Ancient Indian Leaps into Mathematics (eBook)

Contents 
10 
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