Content

PREFACE

ACKNOWLEDGMENT

chapter 1 Hipocrates' quadrature of the Lune (ca. 440 B.C.) 1

The Appearance of Demonstrative Mathematics 1
Some Remarks on Quadrature 11
Great Theorem  17
Epilogue 20

chapter 2 Euclid's Proof of the Pythagorean Theorem (ca. 300 B.C.) 27

The Elements of Euclid 27
Book 1: Preliminaries 32
Book 1: The Early Propositions 37
Book 1: Parallelism and Related Topics 44
Great Theorem 48
Epilogue 53

chapter 3 Euclid and the Infinitude of Primes (ca. 300 B.c.) 61

The Elements, Books II-VI 61
Number Theory in Euclid 68
Great Theorem 73
The Final Books of the Elements 75
Epilogue 81

chapter 4 Archimedes' Determination of Circular Area (ca. 225 B.C.) 84

The Life of Archimedes 84
Great Theorem 89 
Archimedes' Masterpiece: On tbe Sphere and tbe Cylinder 99 
Epilogue 106

chapter 5 Heron's Formula for Triangular Area (ca. A.D. 75) 113

Classical Mathematics after Archimedes 113
Great Theorem 118
Epilogue 127

chapter 6 Cardano and the Solution of the Cubic(1545) 133

A Horatio Algebra Story 133
Great Theorem 142
Further Topics on Solving Equations 147
Epilogue 151

chapter 7 A Gem from Isaac Newton (late 1660s) 155

Mathematics of the Heroic Century 155
A Mind Unleashed 160
Newton's Binomial Theorem 165
Great Theorem 174
Epilogue 177

chapter 8 The Bernoullis and the Harmonic Series (1689) 184

The Contributions of Leibniz 184
The Brothers Bernoulli 191
Great Theorem 196
The Challenge of the Brachistochrone 199
Epilogue 202

chapter 9 The Extraordinary Sums of Leonhard Euler (1734) 207

The Master of All Mathematical Trades 207
Great Theorem 212
Epilogue 218

chapter 10 A Sampler of Euler's Number Theory (1736) 223

The Legacy of Fermat 223
Great Theorem 229
Epilogue 235

chapter 11 The Non-Denumerability of the Continuum (1874) 245

Mathematics of the Nineteenth Century 245
Cantor and the Challenge of the Infinite 251
Great Theorem 259
Epilogue 265

chapter 12 Cantor and the Transfinite Realm (1891) 267

The Nature of Infinite Cardinals 2
Great Theorem 274
Epilogue 281

AFRERWORD 285
CHAPTER NOTES 287
REFERENCES 291
INDEX 295

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Copyright © 1996-2017 Alexander Bogomolny

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