad ibn al-Îasan:mathematician and philosopher ......................................................... 113. The works of al-Îasan ibn al-Haytham on infinitesimal mathematics ......... 25CHAPTER I: THE QUADRATURE OF LUNES AND CIRCLES1.1. INTRODUCTION ....................................................................... 391.2. MATHEMATICAL COMMENTARY ................................................ 421.2.1. Treatise on lunes ................................................................. 421.2.2. Treatise on the quadrature of the circle ........................................ 461.2.3. Exhaustive treatise on the figures of lunes .................................... 491.3. TRANSLATED TEXTS1.3.1. Treatise on Lunes ............................................................... 931.3.2. Treatise on the Quadrature of the Circle .................................... 991.3.3. Exhaustive Treatise on the Figures of Lunes .............................. 107CHAPTER II: CALCULATION OF VOLUMES OF PARABOLOIDS AND SPHERESAND THE EXHAUSTION METHOD2.1. INTRODUCTION ....................................................................... 1432.2. MATHEMATICAL COMMENTARY ................................................ 1442.2.1. Calculation of volumes of paraboloids ........................................ 1442.2.2.1. Arithmetical lemmas ................................................... 1442.2.2.2. Volume of a paraboloid of revolution ................................ 1512.2.2.3. The volume of the second species of paraboloid ................... 1602.2.2.4. Study of surrounding solids .......................................... 1642.2.3. Calculation of the volume of a sphere ......................................... 1682.3. TRANSLATED TEXTS:2.3.1. On the Measurement of the Paraboloid ...................................... 1772.3.2. On the Measurement of the Sphere .......................................... 2212.3.3. On the Division of Two Different Magnitudes as Mentionedin the First Proposition of the Tenth Book of Euclid’s Elements .............. 235CHAPTER III: THE PROBLEMS OF ISOPERIMETRIC AND ISEPIPHANICFIGURES AND THE STUDY OF THE SOLID ANGLE3.1. INTRODUCTION ....................................................................... 2393.2. MATHEMATICAL COMMENTARY ................................................ 242x CONTENTS3.3. TRANSLATED TEXT: On the Sphere which is the Largest of all the SolidFigures having Equal Perimeters and On the Circle which is the Largestof all the Plane Figures having Equal Perimeters ...................................... 305APPENDIX: THE APPROXIMATION OF ROOTS4.1. MATHEMATICAL COMMENTARY ................................................ 3434.2. TRANSLATED TEXTS4.3.1. On the Cause of the Square Root, its Doubling and its Displacement ... 3514.3.2. On the Extraction of the Side of a Cube ........................................ 357SUPPLEMENTARY NOTES1. On the Arithmetic of Transactions ................................................ 3612. The Configuration of the Universe: a Book by al-Îasan ibn al-Haytham ? . 3623. Ibn Sinæn and Ibn al-Haytham on the subject of ‘shadow lines’ ................ 3774. Commentary in the Resolution of Doubts by Ibn al-Haytham onProposition X.1 of the Elements ...................................................... 3815. List of Ibn al-Haytham’s works .................................................... 391BIBLIOGRAPHY ............................................................................. 429INDEXESIndex of names ........................................................................... 439Subject index .............................................................................Index of works .........................................................