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E-raamat: Mathematics of India: Concepts, Methods, Connections

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Mathematics of India: Concepts, Methods, ConnectionsSuurem pilt
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This book identifies three of the exceptionally fruitful periods of the millennia-long history of the mathematical tradition of India: the very beginning of that tradition in the construction of the now-universal system of decimal numeration and of a framework for planar geometry; a classical period inaugurated by Aryabhata’s invention of trigonometry and his enunciation of the principles of discrete calculus as applied to trigonometric functions; and a final phase that produced, in the work of Madhava, a rigorous infinitesimal calculus of such functions. The main highlight of this book is a detailed examination of these critical phases and their interconnectedness, primarily in mathematical terms but also in relation to their intellectual, cultural and historical contexts.

Recent decades have seen a renewal of interest in this history, as manifested in the publication of an increasing number of critical editions and translations of texts, as well as in an informed analytic interpretation of their content by the scholarly community. The result has been the emergence of a more accurate and balanced view of the subject, and the book has attempted to take an account of these nascent insights. As part of an endeavour to promote the new awareness, a special attention has been given to the presentation of proofs of all significant propositions in modern terminology and notation, either directly transcribed from the original texts or by collecting together material from several texts.

Arvustused

This is a very well-written account of the mathematics of India. There is a strong need for further research to fill the gaps that exist in the history of mathematics of India, and I think the current book serves to inspire the younger generation to undertake such an effort. (Gnana B. Tenali, Mathematical Reviews, March 2, 2020)

Preface v
Introduction 1(22)
0.1 Three Key Periods
1(3)
0.2 Sources
4(8)
0.3 Methodology
12(6)
0.4 Sanskrit and its Syllabary
18(5)
I Beginnings
23(118)
1 Background: Culture and Language
25(18)
1.1 The Indus Valley Civilisation
25(4)
1.2 The Vedic Period
29(4)
1.3 The Oral Tradition
33(5)
1.4 Grammar
38(5)
2 Vedic Geometry
43(30)
2.1 The Sulbasutra
43(3)
2.2 The Theorem of the Diagonal
46(9)
2.3 Rectilinear Figures and their Transformations
55(5)
2.4 Circle from Square: The Direct Construction
60(6)
2.5 The Inverse Formula: Square from Circle
66(7)
3 Antecedents? Mathematics in the Indus Valley
73(20)
3.1 Generalities
73(2)
3.2 Measures and Numbers
75(6)
3.3 Geometry
81(5)
3.4 Influences?
86(7)
4 Decimal Numbers
93(20)
4.1 Background
93(4)
4.2 Numbers and Based Numbers
97(6)
4.3 The Place-value Principle and its Realisations
103(5)
4.4 Other Realisations
108(3)
4.5 The Choice of a Base
111(2)
5 Numbers in the Vedic Literature
113(28)
5.1 Origins
113(4)
5.2 Number Names in the Rgveda
117(5)
5.3 Infinity and Zero
122(8)
5.4 Early Arithmetic
130(5)
5.5 Combinatorics
135(6)
II The Aryabhatan Revolution
141(114)
6 From 500 BCE to 500 CE
143(32)
6.1 One Thousand Years of Invasions
143(5)
6.2 The Siddhantas and the Influence of Greek Astronomy
148(9)
6.3 Aryabhatiya -- An Overview
157(3)
6.4 Who was Aryabhata?
160(7)
6.5 The Bakhshali Manuscript: Where Does it Fit in?
167(8)
7 The Mathematics of the Ganitapada
175(38)
7.1 General Survey
175(9)
7.2 The Linear Diophantine Equation -- kuttaka
184(6)
7.3 The Invention of Trigonometry
190(8)
7.4 The Making of the Sine Table: Aryabhata's Rule
198(4)
7.5 Aryabhata's Legacy
202(11)
8 From Brahmagupta to Bhaskara II to Narayana
213(42)
8.1 Mathematics Moves South
213(8)
8.2 The Quadratic Diophantine Problem -- bhavana
221(5)
8.3 Methods of Solution -- cakravala
226(10)
8.4 A Different Circle Geometry: Cyclic Quadrilaterals
236(12)
8.5 The Third Diagonal; Proofs
248(7)
III Madhava and the Invention of Calculus
255(124)
9 The Nila Phenomenon
257(34)
9.1 The Nila School Rediscovered
257(6)
9.2 Mathematicians in their Villages -- and in their Words
263(13)
9.3 The Sanskritisation of Kerala
276(6)
9.4 Who was Madhava? and Narayana?
282(9)
10 Nila Mathematics -- General Survey
291(22)
10.1 The Primary Source: Yuktibhasa
291(6)
10.2 Geometry and Trigonometry; Addition Theorems
297(5)
10.3 The Sine Table; Interpolation
302(5)
10.4 Samskaram: Generating Infinite Series
307(6)
11 The π Series
313(18)
11.1 Calculus and the Gregory-Leibniz Series
313(3)
11.2 The Geometry of Small Angles and their Tangents
316(4)
11.3 Integration: The Power Series
320(3)
11.4 Integrating Powers; Asymptotic Induction
323(4)
11.5 The Arctangent Series
327(4)
12 The Sine and Cosine Series
331(28)
12.1 From Differences to Differentials
331(5)
12.2 Solving the Difference/Differential Equation
336(6)
12.3 The Sphere
342(6)
12.4 The Calculus Debates
348(11)
13 The π Series Revisited: Algebra in Analysis
359(20)
13.1 The Problem
359(4)
13.2 Polynomials: A Primer
363(3)
13.3 Higher Order Corrections
366(6)
13.4 Variations on the π series
372(7)
IV Connections
379(44)
14 What is Indian about the Mathematics of India?
381(18)
14.1 The Geography of Indian Mathematics
381(6)
14.2 The Weight of the Oral Tradition
387(6)
14.3 Geometry with Indian Characteristics
393(6)
15 What is Indian ... ? The Question of Proofs
399(14)
15.1 What is a Proof?
399(5)
15.2 No Reductio ad Absurdum
404(4)
15.3 Recursion, Descent and Induction
408(5)
16 Upasamhara
413(10)
16.1 Towards Modernity
413(3)
16.2 Cross-cultural Currents?
416(3)
16.3 Journey's End
419(4)
Bibliography 423(4)
Index 427
P.P. DIVAKARAN retired as professor from the Tata Institute of Fundamental Research, Mumbai in 1996. Subsequently, he has spent time at several institutions as adjunct or visiting professor. The present book grew out of research done at the Inter-University Centre for Astronomy and Astrophysics, Pune, as the holder of a Senior Award of the Homi Bhabha Fellowships Council.
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