Yadav / Mohan Ancient Indian Leaps into Mathematics

1. Auflage 2011
ISBN: 978-0-8176-4695-0
Verlag: Birkhäuser Boston
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 218 Seiten

This book presents contributions of mathematicians covering topics from ancient India, placing them in the broader context of the history of mathematics. Although the translations of some Sanskrit mathematical texts are available in the literature, Indian contributions are rarely presented in major Western historical works. Yet some of the well-known and universally-accepted discoveries from India, including the concept of zero and the decimal representation of numbers, have made lasting contributions to the foundation of modern mathematics. Through a systematic approach, this book examines these ancient mathematical ideas that were spread throughout India, China, the Islamic world, and Western Europe.

Autoren/Hrsg.

Mohan, Man

Yadav, B. S.

Inhaltsverzeichnis

1;Contents
;10
2;Foreword;12
3;Preface;16
4;List of Contributors;20
5;Indian Calendrical Calculations;22
5.1;1 Introduction;22
5.2;2 Diurnal Calendars;24
5.3;3 Mean Solar Calendars;25
5.3.1;3.1 Single-Cycle Calendars;26
5.3.2;3.2 Generic Single-Cycle Calendars;27
5.3.3;3.3 Indian Mean Solar Calendar;31
5.4;4 True Solar Calendars;32
5.4.1;4.1 Generic Solar Calendars;32
5.4.2;4.2 True Indian Solar Calendar;34
5.4.3;4.3 Indian Astronomical Solar Calendar;35
5.5;5 Lunisolar Calendars;37
5.5.1;5.1 A Generic Dual-Cycle Calendar;38
5.6;6 True Lunisolar Calendar;39
5.7;7 Sunrise;43
5.8;8 Holidays;43
5.9;References;52
6;India's Contributions to Chinese Mathematics Through the Eighth Century C.E.;53
6.1;1 Buddhism: The Medium of Interaction;53
6.2;2 Indian Astronomy and Mathematics in Ancient China;54
6.3;3 Earlier Chinese Parallels of Indian Mathematical Pieces;57
6.4;4 I-Hsing (683--727 C.E.): The Great Chinese Astronomer--Mathematician;63
7;The Influence of Indian Trigonometry on Chinese Calendar-Calculations in the Tang Dynasty;65
7.1;1 The Impact of Indian Trigonometry on Mathematics in Ancient China;65
7.1.1;1.1 The Impact of the Basic Concept and Rsin;66
7.1.1.1;1.1.1 The Basic Concept;66
7.1.1.2;1.1.2 The Table of Rsin;66
7.1.2;1.2 Yi Xing and the Table of Tangents in Dayanli;67
7.1.2.1;1.2.1 A Short Biography of Yi Xing;67
7.1.2.2;1.2.2 The Case of Dayan Plagiarizing Chiuchi;67
7.1.2.3;1.2.3 Yi Xing's Table of Tangents;68
7.1.3;1.3 The Influence of Futianli;71
7.2;2 Conclusions and Some Remarks;72
7.2.1;2.1 A Comparison Between Calendar Systems;72
7.2.2;2.2 Equivalence of the Chinese Gou--Gu Method and Indian Trigonometry;73
7.2.3;2.3 Conclusion for Exchanges;73
7.3;References;74
8;André Weil: His Book on Number Theory and Indian References;75
8.1;1 André Weil;75
8.2;2 His Book Number Theory;77
8.3;3 The Square-Nature (Varga-Prakrti);78
8.4;References;80
9;On the Application of Areas in the Sulbasutras;82
9.1;1 The Sulbasutras;82
9.2;2 Mathematics in the Sulbasutras;82
9.3;3 The Agnicayana;83
9.4;4 Relationship Between the Sulbasutras and Older Literature;85
9.5;5 Application of Areas;86
9.6;6 Transition from Rectangular Falcon to Realistic Falcon;87
9.7;7 The Tail of the Falcon;87
9.8;8 Quadratic Equations in Ancient Mesopotamia;90
9.9;References;92
10;Divisions of Time and Measuring Instruments of Varahmihira;93
10.1;1 Introduction;93
10.2;2 Divisions of Time Prior to Varahmihira;95
10.2.1;2.1 Measures of Time in Vedanga Jyotisa;95
10.2.2;2.2 The Concept of Moment (Ksana);101
10.2.3;2.3 Reckoning of Time in the Arthasastra;102
10.2.4;2.4 Divisions of Time in Aryabhatiya;105
10.3;3 Divisions of Time in the Brhatsamhita;108
10.4;4 Partitions of Time in the Brahmasphuta Siddhanta;110
10.5;5 Reckoning of Time in the Modern Surya-Siddhanta;110
10.6;6 Measurement of Time Prior to Varahmihira;116
10.7;7 The Ambu-Yantra of Varahmihira;119
10.8;8 The Ambu-Yantra After Varahmihira;121
10.9;9 Measurement of Time by Sanku-Yantra;123
10.10;References;126
11;The Golden Mean and the Physics of Aesthetics;129
11.1;1 Introduction;129
11.2;2 Historical Background;130
11.3;3 A Multiplicative Mount Meru and a Multiplicative Sequence of Notes;133
11.4;4 General Recurrence Sequences;134
11.5;5 Wilson's Meru 1 Through Meru 9;134
11.6;6 Structural Considerations;135
11.7;7 Concluding Remarks;136
11.8;References;136
12;Pingala Binary Numbers;138
12.1;1 Introduction;138
12.2;2 Fundamentals;139
12.2.1;2.1 Chandas or Meter;140
12.2.2;2.2 Pada or Quarter;140
12.2.3;2.3 Aksara or Syllable;140
12.2.4;2.4 Laghu or Short Syllables;140
12.2.5;2.5 Guru or Long Syllables;141
12.2.6;2.6 Matra or Metrical Unit;142
12.2.7;2.7 Verse Classification;142
12.3;3 Pratyayas: Methods of Cognitions;144
12.3.1;3.1 Varnic Expansion;144
12.3.2;3.2 Nasta;145
12.3.3;3.3 Uddista: Conversion from a Pingala Binary Number to Decimals;147
12.4;4 Concluding Remarks;149
12.5;References;150
13;The Reception of Ancient Indian Mathematics by Western Historians;152
13.1;1 The Context of Renaissance Humanism;152
13.2;2 The First Descriptions of Indian Algebra;154
13.3;3 A Case Study: The Bloom of Thymaridas;157
13.3.1;3.1 The Original Formulation in Hindu Sources;157
13.3.2;3.2 The Derived Problem in Hindu Sources;158
13.3.3;3.3 The Problem in Greek Sources;160
13.3.3.1;3.3.1 The Bloom of Thymaridas;160
13.3.3.2;3.3.2 Diophantus;161
13.3.3.3;3.3.3 The Extended Rule from Iamblichus;162
13.3.3.4;3.3.4 The Controversy;164
13.4;4 Conclusion: The Ground Was Wet Everywhere;166
13.5;References;166
14;The Indian Mathematical Tradition with Special Reference to Kerala: Methodology and Motivation;170
14.1;1 Introduction;170
14.2;2 Some Significant Developments and Their Motivations;171
14.3;3 Notion of Proof: Forms, Nature, Style, and Purpose;179
14.4;4 The Role of Commentarial Literature in the Dissemination of Mathematical Knowledge;184
14.5;5 Commentarial Literature: A Rich Source for the Study of Proof, Methodology, and Motivation;185
15;The Algorithm of Extraction in Greek and Sino-Indian Mathematical Traditions;188
15.1;1 Introduction;188
15.2;2 The Algorithm of Extraction in Ancient Greece;188
15.2.1;2.1 Heron of Alexandria's Method;189
15.2.2;2.2 Theon of Alexandria's Method;191
15.2.3;2.3 The Influence and Evolution of the Algorithm of Extraction in Western Europe;193
15.3;3 The Algorithm of Extraction in Ancient China;194
15.3.1;3.1 The Pre-Method of the Algorithm of Extraction;194
15.3.2;3.2 The Method of the Algorithm of the Extraction in the Nine Chapters and Thereafter;195
15.3.3;3.3 Liu Hui's Geometrical Explanation of the Algorithm of Extraction;196
15.3.4;3.4 The Influence and Evolution of the Algorithm of Extraction in China;197
15.4;4 The Algorithm of Extraction in Ancient India;198
15.5;5 A Brief Comparison and Conclusions;200
15.5.1;5.1 The Accuracy in the Algorithm, Approximation in the Theorem-Proving System;200
15.5.2;5.2 The Minor Difference;200
15.5.3;5.3 Brief Conclusions;200
15.6;References;201
16;Brahmagupta: The Ancient Indian Mathematician;202
16.1;References;208
17;Mainland Southeast Asia as a Crossroads of Chinese Astronomy and Indian Astronomy;210
17.1;1 Introduction;210
17.2;2 Vietnamese Calendrical Astronomy;210
17.3;3 Mainland Southeast Asian Astronomy (Except for Vietnam);212
17.4;4 Mainland Southeast Asian 19-Year Cycle;213
17.5;5 Conclusion;216
17.6;References;216
18;Mathematical Literature in the Regional Languages of India;218
19;Index;229