Ibn al-Haytham's Geometrical Methods and the Philosophy of Mathematics A History of Arabic Sciences and Mathematics Volume 5 Culture and Civilization in the Middle East Series

This fifth volume of A History of Arabic Sciences and Mathematics is complemented by four preceding volumes which focused on the main chapters of classical mathematics: infinitesimal geometry, theory of conics and its applications, spherical geometry, mathematical astronomy, etc.

This book includes seven main works of Ibn al-Haytham (Alhazen) and of two of his predecessors, Th?bit ibn Qurra and al-Sijz?:

• The circle, its transformations and its properties;





• Analysis and synthesis: the founding of analytical art;





• A new mathematical discipline: the Knowns;





• The geometrisation of place;





• Analysis and synthesis: examples of the geometry of triangles;





• Axiomatic method and invention: Th?bit ibn Qurra;





• The idea of an Ars Inveniendi: al-Sijz?.

Including extensive commentary from one of the world?s foremost authorities on the subject, this fundamental text is essential reading for historians and mathematicians at the most advanced levels of research.

CONTENTS

Foreword

Preface

CHAPTER I: THE PROPERTIES OF THE CIRCLE

INTRODUCTION

1. The concept of homothety

2. Euclid, Pappus and Ibn al-Haytham: on homothety

3. Ibn al-Haytham and homothety as a point by point transformation

4. History of the text

MATHEMATICAL COMMENTARY

TRANSLATED TEXT: On the Properties of Circles

CHAPTER II: THE ANALYTICAL ART IN THE TENTH TO ELEVENTH

CENTURIES

INTRODUCTION

1. The rebirth of a subject

2. Analytical art: discipline and method

3. The analytical art and the new discipline: ‘The Knowns’

4. History of the texts

On Analysis and Synthesis

The Knowns

I. ANALYSIS AND SYNTHESIS: MATHEMATICAL METHOD AND DISCIPLINE

MATHEMATICAL COMMENTARY

1. The double classification of Analysis and Synthesis

Preliminary propositions

Analysis and synthesis in arithmetic

Analysis and synthesis in geometry

Analysis and synthesis in astronomy

Analysis in music

2. Applications of analysis and synthesis in number theory and in geometry

Number theory

Perfect Numbers

Two indeterminate systems of equations of the first degree

Geometrical problems

Problem in plane geometry

Problem solved with the help of transformations

Construction of a circle to touch three given circles

xii CONTENTS

Auxiliary problem

Geometrical commentary on the problem

Algebraic commentary on the auxiliary problem

TRANSLATED TEXT: On Analysis and Synthesis

II. THE KNOWNS: A NEW GEOMETRICAL DISCIPLINE

INTRODUCTION

MATHEMATICAL COMMENTARY

1. Properties of position and of form and geometrical transformations

2. Invariant properties of ge

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 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.

J. V. Field is a historian of science, and is a Visiting Research Fellow in the Department of History of Art and Screen Media, Birkbeck, University of London, UK.