Ancient Indian Leaps into Mathematics

von: B.S. Yadav, Man Mohan

Birkhäuser Basel, 2011

ISBN: 9780817646950 , 218 Seiten

2. Auflage

Format: PDF

Kopierschutz: Wasserzeichen

Windows PC,Mac OSX Apple iPad, Android Tablet PC's

Preis: 109,99 EUR

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Mehr zum Inhalt

Ancient Indian Leaps into Mathematics


 

Contents

Contents

Foreword

12

Preface

16

List of Contributors

20

Indian Calendrical Calculations

22

1 Introduction

22

2 Diurnal Calendars

24

3 Mean Solar Calendars

25

3.1 Single-Cycle Calendars

26

3.2 Generic Single-Cycle Calendars

27

3.3 Indian Mean Solar Calendar

31

4 True Solar Calendars

32

4.1 Generic Solar Calendars

32

4.2 True Indian Solar Calendar

34

4.3 Indian Astronomical Solar Calendar

35

5 Lunisolar Calendars

37

5.1 A Generic Dual-Cycle Calendar

38

6 True Lunisolar Calendar

39

7 Sunrise

43

8 Holidays

43

References

52

India's Contributions to Chinese Mathematics Through the Eighth Century C.E.

53

1 Buddhism: The Medium of Interaction

53

2 Indian Astronomy and Mathematics in Ancient China

54

3 Earlier Chinese Parallels of Indian Mathematical Pieces

57

4 I-Hsing (683--727 C.E.): The Great Chinese Astronomer--Mathematician

63

The Influence of Indian Trigonometry on Chinese Calendar-Calculations in the Tang Dynasty

65

1 The Impact of Indian Trigonometry on Mathematics in Ancient China

65

1.1 The Impact of the Basic Concept and Rsin

66

1.1.1 The Basic Concept

66

1.1.2 The Table of Rsin

66

1.2 Yi Xing and the Table of Tangents in Dayanli

67

1.2.1 A Short Biography of Yi Xing

67

1.2.2 The Case of Dayan Plagiarizing Chiuchi

67

1.2.3 Yi Xing's Table of Tangents

68

1.3 The Influence of Futianli

71

2 Conclusions and Some Remarks

72

2.1 A Comparison Between Calendar Systems

72

2.2 Equivalence of the Chinese Gou--Gu Method and Indian Trigonometry

73

2.3 Conclusion for Exchanges

73

References

74

André Weil: His Book on Number Theory and Indian References

75

1 André Weil

75

2 His Book Number Theory

77

3 The Square-Nature (Varga-Prakrti)

78

References

80

On the Application of Areas in the Sulbasutras

82

1 The Sulbasutras

82

2 Mathematics in the Sulbasutras

82

3 The Agnicayana

83

4 Relationship Between the Sulbasutras and Older Literature

85

5 Application of Areas

86

6 Transition from Rectangular Falcon to Realistic Falcon

87

7 The Tail of the Falcon

87

8 Quadratic Equations in Ancient Mesopotamia

90

References

92

Divisions of Time and Measuring Instruments of Varahmihira

93

1 Introduction

93

2 Divisions of Time Prior to Varahmihira

95

2.1 Measures of Time in Vedanga Jyotisa

95

2.2 The Concept of Moment (Ksana)

101

2.3 Reckoning of Time in the Arthasastra

102

2.4 Divisions of Time in Aryabhatiya

105

3 Divisions of Time in the Brhatsamhita

108

4 Partitions of Time in the Brahmasphuta Siddhanta

110

5 Reckoning of Time in the Modern Surya-Siddhanta

110

6 Measurement of Time Prior to Varahmihira

116

7 The Ambu-Yantra of Varahmihira

119

8 The Ambu-Yantra After Varahmihira

121

9 Measurement of Time by Sanku-Yantra

123

References

126

The Golden Mean and the Physics of Aesthetics

129

1 Introduction

129

2 Historical Background

130

3 A Multiplicative Mount Meru and a Multiplicative Sequence of Notes

133

4 General Recurrence Sequences

134

5 Wilson's Meru 1 Through Meru 9

134

6 Structural Considerations

135

7 Concluding Remarks

136

References

136

Pingala Binary Numbers

138

1 Introduction

138

2 Fundamentals

139

2.1 Chandas or Meter

140

2.2 Pada or Quarter

140

2.3 Aksara or Syllable

140

2.4 Laghu or Short Syllables

140

2.5 Guru or Long Syllables

141

2.6 Matra or Metrical Unit

142

2.7 Verse Classification

142

3 Pratyayas: Methods of Cognitions

144

3.1 Varnic Expansion

144

3.2 Nasta

145

3.3 Uddista: Conversion from a Pingala Binary Number to Decimals

147

4 Concluding Remarks

149

References

150

The Reception of Ancient Indian Mathematics by Western Historians

152

1 The Context of Renaissance Humanism

152

2 The First Descriptions of Indian Algebra

154

3 A Case Study: The Bloom of Thymaridas

157

3.1 The Original Formulation in Hindu Sources

157

3.2 The Derived Problem in Hindu Sources

158

3.3 The Problem in Greek Sources

160

3.3.1 The Bloom of Thymaridas

160

3.3.2 Diophantus

161

3.3.3 The Extended Rule from Iamblichus

162

3.3.4 The Controversy

164

4 Conclusion: The Ground Was Wet Everywhere

166

References

166

The Indian Mathematical Tradition with Special Reference to Kerala: Methodology and Motivation

170

1 Introduction

170

2 Some Significant Developments and Their Motivations

171

3 Notion of Proof: Forms, Nature, Style, and Purpose

179

4 The Role of Commentarial Literature in the Dissemination of Mathematical Knowledge

184

5 Commentarial Literature: A Rich Source for the Study of Proof, Methodology, and Motivation

185

The Algorithm of Extraction in Greek and Sino-Indian Mathematical Traditions

188

1 Introduction

188

2 The Algorithm of Extraction in Ancient Greece

188

2.1 Heron of Alexandria's Method

189

2.2 Theon of Alexandria's Method

191

2.3 The Influence and Evolution of the Algorithm of Extraction in Western Europe

193

3 The Algorithm of Extraction in Ancient China

194

3.1 The Pre-Method of the Algorithm of Extraction

194

3.2 The Method of the Algorithm of the Extraction in the Nine Chapters and Thereafter

195

3.3 Liu Hui's Geometrical Explanation of the Algorithm of Extraction

196

3.4 The Influence and Evolution of the Algorithm of Extraction in China

197

4 The Algorithm of Extraction in Ancient India

198

5 A Brief Comparison and Conclusions

200

5.1 The Accuracy in the Algorithm, Approximation in the Theorem-Proving System

200

5.2 The Minor Difference

200

5.3 Brief Conclusions

200

References

201

Brahmagupta: The Ancient Indian Mathematician

202

References

208

Mainland Southeast Asia as a Crossroads of Chinese Astronomy and Indian Astronomy

210

1 Introduction

210

2 Vietnamese Calendrical Astronomy

210

3 Mainland Southeast Asian Astronomy (Except for Vietnam)

212

4 Mainland Southeast Asian 19-Year Cycle

213

5 Conclusion

216

References

216

Mathematical Literature in the Regional Languages of India

218

Index

229