Ibn al-Haytham's Geometrical Methods and the Philosophy of 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 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.

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