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Home page > Publications > Grandes éditions > A HISTORY OF ARABIC SCIENCES AND MATHEMATICS

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

A HISTORY OF ARABIC SCIENCES AND MATHEMATICS

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

Volume 5, 2017





Roshdi Rashed (SPHERE, UMR 7219)
J. V. Field (Trad.)







"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."



:: Routledge
:: xx, 674 p.
:: Langues : anglais
:: ISBN: 9780415582193
:: avril 2017



Other volumes of the serie:



TABLE OF CONTENTS [to download]



Foreword
Preface


INTRODUCTION: MOTION AND TRANSFORMATIONS IN GEOMETRY


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
      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. Thăbit 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
  • 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 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
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