Historical Roots of Elementary Mathematics

Preface The Greek alphabet 1 EGYPTIAN MATHEMATICS 1-1 Prehistoric mathematics 1-2 The earliest written mathematics 1-3 Numerical notation 1-4 Arithmetic operations 1-5 Multiplication 1-6 Fractions and division 1-7 The red auxiliary numbers 1-8 The 2 ÷ n table 1-9 The leather roll 1-10 Algebraic problems 1-11 Geometry 2 BABYLONIAN MATHEMATICS 2-1 Some historical facts 2-2 Babylonian numerical notation 2-3 The fundamental operations 2-4 Extraction of roots 2-5 Babylonian algebra 2-6 A Babylonian text 2-7 Babylonian geometry 2-8 Approximations to p 2-9 Another problem and a farewell to the Babylonians 3 THE BEGINNING OF GREEK MATHEMATICS 3-1 The earliest records 3-2 Greek numeration systems 3-3 Thales and his importance to mathematics 3-4 Pythagoras and the Pythagoreans 3-5 The Pythagoreans and music 3-6 Pythagorean arithmetica 3-7 Pythagorean numerology 3-8 Pythagorean astronomy 3-9 Pythagorean geometry 3-10 Incommensurable segments and irrational numbers 4 THE FAMOUS PROBLEMS OF GREEK ANTIQUITY 4-1 Introduction 4-2 Hippocrates of Chios and the quadrature of lunes 4-3 Other quadratures 4-4 Hippocrates' geometry 4-5 Duplication of the cube 4-6 The trisection problem 4-7 Hippias and squaring of the circle 4-8 The solutions of the Greek problems 5 EUCLID'S PHILOSOPHICAL FORERUNNERS 5-1 Philosophy and philosophers 5-2 Plato 5-3 Aristotle and his theory of statements 5-4 Concepts and definitions 5-5 Special notations and undefined terms 6 EUCLID 6-1 Elements 6-2 The structure of the Elements of Euclid 6-3 The definitions 6-4 Postulates and common notions 6-5 The meaning of a construction 6-6 The purport of Postulate III 6-7 Congruence 6-8 Congruence (continued) 6-9 The theory of parallels 6-10 The comparison of areas 6-11 The theorem of Pythagoras 6-12 The difference between the Euclidean and the modern method of comparing areas 6-13 Geometric algebra and regular polygons 6-14 Number theory in the Elements 7 GREEK MATHEMATICS AFTER EUCLID. EUCLIDEAN VS. MODERN METHODS 7-1 The span of Greek mathematics 7-2 Archimedes and Eratosthenes 7-3 Apollonius of Perga 7-4 Heron of Alexandria and Diophantus 7-5 Ptolemy and Pappus 7-6 Review of the Greek method 7-7 Objections to the Euclidean system 7-8 The meaning of deduction 7-9 Euclid's system is not purely deductive 7-10 How is geometry built up purely deductively? 7-11 A four-point system 8 NUMERATION AND ARITHMETIC AFTER THE GREEKS 8-1 Roman numerals 8-2 The abacus and tangible arithmetic 8-3 The Hindu-Arabic numerals 8-4 An early American place-value numeration system 8-5 Later developments in positional notation 8-6 Conversions between numeration systems 8-7 Addition and subtraction algorithms in nondecimal bases 8-8 Multiplication alogorithms in nondecimal bases 8-9 "Fractions, rational numbers, and place-value numeration" 8-10 Irrational numbers 8-11 Modern theoretical foundations of arithmetic 8-12 Modern numeration HINTS AND ANSWERS TO SELECTED ANSWERS INDEX