Ibn al-Haytham's Geometrical Methods and the Philosophy of Mathematics: A History of Arabic Sciences and Mathematics Volume 5 (Hardback) book cover

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

A History of Arabic Sciences and Mathematics Volume 5

Edited by Roshdi Rashed

© 2017 – Routledge

674 pages

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Description

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, Thābit ibn Qurra and al-Sijzī:

  • 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: Thābit ibn Qurra;
  • The idea of an Ars Inveniendi: al-Sijzī.

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.

Table of Contents

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 geometrical loci and geometrical transformations

TRANSLATED TEXT: The Knowns

III: ANALYSIS AND SYNTHESIS: EXAMPLES OF THE GEOMETRY OF

TRIANGLES

1. On a geometrical problem: Ibn Sahl, al-Sijzī and Ibn al-Haytham

2. Distances from a point of a triangle to its sides

3. History of the texts

3.1. On a Geometrical Problem

3.2. On the Properties of the Triangle

TRANSLATED TEXTS:

On a Geometrical Problem

On the Properties of the Triangle in Regard to Height

CHAPTER III: IBN AL-HAYTHAM AND THE GEOMETRISATION

OF PLACE

HISTORY OF THE TEXT

TRANSLATED TEXT: On Place

APPENDIX: THE ARS INVENIENDI: THĀBIT IBN QURRA AND AL-SIJZĪ

I. THĀBIT IBN QURRA: AXIOMATIC METHOD AND INVENTION

II. AL-SIJZĪ: THE IDEA OF AN ARS INVENIENDI

1. Introduction

2. A propaedeutic to the ars inveniendi

3. The methods of the ars inveniendi and their applications

3.1. Analysis and point-to-point transformation

3.2 Analysis and variation of one element of the figure

3.3. Analysis and variation of two methods of solution of a single problem

3.4. Analysis and variation of lemmas

3.5. Analysis and variation of constructions carried out using the same figure

3.6. Variations on a problem from Ptolemy

3.7. Variations on the same problem from Ptolemy

in other writings by al-Sijzī

CONTENTS xiii

4. Analysis and synthesis: variation of the auxiliary constructions

5. Two principal methods of the ars inveniendi

III. HISTORY OF THE TEXTS

3.1. Book by Thābit ibn Qurra for Ibn Wahb on the Means of Arriving at

Determining the Construction of Geometrical Problems

3.2. To Smooth the Paths in view of Determining Geometrical Propositions,

by al-Sijzī

3.3. Letter of al-Sijzī to Ibn Yumn on the Construction

of an Acute-angled Triangle

3.4. Two Propositions from the Ancients on the Property of Heights of

an Equilateral Triangle: Ps-Archimedes, Aqāṭun, Menelaus

TRANSLATED TEXTS:

1. Book of Abū al-Ḥasan Thābit ibn Qurra for Ibn Wahb on the Means of

Arriving at Determining the Construction of Geometrical Problems

2. Book of Aḥmad ibn Muḥammad ibn ‘Abd al-Jalīl al-Sijzī to Smooth the Paths

in view of Determining Geometrical Propositions

3. Letter of Aḥmad ibn Muḥammad ibn ‘Abd al-Jalīl <al-Sijzī> to the Physician

Abū ‘Alī Naẓīf ibn Yumn on the Construction of the Acute-angled Triangle

from Two Unequal Straight Lines

4. Two Propositions of the Ancients on the Property of the Heights of

an Equilateral Triangle: Pseudo-Archimedes, Aqāṭun, Menelaus

SUPPLEMENTARY NOTES

I. Fakhr al-Dīn al-Rāzī:

Ibn al-Haytham’s critique of the notion of place as envelope

II. Al-Ḥasan ibn al-Haytham and Muḥammad ibn al-Haytham:

the mathematician and the philosopher – On place

BIBLIOGRAPHY

INDEXES

About the Editor

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.

About the Series

Culture and Civilization in the Middle East

This series studies the Middle East through the twin foci of its diverse cultures and civilizations. Comprising original monographs as well as scholarly surveys, it covers topics in the fields of Middle Eastern literature, archaeology, law, history, philosophy, science, folklore, art, architecture and language. While there is a plurality of views, the series presents serious scholarship in a lucid and stimulating fashion.

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Subject Categories

BISAC Subject Codes/Headings:
HIS026000
HISTORY / Middle East / General
MAT015000
MATHEMATICS / History & Philosophy
SOC053000
SOCIAL SCIENCE / Regional Studies