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Home > Publications > Published books > Books published by researchers of SPHERE: 2009–2014 > Classical Mathematics from al-Khwārizmī to Descartes

Classical Mathematics from al-Khwārizmī to Descartes



Roshdi Rashed



Translated by Michael Shank




This book follows the development of classical mathematics and the relation between work done by the Arab and Islamic worlds and undertaken by the likes of Descartes and Fermat.
’Early modern’ mathematics is a term widely used to refer to the mathematics in the West during the sixteenth and seventeenth century. For many historians and philosophers this the watershed which marks a radical departure from ’classical mathematics’, to more modern mathematics; heralding the arrival of algebra, geometrical algebra, and the mathematics of the continuous. In this book, Roshdi Rashed demonstrates that ’early modern’ mathematics is actually far more composite than previously assumed, with each branch having different traceable origins which span the millenium. Going back to the beginning of these parts, the aim of this book is to identify the concept and practices of key figures in their development, thereby presenting a fuller reality of these mathematics.

This book will be of interest to students and scholars specialising in Islamic science and mathematics, as well as to those with an interest in the more general history of science and mathematics and the transmission of ideas and culture.



:: Culture and Civilization in the Middle East, Londres: Centre for Arab Unity Studies, Routledge
:: 978-0-415-83388-2
:: August 2014
:: xvi-749 p.



TABLE DES MATIERES

Foreword, p. xiii
Translator’s Note, p. xiv
Preface, p. xv


INTRODUCTION: PROBLEMS OF METHOD

1. The history of science: between epistemology and history, p. 3
2. The transmission of Greek heritage into Arabic, p. 19

  • 1. Transmission and translation: setting up the problem, p. 20
    • 1. Towards a new approach, p. 20
    • 2. Cultural transmission, scientific transmission, p. 22
    • 3. Scholarly transmission: one myth and several truths, p. 24
      • 3.1. The rebirth of research, p. 25
      • 3.2. Institution and profession: the age of the Academies, p. 29
      • 3.3. An ideal type of translator: Ḥunayn ibn Isḥāq’s journey, p. 33
      • 3.4. Third phase: from translator-scientist to scientist-translator, p. 35
  • 2. Translation and research: a dialectic with many forms, p. 36
    • 1. Coexisting and overtaking: optics and catoptrics, p. 36
    • 2. Translation and recursive reading: the case of Diophantus, p. 45
    • 3. Translation as a vehicle of research: the Apollonius project, p. 50
    • 4. Ancient evidence of the translation-research dialectic: the case of the Almagest, p. 53
  • 3. Prospective conclusion, p. 55

3. Reading ancient mathematical texts:
the fifth book of Apollonius’s Conics , p. 57
4. The founding acts and main contours of Arabic mathematics, p. 83


PART I
I. ALGEBRA
1. Algebra and its unifying role, p. 105

  • 1. The beginning of algebra: al-Khāwrīzmī, p. 107
  • 2. Al-Khāwrīzmī’s successors: geometrical interpretation and development of algebraic calculation, p. 113
  • 3. The arithmetization of algebra: al-Karajī and his successors, p. 117
  • 4. The geometrization of algebra: al-Khayyām (1048–1131), p. 125
  • 5. The transformation of the theory of algebraic equations: Sharaf al-Dīn al-Ṭūsī, p. 135
  • 6. The destiny of the theory of equations, p. 145

2. Algebra and linguistics: the beginnings of combinatorial analysis, p. 149

  • 1. Linguistics and combinatorics 150
  • 2. Algebraic calculation and combinatorics, p. 160
  • 3. Arithmetic research and combinatorics, p. 162
  • 4. Philosophy and combinatorics, p. 164
  • 5. A treatise on combinatorial analysis, p. 165
  • 6. On the history of combinatorial analysis, p. 169

3. The first classifications of curves, p. 171

  • 1. Introduction, p. 171
  • 2. Simple curves and mixed curves, p. 176
  • 3 Geometrical and mechanical: the characterization of conic sections, p. 188
  • 4. Geometrical transformation
    and the classification of curves, p. 195
  • 5. The intervention of the algebraists: the polynomial equation and the algebraic curve, p. 199
  • 6. The classification of curves as mechanical and geometrical, p. 204
  • 7. Developments of the Cartesian classification of algebraic curves, p. 225
  • 8. Conclusion, p. 234

Appendix: Simplicius: On the Euclidean definition of the straight line and of curved lines, p. 237
4. Descartes’s Géométrie and the distinction between geometrical and mechanical curves, p. 239

  • 1. The geometrical theory of algebraic equations: the completion of al-Khayyām’s program, p. 241
  • 2. From geometry to algebra: the curves and the equations, p. 248

5. Descartes’s ovals, p. 259
6. Descartes and the infinitely small, p. 281
7. Fermat and algebraic geometry, p. 301

  • 1. The geometrical loci and the pointwise transformations, p. 303
  • 2. The equations of geometrical loci, p. 311
  • 3. Solution of equations by the intersection of two curves, p. 316
  • 4. The solution of algebraic equations and the study of algebraic curves, p. 319

II ARITHMETIC
1. Euclidean, neo-Pythagorean and Diophantine arithmetics: new methods in number theory, p. 333

  • 1. Classical number theory, p. 333
    • 1.1. Euclidean and neo-Pythagorean arithmetic, p. 334
    • 1.2. Amicable numbers and the discovery of elementary arithmeticfunctions, p. 336
    • 1.3. Perfect numbers, p. 340
    • 1.4. Equivalent numbers, p. 341
    • 1.5. Polygonal numbers and figurate numbers, p. 342
    • 1.6. The characterization of prime numbers, p. 345
  • 2. Indeterminate analysis, p. 346
    • 2.1. Rational Diophantine analysis, p. 346
    • 2.2. Integer Diophantine analysis, p. 355
    • 2.3. Arithmetic methods in number theory, p. 363

2. Algorithmic methods, p. 365

  • 1. Numerical equations, p. 368
    • 1.1. The extraction of roots, p. 368
    • 1.2. The extraction of roots and the invention of decimal fractions, p. 377
    • 1.3. Numerical polynomial equations, p. 379
  • 2. Interpolation methods, p. 389
  • 3. Thābit ibn Qurra and amicable numbers, p. 399
  • 4. Fibonacci and Arabic mathematics, p. 411
  • 5. Fibonacci and the Latin extension of Arabic mathematics, p. 425
  • 6. Al-Yazdī and the equation
  • 7. Fermat and the modern beginnings of Diophantine analysis, p. 453


PART II:
GEOMETRY
1. The Archimedeans and problems with infinitesimals, p. 473

  • 1. Calculating infinitesimal areas and volumes, p. 475
    • 1.1. The Pioneers, p. 475
    • 1.2. The Heirs, p. 499
    • 1.3. Later developments, p. 506
  • 2. The quadrature of lunes, p. 517
  • 3. Equal perimeters and equal surface areas: a problem of extrema, p. 525
    • 3.1. Al-Khāzin: the mathematics of the Almagest, p. 527
      • 3.1.1. Isoperimeters, p. 528
      • 3.1.2. Equal areas, p. 530
    • 3.2. Ibn al-Haytham: a new theory, p. 531
      • 3.2.1. Isoperimeters, p. 532
      • 3.2.2. Equal surface areas, p. 533
  • 4. The theory of the solid angle, p. 538

2. The traditions of the Conics and the beginning of research on projections, p. 555

  • 1. Cylindrical projections, p. 557
    • 1.1. Al-Bīrūnī’s testimony and his priority claim, p. 557
    • 1.2. Al-Ḥasan ibn Mūsā’s study of the ellipse, p. 559
    • 1.3. Thābit’s treatise on the cylinder, p. 560
    • 1.4. Ibn al-Samḥ’s study of plane sections of a cylinder and the determination of their areas, p. 565
    • 1.5. The theory of projections: al-Qūhī and Ibn Sahl, p. 571
  • 2. Conic projections, p. 575
    • 2.1. Ptolemy’s Planisphere, p. 575
    • 2.2. Al-Farghn’s treatise, al-Kāmil fī ṣanat al-asṭurlāb, p. 578
    • 2.3. Al-Qūhī’s treatise and Ibn Sahl’s commentary on it, p. 583
    • 2.4. Al-ghn’s study of the projection of the sphere, p. 591
    • 2.5. The construction of the sumūt, p. 597

3. The continuous drawing of conic curves and the classification of curves, p. 601

  • 1. Introduction, p. 601
  • 2. Ibn Sahl: a mechanical device to trace conic sections, p. 605
  • 3. Al-Qūhī: the perfect compass, p. 607
  • 4. Al-Sijzī: the improved perfect compass, p. 614
  • 5. Continuous drawing and classification of curves, p. 618

4. Thābit ibn Qurra on Euclid’s fifth postulate, p. 621

  • 1. Introduction, p. 621
  • 2. Thābit ibn Qurra’s first treatise, p. 625
  • 3. Thābit ibn Qurra’s second treatise, p. 630


PART III
APPLICATION OF MATHEMATICS: ASTRONOMY AND OPTICS
1. The celestial kinematics of Ibn al-Haytham, p. 637

  • 1. Introduction, p. 637
    • 1.1. The astronomical work of Ibn al-Haytham, p. 637
    • 1.2. The Configuration of the Motions of the Seven Wandering Stars, p. 645
  • 2. The structure of The Configuration of the Motions, p. 649
    • 2.1. Research on the variations, p. 650
    • 2.2. The planetary theory, p. 659

2. From the geometry of the gaze
to the mathematics of the phenomena of light, p. 681


CONCLUSION: The philosophy of mathematics, p. 695
1. Mathematics as conditions and models of philosophical activity: al-Kindī and Maimonides, p. 699
2. Mathematics in the philosophical synthesis and the ‘formal’ inflection of theontology: Ibn Sn and Nar al-Dīn al-Ṭūsī, p. 708
3. From ars inveniendi to ars analytica, p. 726


INDEX
Index of names, p. 733
Index of works, p. 743





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