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El. knyga: Ibn al-Haytham's Geometrical Methods and the Philosophy of Mathematics: A History of Arabic Sciences and Mathematics Volume 5

Edited by (Centre National de la Recherche Scientifique (CNRS) in Paris, France)
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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, Thbit 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: Thbit ibn Qurra;





The idea of an Ars Inveniendi: al-Sijz.

Including extensive commentary from one of the worlds foremost authorities on the subject, this fundamental text is essential reading for historians and mathematicians at the most advanced levels of research.
Foreword ix
Preface xi
Introduction: Motion And Transformations In Geometry 1(16)
Chapter I The Properties of the Circle
Introduction
17(2)
1 The concept of homothety
19(4)
2 Euclid, Pappus and Ibn al-Haytham: on homothety
23(4)
3 Ibn al-Haytham and homothety as a point by point transformation
27(5)
4 History of the text
32(99)
Mathematical Commentary
35(50)
TRANSLATED TEXT: On the Properties of Circles
85(46)
Chapter II The Analytical art in the Tenth to Eleventh Centuries
Introduction
131(1)
1 The rebirth of a subject
131(7)
2 Analytical art: discipline and method
138(4)
3 The analytical art and the new discipline: `The Knowns'
142(9)
4 History of the texts
151(6)
On Analysis and Synthesis
151(2)
The Knowns
153(4)
I ANALYSIS AND SYNTHESIS: MATHEMATICAL METHOD AND DISCIPLINE MATHEMATICAL COMMENTARY
157(148)
1 The double classification of Analysis and Synthesis
157(17)
Preliminary propositions
157(5)
Analysis and synthesis in arithmetic
162(4)
Analysis and synthesis in geometry
166(5)
Analysis and synthesis in astronomy
171(2)
Analysis in music
173(1)
2 Applications of analysis and synthesis in number theory and in geometry
174(45)
Number theory
175(1)
Perfect numbers
175(3)
Two indeterminate systems of equations of the first degree
178(4)
Geometrical problems
182(1)
Problem in plane geometry
182(3)
Problem solved with the help of transformations
185(2)
Construction of a circle to touch three given circles
187(12)
Auxiliary problem
199(8)
Geometrical commentary on the problem
207(7)
Algebraic commentary on the auxiliary problem
214(5)
TRANSLATED TEXT: On Analysis and Synthesis
219(86)
II THE KNOWNS: A NEW GEOMETRICAL DISCIPLINE
305(130)
Introduction
305(3)
Mathematical Commentary
308(53)
1 Properties of position and of form and geometrical transformations
308(21)
2 Invariant properties of geometrical loci and geometrical transformations
329(32)
TRANSLATED TEXT: On the Knowns
361(74)
III ANALYSIS AND SYNTHESIS: EXAMPLES OF THE GEOMETRY OF TRIANGLES
435(82)
1 On a geometrical problem: Ibn Sahl, al-Sijzi and Ibn al-Haytham
436(17)
2 Distances from a point of a triangle to its sides
453(18)
3 History of the texts
471(6)
3A On a Geometrical Problem
471(2)
3.2 On the Properties of the Triangle
473(4)
TRANSLATED TEXTS: On a Geometrical Problem
477(18)
On the Properties of the Triangle in Regard to Height
485(10)
Chapter III IBN Al-Haytham and the Geometrisation of Place
495(22)
History of the Text
505(2)
TRANSLATED TEXT: On Place
507(10)
APPENDIX: THE ARS INVENIENDI: THABIT IBN QURRA AND AL-SIJZI
517(64)
I Thabit IBN Qurra: Axiomatic Method and Invention
517(4)
II Al-Sijzi: The Idea of An Ars Inveniendi
521(51)
1 Introduction
521(2)
2 A propaedeutic to the ars inveniendi
523(5)
3 The methods of the ars inveniendi and their applications
528(2)
3.1 Analysis and point-to-point transformation
530(3)
3.2 Analysis and variation of one element of the figure
533(1)
3.3 Analysis and variation of two methods of solution of a single problem
534(3)
3.4 Analysis and variation of lemmas
537(1)
3.5 Analysis and variation of constructions carried out using the same figure
537(6)
3.6 Variations on a problem from Ptolemy
543(17)
3.7 Variations on the same problem from Ptolemy in other writings by al-Sijzi
560(6)
4 Analysis and synthesis: variation of the auxiliary constructions
566(2)
5 Two principal methods of the ars inveniendi
568(4)
III History of the Texts
572(9)
3.1 Book by Thabit ibn Qurra for Ibn Wahb on the Means of Arriving at Determining the Construction of Geometrical Problems
572(3)
3.2 To Smooth the Paths for Determining Geometrical Propositions, by al-Sijzi
575(2)
3.3 Letter of al-Sijzi to Ibn Yumn on the Construction of an Acute-angled Triangle
577(1)
3.4 Two Propositions from the Ancients on the Property of Heights of an Equilateral Triangle: Ps-Archimedes, Aqatun, Menelaus
577(4)
TRANSLATED TEXTS
1 Book of Abu al-Hasan Thabit ibn Qurra for Ibn Wahb on the Means of Arriving at Determining the Construction of Geometrical Problems
581(10)
2 Book of Ahmad ibn Muhammad ibn 'Abd al-Jalil al-Sijzi to Smooth the Paths for Determining Geometrical Propositions
591(32)
3 Letter of Ahmad ibn Muhammad ibn 'Abd al-Jalil <al-Sijzi> to the Physician Abu 'Ali Nazif ibn Yumn on the Construction of the Acute-angled Triangle from Two Unequal Straight Lines
623(4)
4 Two Propositions of the Ancients on the Property of the Heights of an Equilateral Triangle: Pseudo-Archimedes, Aqatun, Menelaus
627(6)
SUPPLEMENTARY NOTES
I Fakhr al-Din al-Razi: Ibn al-Haytham's critique of the notion of place as envelope
633(1)
II Al-Hasan ibn al-Haytham and Muhammad ibn al-Haytham: the mathematician and the philosopher - On place
634(5)
BIBLIOGRAPHY
639(10)
INDEXES
Index of names
649(3)
Subject Index
652(7)
Index of works
659(5)
Index of manuscripts
664
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.
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