MATHEMATICAL RECREATIONS AND ESSAYS
TABLE OF CONTENTS PART I MATHEMATICAL RECREATIONS CHAPTER I SOME ARITHMETICAL QUESTIONS ELEMENTARY QUESTIONS ON NUMBERS (MISCELLANEOUS) ARITHMETICAL FALLACIES BACHET’S WEIGHTS PROBLEM PROBLEMS IN HIGHER ARITHMETIC FERMAT’S THEOREM ON BINARY POWERS FERMAT’S LAST THEOREM CHAPTER II SOME GEOMETRICAL QUESTIONS GEOMETRICAL FALLACIES GEOMETRICAL PARADOXES COLOURING MAPS PHYSICAL GEOGRAPHY STATICAL GAMES OF POSITION THREE-IN-A-ROW EXTENSION TO P-IN-A-ROW TESSELATION CROSS-FOURS COLOUR-CUBE PROBLEM DYNAMICAL GAMES OF POSITION SHUNTING PROBLEMS FERRY-BOAT PROBLEMS GEODESIC PROBLEMS PROBLEMS WITH COUNTERS PLACED IN A ROW PROBLEMS ON A CHESS-BOARD WITH COUNTERS OR PAWNS GUARINI’S PROBLEM GEOMETRICAL PUZZLES (RODS, STRINGS, &C) PARADROMIC RINGS CHAPTER III SOME MECHANICAL QUESTIONS PARADOXES ON MOTION FORCE, INERTIA, CENTRIFUGAL FORCE WORK, STABILITY OF EQUILIBRIUM, &C PERPETUAL MOTION MODELS SAILING QUICKER THAN THE WIND BOAT MOVED BY A ROPE INSIDE THE BOAT RESULTS DEPENDENT ON HAUKSBEE’S LAW CUT ON A TENNIS-BALL SPIN ON A CRICKET-BALL FLIGHT OF BIRDS CURIOSA PHYSICA CHAPTER IV SOME MISCELLANEOUS QUESTIONS THE FIFTEEN PUZZLE THE TOWER OF HANO¨I CHINESE RINGS THE EIGHT QUEENS PROBLEM OTHER PROBLEMS WITH QUEENS AND CHESS-PIECES THE FIFTEEN SCHOOL-GIRLS PROBLEM PAGE PROBLEMS CONNECTED WITH A PACK OF CARDS MONGE ON SHU?ING A PACK OF CARDS ARRANGEMENT BY ROWS AND COLUMNS DETERMINATION OF ONE OUT OF 2N(N + 1) GIVEN COUPLES GERGONNE’S PILE PROBLEM THE MOUSE TRAP TREIZE CHAPTER V MAGIC SQUARES NOTES ON THE HISTORY OF MAGIC SQUARES CONSTRUCTION OF ODD MAGIC SQUARES METHOD OF DE LA LOUB`ERE METHOD OF BACHET METHOD OF DE LA HIRE CONSTRUCTION OF EVEN MAGIC SQUARES FIRST METHOD METHOD OF DE LA HIRE AND LABOSNE COMPOSITE MAGIC SQUARES BORDERED MAGIC SQUARES HYPER-MAGIC SQUARES PAN-DIAGONAL OR NASIK SQUARES DOUBLY MAGIC SQUARES MAGIC PENCILS MAGIC PUZZLES CARD SQUARE EULER’S O?CERS PROBLEM DOMINO SQUARES COIN SQUARES CHAPTER VI UNICURSAL PROBLEMS EULER’S PROBLEM DE?NITIONS EULER’S THEOREMS EXAMPLES PAGE MAZES RULES FOR COMPLETELY TRAVERSING A MAZE NOTES ON THE HISTORY OF MAZES GEOMETRICAL TREES THE HAMILTONIAN GAME KNIGHT’S PATH ON A CHESS-BOARD METHOD OF DE MONTMORT AND DE MOIVRE METHOD OF EULER METHOD OF VANDERMONDE METHOD OF WARNSDOR? METHOD OF ROGET METHOD OF MOON METHOD OF JAENISCH NUMBER OF POSSIBLE ROUTES PATHS OF OTHER CHESS-PIECES PART II MISCELLANEOUS ESSAYS AND PROBLEMS CHAPTER VII THE MATHEMATICAL TRIPOS PAGE MEDIEVAL COURSE OF STUDIES: ACTS THE RENAISSANCE AT CAMBRIDGE RISE OF A MATHEMATICAL SCHOOL SUBJECT-MATTER OF ACTS AT DI?ERENT PERIODS DEGREE LISTS ORAL EXAMINATIONS ALWAYS POSSIBLE ADDITIONAL WORK THROWN ON MODERATORS STIPENDS RAISED FACILITATES ORDER OF MERIT SCHEME OF EXAMINATION IN 1750 RIGHT OF MAS TO TAKE PART IN IT SCHEME OF EXAMINATION IN 1763 FOUNDATIONS OF SMITH’S PRIZES, 1768 INTRODUCTION OF A WRITTEN EXAMINATION, CIRC 1770 DESCRIPTION OF THE EXAMINATION IN 1772 SCHEME OF EXAMINATION IN 1779 SYSTEM OF BRACKETS PROBLEM PAPERS IN 1785 AND 1786 DESCRIPTION OF THE EXAMINATION IN 1791 THE POLL PART OF THE EXAMINATION A PASS STANDARD INTRODUCED PROBLEM PAPERS FROM 1802 ONWARDS DESCRIPTION OF THE EXAMINATION IN 1802 SCHEME OF READING IN 1806 INTRODUCTION OF MODERN ANALYTICAL NOTATION ALTERATIONS IN SCHEMES OF STUDY, SCHEME OF EXAMINATION IN 1827 SCHEME OF EXAMINATION IN 1833 ALL THE PAPERS MARKED SCHEME OF EXAMINATION IN 1839 PAGE SCHEME OF EXAMINATION IN 1848 CREATION OF A BOARD OF MATHEMATICAL STUDIES SCHEME OF EXAMINATION IN 1873 SCHEME OF EXAMINATION IN 1882 FALL IN NUMBER OF STUDENTS READING MATHEMATICS ORIGIN OF TERM TRIPOS TRIPOS VERSES CHAPTER VIII THREE GEOMETRICAL PROBLEMS THE THREE PROBLEMS THE DUPLICATION OF THE CUBE LEGENDARY ORIGIN OF THE PROBLEM LEMMA OF HIPPOCRATES SOLUTIONS OF ARCHYTAS, PLATO, MENAECHMUS, APOLLONIUS, AND SPORUS SOLUTIONS OF VIETA, DESCARTES, GREGORY OF ST VINCENT, AND NEWTON THE TRISECTION OF AN ANGLE SOLUTIONS QUOTED BY PAPPUS (THREE) SOLUTIONS OF DESCARTES, NEWTON, CLAIRAUT, AND CHASLES THE QUADRATURE OF THE CIRCLE INCOMMENSURABILITY OF ? DE?NITIONS OF ? ORIGIN OF SYMBOL ? METHODS OF APPROXIMATING TO THE NUMERICAL VALUE OF ? GEOMETRICAL METHODS OF APPROXIMATION RESULTS OF EGYPTIANS, BABYLONIANS, JEWS RESULTS OF ARCHIMEDES AND OTHER GREEK WRITERS RESULTS OF ROMAN SURVEYORS AND GERBERT RESULTS OF INDIAN AND EASTERN WRITERS RESULTS OF EUROPEAN WRITERS, 1200–1630 THEOREMS OF WALLIS AND BROUNCKER ANALYTICAL METHODS OF APPROXIMATION GREGORY’S SERIES RESULTS OF EUROPEAN WRITERS, 1699–1873 GEOMETRICAL APPROXIMATIONS APPROXIMATIONS BY THE THEORY OF PROBABILITY CHAPTER IX MERSENNE’S NUMBERS PAGE MERSENNE’S ENUNCIATION OF THE THEOREM LIST OF KNOWN RESULTS CASES AWAITING VERI?CATION HISTORY OF INVESTIGATIONS METHODS USED IN ATTACKING THE PROBLEM BY TRIAL OF DIVISORS OF KNOWN FORMS BY INDETERMINATE EQUATIONS BY PROPERTIES OF QUADRATIC FORMS BY THE USE OF A CANON ARITHMETICUS BY PROPERTIES OF BINARY POWERS BY THE USE OF THE BINARY SCALE BY THE USE OF FERMAT’S THEOREM MECHANICAL METHODS OF FACTORIZING NUMBERS CHAPTER X ASTROLOGY ASTROLOGY TWO BRANCHES: NATAL AND HORARY ASTROLOGY RULES FOR CASTING AND READING A HOROSCOPE HOUSES AND THEIR SIGNI?CATIONS PLANETS AND THEIR SIGNI?CATIONS ZODIACAL SIGNS AND THEIR SIGNI?CATIONS KNOWLEDGE THAT RULES WERE WORTHLESS NOTABLE INSTANCES OF HOROSCOPY LILLY’S PREDICTION OF THE GREAT FIRE AND PLAGUE FLAMSTEED’S GUESS CARDAN’S HOROSCOPE OF EDWARD VI CHAPTER XI CRYPTOGRAPHS AND CIPHERS A CRYPTOGRAPH DE?NITION ILLUSTRATION A CIPHER DE?NITION ILLUSTRATION ESSENTIAL FEATURES OF CRYPTOGRAPHS AND CIPHERS CRYPTOGRAPHS OF THREE TYPES ILLUSTRATIONS ORDER OF LETTERS RE-ARRANGED USE OF NON-SIGNI?CANT SYMBOLS THE GRILLE USE OF BROKEN SYMBOLS THE SCYTALE CIPHERS USE OF ARBITRARY SYMBOLS UNNECESSARY PAGE CIPHERS OF FOUR TYPES CIPHERS OF THE FIRST TYPE ILLUSTRATIONS CIPHERS OF THE SECOND TYPE ILLUSTRATIONS CIPHERS OF THE THIRD TYPE ILLUSTRATIONS CIPHERS OF THE FOURTH TYPE ILLUSTRATIONS REQUISITES IN A GOOD CIPHER CIPHER MACHINES HISTORICAL CIPHERS JULIUS CAESAR, AUGUSTUS BACON CHARLES I PEPYS DE ROHAN MARIE ANTOINETTE THE CODE DICTIONARY POE’S WRITINGS CHAPTER XII HYPER-SPACE TWO SUBJECTS OF SPECULATION ON HYPER-SPACE SPACE OF TWO DIMENSIONS AND OF ONE DIMENSION SPACE OF FOUR DIMENSIONS EXISTENCE IN SUCH A WORLD ARGUMENTS IN FAVOUR OF THE EXISTENCE OF SUCH A WORLD NON-EUCLIDEAN GEOMETRIES EUCLID’S AXIOMS AND POSTULATES THE PARALLEL POSTULATE HYPERBOLIC GEOMETRY OF TWO DIMENSIONS ELLIPTIC GEOMETRY OF TWO DIMENSIONS ELLIPTIC, PARABOLIC AND HYPERBOLIC GEOMETRIES COMPARED NON-EUCLIDEAN GEOMETRIES OF THREE OR MORE DIMENSIONS CHAPTER XIII TIME AND ITS MEASUREMENT UNITS FOR MEASURING DURATIONS (DAYS, WEEKS, MONTHS, YEARS) THE CIVIL CALENDAR (JULIAN, GREGORIAN, &C) THE ECCLESIASTICAL CALENDAR (DATE OF EASTER) DAY OF THE WEEK CORRESPONDING TO A GIVEN DATE PAGE MEANS OF MEASURING TIME STYLES, SUN-DIALS, SUN-RINGS WATER-CLOCKS, SAND-CLOCKS, GRADUATED CANDLES CLOCKS AND WATCHES WATCHES AS COMPASSES CHAPTER XIV MATTER AND ETHER THEORIES HYPOTHESIS OF CONTINUOUS MATTER ATOMIC THEORIES POPULAR ATOMIC HYPOTHESIS BOSCOVICH’S HYPOTHESIS HYPOTHESIS OF AN ELASTIC SOLID ETHER LABILE ETHER DYNAMICAL THEORIES THE VORTEX RING HYPOTHESIS THE VORTEX SPONGE HYPOTHESIS THE ETHER-SQUIRTS HYPOTHESIS THE ELECTRON HYPOTHESIS SPECULATIONS DUE TO INVESTIGATIONS ON RADIO-ACTIVITY THE BUBBLE HYPOTHESIS CONJECTURES AS TO THE CAUSE OF GRAVITY CONJECTURES TO EXPLAIN THE ?NITE NUMBER OF SPECIES OF ATOMS SIZE OF THE MOLECULES OF BODIES