How it All Began A Thematic History of

Intended for undergraduate mathematics students and teachers, How it All Began provides an introduction to the history of various branches of mathematics, the lives of the mathematicians, and the challenges they faced. This new work explores the great Greek legacy, Indian arithmetic, Integral Calculus, the Analytic Geometry of Rene Descartes, Differential Calculus, the Non-Commutative Algebra of Hamilton, the arithmetization of analysis, and the beginnings of Algebra.

Preface

1 THE GREAT GREEK LEGACY

1

(38)

1 Introduction

1

(1)

2 Food-gatherers become food-growers

1

(4)

3 Egyptian mathematics

5

(2)

4 The beginnings of Greek Geometry

7

(1)

5 The Greek Geometry

8

(2)

6 Squaring of rectilinear figures

10

(6)

7 Number Theory in the Elements: Priliminary notions

16

(5)

8 Some number-theoretic results in the Elements

21

(1)

9 Euclid's proofs

21

(7)

10 Greek mathematics after Euclid

28

(1)

11 Conic sections of Apollonius

28

(2)

12 The Arithmetica of Diophantus

30

(5)

13 The greatest Greek gift

35

(4)

2 THE INDIAN ARITHMETIC: ITS ENTRY AND ACCEPTANCE IN EUROPE

39

(18)

1 Indian Numerals and Algorithms

39

(1)

2 Fibonacci: Liber Abbaci

40

(1)

3 Abacus: The prevalent system in Europe

41

(1)

4 Operations on the Abacus

42

(3)

5 Multiplication and Division on an Abacus

45

(4)

6 Abacus versus the Decimal System

49

(2)

7 Other contributions from India

51

(2)

8 Entry into Arabia and thence into Europe

53

(4)

3 THE METHOD OF INTEGRAL CALCULUS

57

(30)

1 The Greek roots of the method

57

(1)

2 Measurement of the circle

58

(7)

3 Quadrature of the Parabola

65

(7)

4 The work of Kepler

72

(4)

5 The contribution of Cavalieri

76

(3)

6 The work of Fermat

79

(2)

7 The work of Newton

81

(6)

4 ANALYTIC GEOMETRY OF RENE DESCARTES

87

(34)

1 La Geometrie

87

(1)

2 Two illustrations

88

(4)

3 Demonstration of arithmetic operations in geometry

92

(2)

4 The method of Analytic Geometry

94

(5)

5 The problem of Pappus

99

(5)

6 Standardized Analytic Geometry

104

(5)

7 Analytic Geometry: the Method

109

(1)

8 The Transform-Solve-Invert method

110

(11)

5 THE METHOD OF DIFFERENTIAL CALCULUS

121

(22)

1 Apollonius: the concept of the minimum

121

(4)

2 Roberval and Descartes

125

(3)

3 Fermat: the method of differential calculus

128

(6)

4 Professor Isaac Barrow and his method

134

(4)

5 Newton and Leibnitz

138

(5)

6 THE FIRST NON-COMMUTATIVE ALGEBRA

143

(16)

1 The first great crisis in mathematics

143

(1)

2 The second great crisis

144

(3)

3 The number-pair episode revisited

147

(1)

4 Hamilton's discovery of the Quaternions

148

(4)

5 The Quaternion Algebra

152

(7)

7 ARITHMETIZATION OF ANALYSIS

159

(32)

1 Imperatives for the growth of Mathematics

159

(3)

2 The crisis in Calculus

162

(4)

3 The awareness

166

(3)

4 The lead

169

(3)

5 The Search

172

(1)

6 Construction of I from N and Q from I

173

(2)

7 The search continued

175

(16)

A Cantor's Theory

175

(6)

B Dedekind Theory of Cuts

181

(10)

8 BEGINNINGS OF ALGEBRA

191

(48)

1 Al-Khowarizmi and Emmy Noether

191

(1)

2 Al-Khowarizmi's book

191

(2)

3 Al-jabr and Al-muqabala

193

(1)

4 Six basic formulae

194

(5)

5 Standard Formulae

199

(1)

6 Illustrations of the solution of problems

200

(1)

7 Mensuration Problems

201

(2)

8 Problems of legacies

203

(2)

9 Omar Khayyam

205

(7)

10 Liber Abbaci

212

(6)

11 The next stage

218

(1)

12 The Cubic

219

(1)

13 The Cubic x³ + ax = b

220

(5)

14 The Cubics x³ + ax = b and x³ + b = ax

225

(2)

15 The Quartic

227

(3)

16 A look-back

230

(1)

17 The new era

231

(8)

ANSWERS

239

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How it All Began A Thematic History of Mathematics

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Table of Contents

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