| Introduction | vii |
| VOLUME ONE | |
| PART I: General Survey | |
| Philip E. B. Jourdain: Commentary | 2 |
| 1. The Nature of Mathematics | |
| by PHILIP E. B. JOURDAIN | 4 |
| PART Il: Historical and Biographical | |
| Commentary | 74 |
| 1. The Great Mathematicians | |
| by HERBERT WESTREN TURNBULL | 75 |
| Commentary | 169 |
| 2. The Rhind Papyrus | |
| by JAMES R. NEWMAN | 170 |
| Commentary | 179 |
| 3. Archimedes | |
| by PLUTARCH, VITRUVIUS, TZETZES | 180 |
| Commentary | 188 |
| 4. Greek Mathematics | |
| by IVOR THOMAS | 189 |
| Robert Recorde: Commentary | 210 |
| 5. The Declaration of the Profit of Arithmeticke | |
| by ROBERT RECORDE | 212 |
| Kepler and Lodge: Commentary | 218 |
| 6. Johann Kepler | |
| by SIR OLIVER LODGE | 220 |
| Descartes and Analytical Geometry: Commentary | 235 |
| 7. The Geometry | |
| by RENE DESCARTES | 239 |
| Commentary | 254 |
| 8. Isaac Newton | |
| by E. N. DA C. ANDRADE | 255 |
| 9. Newton, the Man | |
| by JOHN MAYNARD KEYNES | 277 |
| Bishop Berkeley and Infinitesimals: Commentary | 286 |
| 10. The Analyst | |
| by BISHOP BERKELEY | 288 |
| Gauss: Commentary | 294 |
| 11. Gauss, the Prince of Mathematicians | |
| by ERIC TEMPLE BELL | 295 |
| Cayley and Sylvester: Commentary | 340 |
| 12. Invariant Twins, Cayley and Sylvester | |
| by ERIC TEMPLE BELL | 341 |
| Commentary | 366 |
| 13. Srinivasa Ramanujan | |
| by JAMES R. NEWMAN | 368 |
| Bertrand Russell: Commentary | 377 |
| 14. My Mental Development | |
| by BERTRAND RUSSELL | 381 |
| Alfred North Whitehead: Commentary | 395 |
| 15. Mathematics as an Element in the History of Thought | |
| by ALFRED NORTH WHITEHEAD | 402 |
| PART III: Arithmetic, Numbers and the Art of Counting | |
| Poppy Seeds and Large Numbers: Commentary | 418 |
| 1. The Sand Reckoner | |
| by ARCHIMEDES | 420 |
| The Art of Counting: Commentary | 430 |
| 2. Counting | |
| by LEVI LEONARD CONANT | 432 |
| 3. From Numbers to Numerals and From Numerals to Computation | |
| by DAVID EUGENE SMITH and JEKUTHIEL GINSBURG | 442 |
| Idiot Savants: Commentary | 465 |
| 4. Calculating Prodigies | 467 |
| by W. W. ROUSE BALL | |
| Gifted Birds: Commentary | 488 |
| 5. The Ability of Birds to "Count" | |
| by O. KOEHLER | 489 |
| The Mysteries of Arithmetic: Commentary | 497 |
| 6. The Queen of Mathematics | |
| by ERIC TEMPLE BELL | 498 |
| (P + PQ)m/n Commentary | 519 |
| 7. On the Binomial Theorem for Fractional and Negative Exponents | |
| by ISAAC NEWTON | 521 |
| The Number Concept: Commentary | 525 |
| 8. Irrational Numbers | |
| by RICHARD DEDEKIND | 528 |
| 9. Definition of Number | |
| by BERTRAND RUSSELL | 537 |
| PART IV: Mathematics of Space and Motion | |
| William Kingdon Clifford: Commentary | 546 |
| 1. The Exactness of Mathematical Laws | |
| by WILLIAM KINGDON CLIFFORD | 548 |
| 2. The Postulates of the Science of Space | |
| by WILLIAM KINGDON CLIFFORD | 552 |
| 3. On the Space Theory of Matter | |
| by WILLIAM KINGDON CLIFFORD | 568 |
| A Famous. Problem: Commentary | 570 |
| 4. The Seven Bridges of Konigsberg | |
| by LEONHARD EULER | 573 |
| 5. Topology | |
| by RICHARD COURANT and HERBERT ROBBINS | 581 |
| Dürer and the Mathematics of Painting: Commentary | 600 |
| 6. Durer as a Mathematician | |
| by ERWIN PANOFSKY | 603 |
| 7. Projective Geometry | |
| by MORRIS KLINE | 622 |
| Hermann von Helmholtz: Commentary | 642 |
| 8. On the Origin and Significance of Geometrical Axioms | |
| by HERMANN VON HELMHOLTZ | 647 |
| Commentary | 669 |
| 9. Symmetry | |
| by HERMANN WEYL | 671 |
| VOLUME TWO | |
| PART V: Mathematics and the Physical World | |
| Galileo Galilei: Commentary | 726 |
| 1. Mathematics of Motion | |
| by GALILEO GALILEI | 734 |
| The Bernoullis: Commentary | 771 |
| 2. Kinetic Theory of Gases | |
| by DANIEL BERNOULLI | 774 |
| A Great Prize, a Long-Suffering Inventor and the First Accurate Clock: Commentary | 778 |
| 3. The Longitude | |
| by LLOYD A. BROWN | 780 |
| John Couch Adams: Commentary | 820 |
| 4. John Couch Adams and the Discovery of Neptune | |
| by SIR HAROLD SPENCER JONES | 822 |
| H. G. J. Moseley: Commentary | 840 |
| 5. Atomic Numbers | |
| by H. G. J. MOSELEY | 842 |
| The Small Furniture of Earth: Commentary | 851 |
| 6. The Röntgen Rays | |
| by SIR WILLIAM BRAGG | 854 |
| 7. Crystals and the Future of Physics | |
| by PHILIPPE LE CORBEILLER | 871 |
| Queen Dido, Soap Bubbles, and a Blind Mathematician: Commentary | 882 |
| 8. What Is Calculus of Variations and What Are Its Applications? | |
| by KARL MENGER | 886 |
| 9. The Soap-bubble | |
| by C. VERNON BOYS | 891 |
| 10. Plateau's Problem | |
| by RICHARD COURANT and HERBERT ROBBINS | 901 |
| The Periodic Law and Mendeléeff: Commentary | 910 |
| 11. Periodic Law of the Chemical Elements | |
| by DMITRI MENDELÉEFF | 913 |
| 12. Mendeléeff | |
| by Bernard Jaffe | 919 |
| Gregor Mendel: Commentary | 932 |
| 13. Mathematics of Heredity | |
| by GREGOR MENDEL | 937 |
| J. B. S. Haldane: Commentary | 950 |
| 14. On Being the Right Size | |
| by J. B. S. HALDANE | 952 |
| 15. Mathematics of Natural Selection | |
| by J. B. S. HALDANE | 958 |
| Erwin Schrödinger: Commentary | 973 |
| 16. Heredity and the Quantum Theory | |
| by ERWIN SCHRÖDINGER | 975 |
| D'Arcy Wentworth Thompson: Commentary | 996 |
| 17. On Magnitude | |
| by D'ARCY WENTWORTH THOMPSON | 1001 |
| Uncertainty: Commentary | 1047 |
| 18. The Uncertainty Principle | |
| by WERNER HEISENBERG | 1051 |
| 19. Causality and Wave Mechanics | |
| by ERWIN SCHRÖDINGER | 1056 |
| Sir Arthur Stanley Eddington: Commentary | 1069 |
| 20. The Constants of Nature | |
| by SIR ARTHUR STANLEY EDDINGTON | 1074 |
| 21. The New Law of Gravitation and the Old Law | |
| by SIR ARTHUR STANLEY EDDINGTON | 1094 |
| Commentary | 1105 |
| 22. The Theory of Relativity | |
| by CLEMENT V. DURELL | 1107 |
| PART VI: Mathematics and Social Science | |
| The Founder of Psychophysics: Commentary | 1146 |
| 1. Gustav Theodor Fechner | |
| by EDWIN G. BORING | 1148 |
| Sir Francis Galton: Commentary | 1167 |
| 2. Classification of Men According to Their Natural Gifts | |
| by SIR FRANCIS GALTON | 1173 |
| Thomas Robert Malthus: Commentary | 1189 |
| 3. Mathematics of Population and Food | |
| by THOMAS ROBERT MALTHUS | 1192 |
| Cournot, Jevons, and the Mathematics of Money: Commentary | 1200 |
| 4. Mathematics of Value and Demand | |
| by AUGUSTIN COURNOT | 1203 |
| 5. Theory of Political Economy | |
| by WILLIAM STANLEY JEVONS | 1217 |
| A Distinguished Quaker and War: Commentary | 1238 |
| 6. Mathematics of War and Foreign Politics | |
| by LEWIS FRY RICHARDSON | 1240 |
| 7. Statistics of Deadly Quarrels | |
| by LEWIS FRY RICHARDSON | 1254 |
| The Social Application of Mathematics: Commentary | 1264 |
| 8. The Theory of Economic Behavior | |
| by LEONID HURWICZ | 1267 |
| 9. Theory of Games | |
| by S. VAJDA | 1285 |
| 10. Sociology Learns the Language of Mathematics | |
| by ABRAHAM KAPLAN | 1294 |
| PART VII: The Laws of Chance | |
| Pierre Simon de Laplace: Commentary | 1316 |
| 1. Concerning Probability | |
| by PIERRE SIMON DE LAPLACE | 1325 |
| 2. The Red and the Black | |
| by CHARLES SANDERS PEIRCE | 1334 |
| 3. The Probability of Induction | |
| by CHARLES SANDERS PEIRCE | 1341 |
| Lord Keynes: Commentary | 1355 |
| 4. The Application of Probability to Conduct | |
| by JOHN MAYNARD KEYNES | 1360 |
| An Absent-minded Genius and the Laws of Chance: Commentary | 1374 |
| 5. Chance | |
| by HENRI POINCARÉ | 1380 |
| Ernest Nagel and the Laws of Probability: Commentary | 1395 |
| 6. The Meaning of Probability | |
| by ERNEST NAGEL | 1398 |
| VOLUME THREE | |
| PART VIII: Statistics and the Design of Experiments | |
| An Ingenious Army Captain and a Generous and Many-sided Man: Commentary | 1416 |
| l. Foundations of Vital Statistics | |
| by JOHN GRAUNT | 1421 |
| 2. First Life Insurance Tables | |
| by EDMUND HALLEY | 1437 |
| Commentary | 1448 |
| 3. The Law of Large Numbers | |
| by JACOB BERNOULLI | 1452 |
| Statistics and the Lady with a Fine Palate: Commentary | 1456 |
| 4. Sampling and Standard Error | |
| by L. C. TIPPETT | 1459 |
| 5. On the Average and Scatter | |
| by M. J. MORONEY | 1487 |
| 6. Mathematics of a Lady Tasting Tea | |
| by SIR RONALD A. FISHER | 1512 |
| The Scientific Aptitude of George Bernard Shaw: Commentary | 1522 |
| 7. The Vice of Gambling andthe Virtue of Insurance | |
| by GEORGE BERNARD SHAW | 1524 |
| PART IX: The Supreme Art of Abstraction: Group Theory | |
| Certain Important Abstractions: Commentary | 1534 |
| 1. The Group Concept | |
| by CASSIUS J. KEYSER | 1538 |
| 2. The Theory of Groups | |
| by SIR ARTHUR STANLEY EDDINGTON | 1558 |
| PART X: Mathematics of Infinity | |
| 1. Mathematics and the Metaphysicians | |
| by BERTRAND RUSSELL | 1576 |
| Hans Hahn: Commentary | 1591 |
| 2. Infinity | |
| by HANS HAHN | 1593 |
| PART XI: Mathematical Truth and the Structure of Mathematics | |
| The Foundations of Mathematics: Commentary | 1614 |
| l. On the Nature of Mathematical Truth | |
| by CARL G. HEMPEL | 1619 |
| 2. Geometry and Empirical Science | |
| by CARL G. HEMPEL | 1635 |
| 3. The Axiomatic Method | |
| by RAYMOND L. WILDER | 1647 |
| 4. Goedel s Proof | |
| by ERNEST NAGEL and JAMES R. NEWMAN | 1668 |
| 5. A Mathematical Science | |
| by OSWALD VEBLEN and JOHN WESLEY YOUNG | 1696 |
| 6. Mathematics and the World | |
| by DOUGLAS GASKING | 1708 |
| 7. Mathematical Postulates and Human Understanding | |
| by RICHARD VON MISES | 1723 |
| PART XII: The Mathematical Way of Thinking | |
| An Eristic Controversy: Commentary | 1756 |
| 1. The Study That Knows Nothing of Observation | |
| by JAMES JOSEPH SYLVESTER | 1758 |
| Charles Sanders Peirce: Commentary | 1767 |
| 2. The Essence of Mathematics | |
| by CHARLES SANDERS PEIRCE | 1773 |
| Ernst Mach: Commentary | 1784 |
| 3. The Economy of Science | |
| by ERNST MACH | 1787 |
| Norman Robert Campbell: Commentary | 1796 |
| 4. Measurement | |
| by NORMAN CAMPBELL | 1797 |
| 5. Numerical Laws and the Use of Mathematics in Science | |
| by NORMAN CAMPBELL | 1814 |
| Hermann Weyl: Commentary | 1830 |
| 6. The Mathematical Way of Thinking | |
| by HERMANN WEYL | 1832 |
| PART XIII: Mathematics and Logic | |
| Symbolic Logic, George Boole and a Horrible Dream: Commentary | 1852 |
| 1. Mathematical Analysis of Logic | |
| by GEORGE BOOLE | 1856 |
| 2. History of Symbolic Logic | |
| by CLARENCE IRVING LEWIS and COOPER HAROLD LANGFORD | 1859 |
| 3. Symbolic Notation, Haddocks' Eyes and the Dog-Walking Ordinance | |
| by ERNEST NAGEL | 1878 |
| 4. Symbolic Logic | |
| by ALFRED TARSKI | 1901 |
| PART XIV: The Unreasonableness of Mathematics | |
| The Importance of Being Absurd: Commentary | 1934 |
| 1. Paradox Lost and Paradox Regained | |
| by EDWARD KASNER and JAMES R. NEWMAN | 1936 |
| 2. The Crisis in Intuition | |
| by HANS HAHN | 1956 |
| PART XV: How to Solve It | |
| The Tears of Mathematics: Commentary | 1978 |
| 1. How to Solve It | |
| by G. POLYA | 1980 |
| PART XVI: The Vocabulary of Mathematics | |
| Double Infinite Rapport and Other Mathematical Jargon: Commentary | 1994 |
| 1. New Names for Old | |
| by EDWARD KASNER and JAMES R. NEWMAN | 1996 |
| PART XVII: Mathematics as an Art | |
| lohn William Navin Sullivan: Commentary | 2012 |
| 1. Mathematics as an Art | |
| by JOHN WILLIAM NAVIN SULLIVAN | 2015 |
| VOLUME FOUR | |
| PART XVIII: The Mathematician | |
| G. H. Hardy: Commentary | 2024 |
| 1. A Mathematician's Apology | |
| by G. H. HARDY | 2027 |
| The Elusiveness of Invention: Commentary | 2039 |
| 2. Mathematical Creation | |
| by HENRI POINCARÉ | 2041 |
| The Use of a Top Hat as a Water Bucket: Commentary | 2051 |
| 3. The Mathematician | |
| by JOHN VON NEUMANN | 2053 |
| PART XIX: Mathematical Machines: Can a Machine Think? | |
| A utomatic Computers: Commentary | 2066 |
| 1. The General and Logical Theory of Automata | |
| by JOHN VON NEUMANN | 2070 |
| 2. Can a Machine Think? | |
| by A. M. TURING | 2099 |
| 3. A Chess-Playing Machine | |
| by CLAUDE SHANNON | 2124 |
| PART XX: Mathematics in Warfare | |
| Frederick William Lanchester: Commentary | 2136 |
| 1. Mathematics in Warfare | |
| by FREDERICK WILLLAM LANCHESTER | 2138 |
| Operations Research: Commentary | 2158 |
| 2. How to Hunt a Submarine | |
| by PHILLIP M. MORSE and GEORGE E. KIMBALL | 2160 |
| PART XXI: A Mathematical Theory of Art | |
| George David Birkhoff: Commentary | 2182 |
| 1. Mathematics of Aesthetics | |
| by GEORGE DAVID BIRKHOFF | 2185 |
| PART XXII: Mathematics of the Good | |
| 1. A Mathematical Approach to Ethics | |
| by GEORGE DAVID BIRKHOFF | 2198 |
| PART XXIII: Mathematics in Literature | |
| The Island of Laputa: Commentary | 2210 |
| 1. Cycloid Pudding by JONATHAN SWIFT | 2214 |
| Aldous Huxley: Commentary | 2221 |
| 2. Young Archimedes | |
| by ALDOUS HUXLEY | 2223 |
| Mr. Fortune: Commentary | 2250 |
| 3. Geometry in the South Pacific | |
| by SYLVIA TOWNSEND WARNER | 2252 |
| Statistics as a Literary Stimulus: Commentary | 2261 |
| 4. Inflexible Logic | |
| by RUSSELL MALONEY | 2262 |
| 5. The Law | |
| by ROBERT M. COATES | 2268 |
| PART XXIV: Mathematics and Music | |
| Sir James Jeans: Commentary | 2274 |
| 1. Mathematics of Music | |
| by SIR JAMES JEANS | 2278 |
| PART XXV: Mathematics as a Culture Clue | |
| Oswald Spengler: Commentary | 2312 |
| 1. Meaning of Numbers | |
| by OSWALD SPENGLER | 2315 |
| 2. The Locus of Mathematical Reality: | |
| An Anthropological Footnote | |
| by LESLIE A. WHITE | 2348 |
| PART XXVI: Amusements, Puzzles, Fancies | |
| Augustus De Morgan, an Estimable Man: Commentary | 2366 |
| 1. Assorted Paradoxes | |
| by AUGUSTUS DE MORGAN | 2369 |
| A Romance of Many Dimensions: Commentary | 2383 |
| 2. Flatland | |
| by EDWIN A. ABBOTT | 2385 |
| Lewis Carroll: Commentary | 2397 |
| 3. What the Tortoise Said to Achilles and Other Riddles | |
| by LEWIS CARROLL | 2402 |
| Continuity: Commentary | 2410 |
| 4. The Lever of Mahomet | |
| by RICHARD COURANT and HERBERT ROBBINS | 2412 |
| Games and Puzzles: Commentary | 2414 |
| 5. Pastimes of Past and Present Times | |
| by EDWARD KASNER and JAMES R. NEWMAN | 2416 |
| 6. Arithmetical Restorations | |
| by W. W. ROUSE BALL | 2439 |
| 7. The Seven Seven's | |
| by W. E. H. BERWICK | 2444 |
| Thomas John l'Anson Bromwich: Commentary | 2449 |
| 8. Easy Mathematics and Lawn Tennis | |
| by T. J. l'A. BROMWICH | 2450 |
| Stephen Butler Leacock: Commentary | 2455 |
| 9. Mathematics for Golfers | |
| by STEPHEN LEACOCK | 2456 |
| 10. Common Sense and the Universe | |
| by STEPHEN LEACOCK | 2460 |
| Index | 2471 |