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 LongSuffering 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 Soapbubble 

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 Absentminded 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 Manysided 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 DogWalking 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 ChessPlaying 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 