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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