| 1 | Mathematics in Mesopotamia | 3 |
| 1 | Where Did Mathematics Begin? | 3 |
| 2 | Political History in Mesopotamia | 4 |
| 3 | The Number Symbols | 5 |
| 4 | Arithmetic Operations | 7 |
| 5 | Babylonian Algebra | 8 |
| 6 | Babylonian Geometry | 10 |
| 7 | The Uses of Mathematics in Babylonia | 11 |
| 8 | Evaluation of Babylonian Mathematics | 13 |
| 2 | Egyptian Mathematics | 15 |
| 1 | Background | 15 |
| 2 | The Arithmetic | 16 |
| 3 | Algebra and Geometry | 18 |
| 4 | Egyptian Uses of Mathematics | 21 |
| 5 | Summary | 22 |
| 3 | The Creation of Classical Greek Mathematics | 24 |
| 1 | Background | 24 |
| 2 | The General Sources | 25 |
| 3 | The Major Schools of the Classical Period | 27 |
| 4 | The Ionian School | 28 |
| 5 | The Pythagoreans | 28 |
| 6 | The Eleatic School | 34 |
| 7 | The Sophist School | 37 |
| 8 | The Platonic School | 42 |
| 9 | The School of Eudoxus | 48 |
| 10 | Aristotle and His School | 51 |
| 4 | Euclid and Apollonius | 56 |
| 1 | Introduction | 56 |
| 2 | The Background of Euclid's Elements | 57 |
| 3 | The Definitions and Axioms of the Elements | 58 |
| 4 | Books I to IV of the Elements | 60 |
| 5 | Book V: The Theory of Proportion | 68 |
| 6 | Book VI: Similar Figures | 73 |
| 7 | Books VII, VIII, and IX: The Theory of Numbers | 77 |
| 8 | Book X: The Classification of Incommensurables | 80 |
| 9 | Books XI, XII, and XIII: Solid Geometry and the Method of Exhaustion | 81 |
| 10 | The Merits and Defects of the Elements | 86 |
| 11 | Other Mathematical Works by Euclid | 88 |
| 12 | The Mathematical Work of Apollonius | 89 |
| 5 | The Alexandrian Greek Period: Geometry and Trigonometry | 101 |
| 1 | The Founding of Alexandria | 101 |
| 2 | The Character of Alexandrian Greek Mathematics | 103 |
| 3 | Areas and Volumes in the Work of Archimedes | 105 |
| 4 | Areas and Volumes in the Work of Heron | 116 |
| 5 | Some Exceptional Curves | 117 |
| 6 | The Creation of Trigonometry | 119 |
| 7 | Late Alexandrian Activity in Geometry | 126 |
| 6 | The Alexandrian Period: The Reemergence of Arithmetic and Algebra | 131 |
| 1 | The Symbols and Operations of Greek Arithmetic | 131 |
| 2 | Arithmetic and Algebra as an Independent Development | 135 |
| 7 | The Greek Rationalization of Nature | 145 |
| 1 | The Inspiration for Greek Mathematics | 145 |
| 2 | The Beginnings of a Rational View of Nature | 146 |
| 3 | The Development of the Belief in Mathematical Design | 147 |
| 4 | Greek Mathematical Astronomy | 154 |
| 5 | Geography | 160 |
| 6 | Mechanics | 162 |
| 7 | Optics | 166 |
| 8 | Astrology | 168 |
| 8 | The Demise of the Greek World | 171 |
| 1 | A Review of the Greek Achievements | 171 |
| 2 | The Limitations of Greek Mathematics | 173 |
| 3 | The Problems Bequeathed by the Greeks | 176 |
| 4 | The Demise of the Greek Civilization | 177 |
| 9 | The Mathematics of the Hindus and Arabs | 183 |
| 1 | Early Hindu Mathematics | 183 |
| 2 | Hindu Arithmetic and Algebra of the Period A.D. 200-1200 | 184 |
| 3 | Hindu Geometry and Trigonometry of the Period A.D. 200-1200 | 188 |
| 4 | The Arabs | 190 |
| 5 | Arabic Arithmetic and Algebra | 191 |
| 6 | Arabic Geometry and Trigonometry | 195 |
| 7 | Mathematics circa 1300 | 197 |
| 10 | The Medieval Period in Europe | 200 |
| 1 | The Beginnings of a European Civilization | 200 |
| 2 | The Materials Available for Learning | 201 |
| 3 | The Role of Mathematics in Early Medieval Europe | 202 |
| 4 | The Stagnation in Mathematics | 203 |
| 5 | The First Revival of the Greek Works | 205 |
| 6 | The Revival of Rationalism and Interest in Nature | 206 |
| 7 | Progress in Mathematics Proper | 209 |
| 8 | Progress in Physical Science | 211 |
| 9 | Summary | 213 |
| 11 | The Renaissance | 216 |
| 1 | Revolutionary Influences in Europe | 216 |
| 2 | The New Intellectual Outlook | 218 |
| 3 | The Spread of Learning | 220 |
| 4 | Humanistic Activity in Mathematics | 221 |
| 5 | The Clamor for the Reform of Science | 223 |
| 6 | The Rise of Empiricism | 227 |
| 12 | Mathematical Contributions in the Renaissance | 231 |
| 1 | Perspective | 231 |
| 2 | Geometry Proper | 234 |
| 3 | Algebra | 236 |
| 4 | Trigonometry | 237 |
| 5 | The Major Scientific Progress in the Renaissance | 240 |
| 6 | Remarks on the Renaissance | 247 |
| 13 | Arithmetic and Algebra in the Sixteenth and Seventeenth Centuries | 250 |
| 1 | Introduction | 250 |
| 2 | The Status of the Number System and Arithmetic | 251 |
| 3 | Symbolism | 259 |
| 4 | The Solution of Third and Fourth Degree Equations | 263 |
| 5 | The Theory of Equations | 270 |
| 6 | The Binomial Theorem and Allied Topics | 272 |
| 7 | The Theory of Numbers | 274 |
| 8 | The Relationship of Algebra to Geometry | 278 |
| 14 | The Beginnings of Projective Geometry | 285 |
| 1 | The Rebirth of Geometry | 285 |
| 2 | The Problems Raised by the Work on Perspective | 286 |
| 3 | The Work of Desargues | 288 |
| 4 | The Work of Pascal and La Hire | 295 |
| 5 | The Emergence of New Principles | 299 |
| 15 | Coordinate Geometry | 302 |
| 1 | The Motivation for Coordinate Geometry | 302 |
| 2 | The Coordinate Geometry of Fermat | 303 |
| 3 | Rene Descartes | 304 |
| 4 | Descartes's Work in Coordinate Geometry | 308 |
| 5 | Seventeenth-Century Extensions of Coordinate Geometry | 317 |
| 6 | The Importance of Coordinate Geometry | 321 |
| 16 | The Mathematization of Science | 325 |
| 1 | Introduction | 325 |
| 2 | Descartes's Concept of Science | 325 |
| 3 | Galileo's Approach to Science | 327 |
| 4 | The Function Concept | 335 |
| 17 | The Creation of the Calculus | 342 |
| 1 | The Motivation for the Calculus | 342 |
| 2 | Early Seventeenth-Century Work on the Calculus | 344 |
| 3 | The Work of Newton | 356 |
| 4 | The Work of Leibniz | 370 |
| 5 | A Comparison of the Work of Newton and Leibniz | 378 |
| 6 | The Controversy over Priority | 380 |
| 7 | Some Immediate Additions to the Calculus | 381 |
| 8 | The Soundness of the Calculus | 383 |
| List of Abbreviations | |
| Index | |
