Founding Figures and Commentators in Arabic Mathematics: A History of Arabic Sciences and Mathematics - Volume 1

 

Roshdi Rashed e Nader El-Bizri

Routledge |  2011 | 813 páginas | rar - pdf | 4,2 Mb

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In this unique insight into the history and philosophy of mathematics and science in the mediaeval Arab world, the eminent scholar Roshdi Rashed illuminates the various historical, textual and epistemic threads that underpinned the history of Arabic mathematical and scientific knowledge up to the seventeenth century. The first of five wide-ranging and comprehensive volumes, this book provides a detailed exploration of Arabic mathematics and sciences in the ninth and tenth centuries.

Extensive and detailed analyses and annotations support a number of key Arabic texts, which are translated here into English for the first time. In this volume Rashed focuses on the traditions of celebrated polymaths from the ninth and tenth centuries ‘School of Baghdad’ - such as the Banū Mūsā, Thābit ibn Qurra, Ibrāhīm ibn Sinān, Abū Ja´far al-Khāzin, Abū Sahl Wayjan ibn Rustām al-Qūhī - and eleventh-century Andalusian mathematicians like Abū al-Qāsim ibn al-Samh, and al-Mu’taman ibn Hūd. The Archimedean-Apollonian traditions of these polymaths are thematically explored to illustrate the historical and epistemological development of ‘infinitesimal mathematics’ as it became more clearly articulated in the eleventh-century influential legacy of al-Hasan ibn al-Haytham (‘Alhazen’).

Contributing to a more informed and balanced understanding of the internal currents of the history of mathematics and the exact sciences in Islam, and of its adaptive interpretation and assimilation in the European context, this fundamental text will appeal to historians of ideas, epistemologists, mathematicians at the most advanced levels of research.

CONTENTS

Editor’s Foreword .......... xiii

Preface........ xix

Note ......... xxiv

CHAPTER I : BANU MUSA AND THE CALCULATION OF THE VOLUME OF THE SPHERE AND THE CYLINDER

1.1. INTRODUCTION ....... 1

1.1.1. The Banu Musa: dignitaries and learned ..... 1

1.1.2. The mathematical works of the Banu Musa ....... 7

1.1.3. Treatise on the measurement of plane and spherical figures: a Latin translation and a rewritten version by al-Tusî ...... 10

1.1.4. Title and date of the  Banu Musa treatise .......... 34

1.2. MATHEMATICAL COMMENTARY .......... 38

1.2.1. Organization and structure of the Banu Musa book ..... 38

1.2.2. The area of the circle ......... 40

1.2.3. The area of the triangle and Hero’s formula ..... 46

1.2.4. The surface area of a sphere and its volume ...... 47

1.2.5. The two-means problem and its mechanical construction ...... 60

1.2.6a. The trisection of angles and Pascal’s Limaçon ........... 66

1.2.6b. Approximating cubic roots ..... 69

1.3. TRANSLATED TEXT: On the Knowledge of the Measurement of Plane and Spherical Figures.... 73

CHAPTER II: THABIT IBN QURRA AND HIS WORKS IN INFINITESIMAL MATHEMATICS

2.1. INTRODUCTION ............ 113

2.1.1. Thabit ibn Qurra: from Harran to Baghdad ............ 113

2.1.2. The works of Thabitibn Qurra in infinitesimal mathematics ....... 122

2.1.3. History of the texts and their translations ....... 124

2.2. MEASURING THE PARABOLA ........ 130

2.2.1. Organization and structure of Ibn Qurra’s treatise ......... 130

2.2.2. Mathematical commentary ....... 133

2.2.2.1. Arithmetical propositions ......... 133

2.2.2.2. Sequence of segments and bounding ...... 142

2.2.2.3. Calculation of the area of a portion of a parabola ......... 154

2.2.3. Translated text: On the Measurement of the Conic Section Called Parabola .. 169

2.3. MEASURING THE PARABOLOID ... 209

2.3.1. Organization and structure of Ibn Qurra’s treatise ... 209

2.3.2. Mathematical commentary ....... 214

2.3.2.1. Arithmetical propositions ..... 214

2.3.2.2. Extension to sequences of segments ...... 218

2.3.2.3. Volumes of cones, rhombuses and other solids ..... 223

2.3.2.4. Property of four segments ........ 230

2.3.2.5. Arithmetical propositions ... 231
2.3.2.6. Sequence of segments and bounding ..... 233
2.3.2.7. Calculation of the volumes of paraboloids ...... 244
2.3.2.8. Parallel between the treatise on the area of the parabola and the treatise on the volume of the paraboloid ....256
2.3.3. Translated text: On the Measurement of the Paraboloids .... 261
2.4. ON THE SECTIONS OF THE CYLINDER AND ITS LATERAL SURFACE .... 333
2.4.1. Introduction ...... 333
2.4.2. Mathematical commentary ........ 337
2.4.2.1. Plane sections of the cylinder . 337
2.4.2.2. Area of an ellipse and elliptical sections . 341
2.4.2.3. Concerning the maximal section of the cylinder and concerning its minimal sections .... 356
2.4.2.4. Concerning the lateral area of the cylinder and the lateral area of portions of the cylinder lying between the plane sections touching all sides ..... 363
2.4.3. Translated text: On the Sections of the Cylinder and its Lateral Surface . 381
CHAPTER III: IBN SINAN, CRITIQUE OF AL-MAHANï: THE AREA OF THE PARABOLA
3.1. INTRODUCTION ................. 459
3.1.1. Ibrahîm ibn Sinan: ‘heir’ and ‘critic’ .... 459
3.1.2. The two versions of The Measurement of the Parabola: texts and translations . 463
3.2. MATHEMATICAL COMMENTARY ....... 466
3.3. TRANSLATED TEXTS
3.3.1. On the Measurement of the Parabola .... 483
3.3.2. On the Measurement of a Portion of the Parabola ......... 495
CHAPTER IV: ABU JA‘FAR AL-KHAZIN: ISOPERIMETRICS AND ISEPIPHANICS
4.1. INTRODUCTION ... 503
4.1.1. Al-Khazin: his name, life and works ...... 503
4.1.2. The treatises of al-Khazin on isoperimeters and isepiphanics... 506
4.2. MATHEMATICAL COMMENTARY ..... 507
4.2.1. Introduction ..... 507
4.2.2. Isoperimetrics ... 509
4.2.3. Isepiphanics ....... 524
4.2.4. The opuscule of al-Sumaysa†î ... 546
4.3. TRANSLATED TEXTS
4.3.1. Commentary on the First Book of the Almagest ..... 551
4.3.2. The Surface of any Circle is Greater than the Surface of any Regular Polygon with the Same Perimeter (al-Sumaysa†î) ... 577
CHAPTER V: AL-QUHï, CRITIQUE OF THABIT: VOLUME OF THE PARABOLOID OF REVOLUTION
5.1. INTRODUCTION ........ 579
5.1.1. The mathematician and the artisan ...... 579
5.1.2. The versions of the volume of a paraboloid... 583
5.2. MATHEMATICAL COMMENTARY ... 588
5.3. TRANSLATION TEXTS
5.3.1. On the Determination of the Volume of a Paraboloid .. 599
5.3.2. On the Volume of a Paraboloid .... 609
CHAPTER VI: IBN AL-SAMÎ: THE PLANE SECTIONS OF A CYLINDER AND THE DETERMINATION OF THEIR AREAS
6.1. INTRODUCTION ...... 615
6.1.1. Ibn al-SamÌ and Ibn Qurra, successors to al-Îasan ibn Musa ..... 615
6.1.2. Serenus of Antinoupolis, al-Îasæn ibn Musa , Thabit ibn Qurra and Ibn al-SamÌ ... 618
6.1.3. The structure of the study by Ibn al-SamÌ ..... 622
6.2. MATHEMATICAL COMMENTARY .. 623
6.2.1. Definitions and accepted results ... 623
6.2.2. The cylinder .... 626
6.2.3. The plane sections of a cylinder ....... 627
6.2.4. The properties of a circle ...... 628
6.2.5. Elliptical sections of a right cylinder .... 632
6.2.6. The ellipse as a plane section of a right cylinder .. 639
6.2.7. The area of an ellipse ......... 645
6.2.8. Chords and sagittas of the ellipse ....... 653
6.3. TRANSLATED TEXT: On the Cylinder and its Plane Section ........... 667
CHAPTER VII: IBN HUD: THE MEASUREMENT OF THE PARABOLA AND THE ISOPERIMETRIC PROBLEM
7.1. INTRODUCTION ...... 721
7.1.1. Kitab al-Istikmal, a mathematical compendium ....... 721
7.1.2. Manuscript transmission of the texts .......... 727
7.2. THE MEASUREMENT OF THE PARABOLA ...... 729
7.2.1. Infinitesimal property or conic property ...... 729
7.2.2. Mathematical commentary on Propositions 18–21 ..... 733
7.2.3. Translation: Kitab al-Istikmal........ 749
7.3. THE ISOPERIMETRIC PROBLEM ........... 755
7.3.1. An extremal property or a geometric property ...... 755
7.3.2. Mathematical commentary on Propositions 16 and 19 ..... 758
7.3.3. Translation: Kitab al-Istikmal...... 764
SUPPLEMENTARY NOTES
The Formula of Hero of Alexandria according to Thæbit ibn Qurra.... 767
Commentary of Ibn Abî Jarræda on The Sections of the Cylinder by Thabit ibn Qurra .767
BIBLIOGRAPHY .............. 779
INDEXES
Index of names .... 793
Subject index ........ 797
Index of works .......... 805

Outro livro da mesma colecção:

Ibn al-Haytham's theory of conics, geometrical constructions and practical geometry : a history of Arabic sciences and mathematics. Volume 3 por Rushdī Rāshid; Judith Veronica Field Idioma: Inglês   Editora: London : Routledge, 2013.