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[PDF page 1] The Project Gutenberg eBook of Mathematical Recreations and Essays, by W. W. Rouse Ball This eBook is for the use of anyone anywhere in the United States and most other parts of the world at no cost and with almost no restrictions whatsoever. Y ou may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this eBook or online at www.gutenberg.org. If you are not located in the United States, you will have to check the laws of the country where you are located before using this eBook. Title: Mathematical Recreations and Essays Author: W. W. Rouse Ball Release Date: October 8, 2008 [EBook #26839] [Most recently updated: October 19, 2021] Language: English Character set encoding: UTF-8 Produced by Joshua Hutchinson, David Starner, David Wilson and the Online Distributed Proofreading Team at https://www.pgdp.net Revised by David Wilson *** START OF THE PROJECT GUTENBERG EBOOK MATHEMATICAL RECREATIONS ***

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[PDF page 2] First Edition, Feb. 1892. Reprinted, May, 1892. Second Edition, 1896. Reprinted, 1905.

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[PDF page 3] MATHEMATICAL RECREATIONS AND ESSAYS BY W.W. ROUSE BALL Fellow and Tutor of Trinity College, Cambridge. FOURTH EDITION London: MACMILLAN AND CO., Limited NEW YORK: THE MACMILLAN COMPANY 1905 [All rights reserved. ]

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[PDF page 4] Produced by Joshua Hutchinson, David Starner, David Wilson and the Online Distributed Proofreading Team at https://www.pgdp.net Transcriber’s notes Most of the open questions discussed by the author were settled during the twentieth century. The author’s footnotes are labelled using printer’s marks*; footnotes showing where corrections to the text have been made are labelled numerically1. Minor typographical corrections are documented in the L ATEX source. This document is designed for two-sided printing. Consequently, the many hyperlinked cross-references are not visually distinguished. The document can be recompiled for more comfortable on-screen viewing: see comments in source L ATEX code. ‘Character set encoding’ refers to the LATEX source.

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[PDF page 5] PREF ACE TO THE FIRST EDITION. The following pages contain an account of certain mathematical recreations, problems, and speculations of past and present times. I hasten to add that the conclusions are of no practical use, and most of the results are not new. If therefore the reader proceeds further he is at least forewarned. At the same time I think I may assert that many of the diversions— particularly those in the latter half of the book—are interesting, not a few are associated with the names of distinguished mathematicians, while hitherto several of the memoirs quoted have not been easily ac- cessible to English readers. The book is divided into two parts, but in both parts I have in- cluded questions which involve advanced mathematics. The first partconsists of seven chapters, in which are included var- ious problems and amusements of the kind usually calledmathematical recreations. The questions discussed in the first of these chapters are connected with arithmetic; those in the second with geometry; and those in the third relate to mechanics. The fourth chapter contains an account of some miscellaneous problems which involve both num- ber and situation; the fifth chapter contains a concise account of magic squares; and the sixth and seventh chapters deal with some unicursal iii

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[PDF page 6] iv PREF ACE problems. Several of the questions mentioned in the first three chap- ters are of a somewhat trivial character, and had they been treated in any standard English work to which I could have referred the reader, I should have pointed them out. In the absence of such a work, I thought it best to insert them and trust to the judicious reader to omit them altogether or to skim them as he feels inclined. The second partconsists of five chapters, which are mostlyhistori- cal. They deal respectively with three classical problems in geometry— namely, the duplication of the cube, the trisection of an angle, and the quadrature of the circle—astrology, the hypotheses as to the nature of space and mass, and a means of measuring time. Ihaveinserteddetailedreferences, asfarasIknow, astothesources of the various questions and solutions given; also, wherever I have given only the result of a theorem, I have tried to indicate authorities where a proof may be found. In general, unless it is stated otherwise, I have taken the references direct from the original works; but, in spite of considerable time spent in verifying them, I dare not suppose that they are free from all errors or misprints. I shall be grateful for notices of additions or corrections which may occur to any of my readers. W.W. ROUSE BALL Trinity College, Cambridge. February, 1892.

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[PDF page 7] NOTE TO THE FOURTH EDITION. In this edition I have inserted in the earlier chapters descriptions of several additional Recreations involving elementary mathematics, and I have added in the second part chapters on theHistory of the Mathe- matical Tripos at Cambridge, Mersenne’s Numbers, andCryptography and Ciphers. It is with some hesitation that I include in the book the chapters on Astrology and Ciphers, for these subjects are only remotely connected with Mathematics, but to afford myself some latitude I have altered the title of the second part toMiscellaneous Essays and Problems. W.W.R.B. Trinity College, Cambridge. 13 May, 1905. v

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[PDF page 8] TABLE OF CONTENTS. PART I. Mathematical Recreations. Chapter I. Some Arithmetical Questions. PAGE Elementary Questions on Numbers (Miscellaneous) . . . . . . 4 Arithmetical Fallacies . . . . . . . . . . . . . . . . . . . . . . 20 Bachet’s Weights Problem . . . . . . . . . . . . . . . . . . . . 27 Problems in Higher Arithmetic . . . . . . . . . . . . . . . . . 29 Fermat’sTheorem on Binary Powers. . . . . . . . . . . . 31 Fermat’sLast Theorem. . . . . . . . . . . . . . . . . . . . 32 Chapter II. Some Geometrical Questions. Geometrical Fallacies . . . . . . . . . . . . . . . . . . . . . . . 35 Geometrical Paradoxes . . . . . . . . . . . . . . . . . . . . . . 42 Colouring Maps . . . . . . . . . . . . . . . . . . . . . . . . . . 44 Physical Geography . . . . . . . . . . . . . . . . . . . . . . . 46 Statical Games of Position . . . . . . . . . . . . . . . . . . . . 48 Three-in-a-row. Extension to p-in-a-row . . . . . . . . . 48 Tesselation. Cross-Fours . . . . . . . . . . . . . . . . . . 50 Colour-Cube Problem . . . . . . . . . . . . . . . . . . . . 51 vi

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[PDF page 9] TABLE OF CONTENTS. vii PAGE Dynamical Games of Position . . . . . . . . . . . . . . . . . . 52 Shunting Problems . . . . . . . . . . . . . . . . . . . . . . 53 Ferry-Boat Problems . . . . . . . . . . . . . . . . . . . . . 55 Geodesic Problems . . . . . . . . . . . . . . . . . . . . . . 57 Problems with Counters placed in a row . . . . . . . . . . 58 Problems on a Chess-board with Counters or Pawns . . . . 60 Guarini’s Problem . . . . . . . . . . . . . . . . . . . . . . 63 Geometrical Puzzles (rods, strings, &c.) . . . . . . . . . . . . 64 Paradromic Rings . . . . . . . . . . . . . . . . . . . . . . . . . 64 Chapter III. Some Mechanical Questions. Paradoxes on Motion . . . . . . . . . . . . . . . . . . . . . . . 67 Force, Inertia, Centrifugal Force . . . . . . . . . . . . . . . . . 70 Work, Stability of Equilibrium, &c. . . . . . . . . . . . . . . . 72 Perpetual Motion . . . . . . . . . . . . . . . . . . . . . . . . . 75 Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 Sailing quicker than the Wind . . . . . . . . . . . . . . . . . . 79 Boat moved by a rope inside the boat . . . . . . . . . . . . . 81 Results dependent on Hauksbee’s Law . . . . . . . . . . . . . 82 Cut on a tennis-ball. Spin on a cricket-ball . . . . . . . 83 Flight of Birds . . . . . . . . . . . . . . . . . . . . . . . . . . 85 Curiosa Physica . . . . . . . . . . . . . . . . . . . . . . . . . . 86 Chapter IV. Some Miscellaneous Questions. The Fifteen Puzzle . . . . . . . . . . . . . . . . . . . . . . . . 88 The Tower of Hanoï . . . . . . . . . . . . . . . . . . . . . . . 91 Chinese Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 The Eight Queens Problem . . . . . . . . . . . . . . . . . . . 97 Other Problems with Queens and Chess-pieces . . . . . . . . . 102 The Fifteen School-Girls Problem . . . . . . . . . . . . . . . . 103

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[PDF page 10] viii TABLE OF CONTENTS. PAGE Problems connected with a pack of cards . . . . . . . . . . . . 109 Monge on shuffling a pack of cards . . . . . . . . . . . . . 109 Arrangement by rows and columns . . . . . . . . . . . . . 111 Determination of one out of1 2 n(n + 1) given couples . . . . 113 Gergonne’s Pile Problem . . . . . . . . . . . . . . . . . . . 115 The Mouse Trap. Treize . . . . . . . . . . . . . . . . . . 119 Chapter V. Magic Squares. Notes on the History of Magic Squares . . . . . . . . . . . . . 122 Construction of Odd Magic Squares . . . . . . . . . . . . . . . 123 Method of De la Loubère . . . . . . . . . . . . . . . . . . . 124 Method of Bachet . . . . . . . . . . . . . . . . . . . . . . . 125 Method of De la Hire . . . . . . . . . . . . . . . . . . . . . 126 Construction of Even Magic Squares . . . . . . . . . . . . . . 128 First Method . . . . . . . . . . . . . . . . . . . . . . . . . 129 Method of De la Hire and Labosne . . . . . . . . . . . . . 132 Composite Magic Squares . . . . . . . . . . . . . . . . . . . . 134 Bordered Magic Squares . . . . . . . . . . . . . . . . . . . . . 135 Hyper-Magic Squares . . . . . . . . . . . . . . . . . . . . . . . 136 Pan-diagonal or Nasik Squares . . . . . . . . . . . . . . . . 136 Doubly Magic Squares . . . . . . . . . . . . . . . . . . . . 137 Magic Pencils . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 Magic Puzzles . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 Card Square . . . . . . . . . . . . . . . . . . . . . . . . . . 140 Euler’s Officers Problem . . . . . . . . . . . . . . . . . . . 140 Domino Squares . . . . . . . . . . . . . . . . . . . . . . . . 141 Coin Squares . . . . . . . . . . . . . . . . . . . . . . . . . 141 Chapter VI. Unicursal Problems. Euler’s Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 143 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 Euler’s Theorems . . . . . . . . . . . . . . . . . . . . . . . 145 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

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[PDF page 11] TABLE OF CONTENTS. ix PAGE Mazes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 Rules for completely traversing a Maze . . . . . . . . . . . 149 Notes on the History of Mazes . . . . . . . . . . . . . . . . 150 Geometrical Trees . . . . . . . . . . . . . . . . . . . . . . . . 154 The Hamiltonian Game . . . . . . . . . . . . . . . . . . . . . 155 Knight’s Path on a Chess-Board . . . . . . . . . . . . . . . . . 158 Method of De Montmort and De Moivre . . . . . . . . . . 159 Method of Euler . . . . . . . . . . . . . . . . . . . . . . . 160 Method of Vandermonde . . . . . . . . . . . . . . . . . . . 163 Method of Warnsdorff . . . . . . . . . . . . . . . . . . . . 163 Method of Roget . . . . . . . . . . . . . . . . . . . . . . . 164 Method of Moon . . . . . . . . . . . . . . . . . . . . . . . 167 Method of Jaenisch . . . . . . . . . . . . . . . . . . . . . . 168 Number of possible routes . . . . . . . . . . . . . . . . . . 168 Paths of other Chess-Pieces . . . . . . . . . . . . . . . . . . . 168

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[PDF page 12] x TABLE OF CONTENTS. PART II. Miscellaneous Essays and Problems. Chapter VII. The Mathematical Tripos. PAGE Medieval Course of Studies: Acts . . . . . . . . . . . . . . . . 170 The Renaissance at Cambridge . . . . . . . . . . . . . . . . . 171 Rise of a Mathematical School . . . . . . . . . . . . . . . . 171 Subject-Matter of Acts at different periods . . . . . . . . . . . 171 Degree Lists . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 Oral Examinations always possible . . . . . . . . . . . . . . . 173 Public Oral Examinations become customary, 1710–30 . . . . 174 Additional work thrown on Moderators. Stipends raised . 174 Facilitates order of merit . . . . . . . . . . . . . . . . . . . 175 Scheme of Examination in 1750 . . . . . . . . . . . . . . . . . 175 Right of M.A.s to take part in it . . . . . . . . . . . . . . . 175 Scheme of Examination in 1763 . . . . . . . . . . . . . . . . . 176 Foundations of Smith’s Prizes, 1768 . . . . . . . . . . . . . . . 177 Introduction of a Written Examination, circ. 1770 . . . . . . . 178 Description of the Examination in 1772 . . . . . . . . . . . . . 178 Scheme of Examination in 1779 . . . . . . . . . . . . . . . . . 181 System of Brackets . . . . . . . . . . . . . . . . . . . . . . 181 Problem Papers in 1785 and 1786 . . . . . . . . . . . . . . . . 182 Description of the Examination in 1791 . . . . . . . . . . . . . 183 The Poll Part of the Examination . . . . . . . . . . . . . . 184 A Pass Standard introduced . . . . . . . . . . . . . . . . . . . 185 Problem Papers from 1802 onwards . . . . . . . . . . . . . . . 185 Description of the Examination in 1802 . . . . . . . . . . . . . 186 Scheme of Reading in 1806 . . . . . . . . . . . . . . . . . . . . 188 Introduction of modern analytical notation . . . . . . . . . . . 191 Alterations in Schemes of Study, 1824 . . . . . . . . . . . . . 194 Scheme of Examination in 1827 . . . . . . . . . . . . . . . . . 194 Scheme of Examination in 1833 . . . . . . . . . . . . . . . . . 196 All the papers marked . . . . . . . . . . . . . . . . . . . . 196 Scheme of Examination in 1839 . . . . . . . . . . . . . . . . . 196

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[PDF page 13] TABLE OF CONTENTS. xi PAGE Scheme of Examination in 1848 . . . . . . . . . . . . . . . . . 197 Creation of a Board of Mathematical Studies . . . . . . . . . 197 Scheme of Examination in 1873 . . . . . . . . . . . . . . . . . 198 Scheme of Examination in 1882 . . . . . . . . . . . . . . . . . 199 Fall in number of students reading mathematics . . . . . . 200 Origin of term Tripos . . . . . . . . . . . . . . . . . . . . . . . 200 Tripos Verses . . . . . . . . . . . . . . . . . . . . . . . . . 201 Chapter VIII. Three Geometrical Problems. The Three Problems . . . . . . . . . . . . . . . . . . . . . . . 203 The Duplication of the Cube . . . . . . . . . . . . . . . . . . 204 Legendary origin of the problem . . . . . . . . . . . . . . . 204 Lemma of Hippocrates . . . . . . . . . . . . . . . . . . . . . . 205 Solutions of Archytas, Plato, Menaechmus, Apollonius, and Sporus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 Solutions of Vieta, Descartes, Gregory of St Vincent, and Newton . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 The Trisection of an Angle . . . . . . . . . . . . . . . . . . . . 209 Solutions quoted by Pappus (three) . . . . . . . . . . . . . . . 209 Solutions of Descartes, Newton, Clairaut, and Chasles . . . . 210 The Quadrature of the Circle . . . . . . . . . . . . . . . . . . 211 Incommensurability of π . . . . . . . . . . . . . . . . . . . . . 211 Definitions of π . . . . . . . . . . . . . . . . . . . . . . . . . . 212 Origin of symbolπ . . . . . . . . . . . . . . . . . . . . . . . . 213 Methods of approximating to the numerical value ofπ . . . . 213 Geometrical methods of approximation . . . . . . . . . . . . . 213 Results of Egyptians, Babylonians, Jews . . . . . . . . . . 214 Results of Archimedes and other Greek writers . . . . . . . 214 Results of Roman surveyors and Gerbert . . . . . . . . . . 215 Results of Indian and Eastern writers . . . . . . . . . . . . 215 Results of European writers, 1200–1630 . . . . . . . . . . . 216 Theorems of Wallis and Brouncker . . . . . . . . . . . . . . . 219 Analytical methods of approximation. Gregory’s series . . . . 219 Results of European writers, 1699–1873 . . . . . . . . . . . 219 Geometrical approximations . . . . . . . . . . . . . . . . . . . 221 Approximations by the theory of probability . . . . . . . . . . 221

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[PDF page 14] xii TABLE OF CONTENTS. Chapter IX. Mersenne’s Numbers. PAGE Mersenne’s Enunciation of the Theorem . . . . . . . . . . . . 223 List of known results . . . . . . . . . . . . . . . . . . . . . . . 224 Cases awaiting verification . . . . . . . . . . . . . . . . . . . . 224 History of Investigations . . . . . . . . . . . . . . . . . . . . . 225 Methods used in attacking the problem . . . . . . . . . . . . . 229 By trial of divisors of known forms . . . . . . . . . . . . . 230 By indeterminate equations . . . . . . . . . . . . . . . . . 232 By properties of quadratic forms . . . . . . . . . . . . . . 233 By the use of aCanon Arithmeticus . . . . . . . . . . . . 233 By properties of binary powers . . . . . . . . . . . . . . . 234 By the use of the binary scale . . . . . . . . . . . . . . . . 234 By the use of Fermat’s Theorem . . . . . . . . . . . . . . . 235 Mechanical methods of Factorizing Numbers . . . . . . . . . . 235 Chapter X. Astrology. Astrology. Two branches: natal and horary astrology . . . . . 237 Rules for casting and reading a horoscope . . . . . . . . . . . 237 Houses and their significations . . . . . . . . . . . . . . . . 237 Planets and their significations . . . . . . . . . . . . . . . 239 Zodiacal signs and their significations . . . . . . . . . . . . 241 Knowledge that rules were worthless . . . . . . . . . . . . . . 242 Notable instances of horoscopy . . . . . . . . . . . . . . . . . 245 Lilly’s prediction of the Great Fire and Plague . . . . . . . 245 Flamsteed’s guess . . . . . . . . . . . . . . . . . . . . . . . 245 Cardan’s horoscope of Edward VI . . . . . . . . . . . . . . 246 Chapter XI. Cryptographs and Ciphers. A Cryptograph. Definition. Illustration . . . . . . . . . . . . . 250 A Cipher. Definition. Illustration . . . . . . . . . . . . . . . . 251 Essential Features of Cryptographs and Ciphers . . . . . . . . 251 Cryptographs of Three Types. Illustrations . . . . . . . . . . 252 Order of letters re-arranged . . . . . . . . . . . . . . . . . 252 Use of non-significant symbols. The Grille . . . . . . . . . 255 Use of broken symbols. The Scytale . . . . . . . . . . . . . 257

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[PDF page 15] TABLE OF CONTENTS. xiii PAGE Ciphers. Use of arbitrary symbols unnecessary . . . . . . . . . 258 Ciphers of Four Types . . . . . . . . . . . . . . . . . . . . . . 258 Ciphers of the First Type. Illustrations . . . . . . . . . . . 259 Ciphers of the Second Type. Illustrations . . . . . . . . . . 262 Ciphers of the Third Type. Illustrations . . . . . . . . . . 264 Ciphers of the Fourth Type. Illustrations . . . . . . . . . . 266 Requisites in a good Cipher . . . . . . . . . . . . . . . . . . . 267 Cipher Machines . . . . . . . . . . . . . . . . . . . . . . . . . 268 Historical Ciphers . . . . . . . . . . . . . . . . . . . . . . . . . 268 Julius Caesar, Augustus . . . . . . . . . . . . . . . . . . . 268 Bacon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268 Charles I . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268 Pepys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270 De Rohan . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 Marie Antoinette . . . . . . . . . . . . . . . . . . . . . . . 272 The Code Dictionary . . . . . . . . . . . . . . . . . . . . . 273 Poe’s Writings . . . . . . . . . . . . . . . . . . . . . . . . . 274 Chapter XII. Hyper-space. Two subjects of speculation on Hyper-space . . . . . . . . . . 277 Space of two dimensions and of one dimension . . . . . . . . . 277 Space of four dimensions . . . . . . . . . . . . . . . . . . . . . 278 Existence in such a world . . . . . . . . . . . . . . . . . . 278 Arguments in favour of the existence of such a world . . . 279 Non-Euclidean Geometries . . . . . . . . . . . . . . . . . . . . 283 Euclid’s axioms and postulates. The parallel postulate . . . . 283 Hyperbolic Geometry of two dimensions . . . . . . . . . . . . 284 Elliptic Geometry of two dimensions . . . . . . . . . . . . . . 284 Elliptic, Parabolic and Hyperbolic Geometries compared . 284 Non-Euclidean Geometries of three or more dimensions . . . . 286 Chapter XIII. Time and its Measurement. Units for measuring durations (days, weeks, months, years) . . 288 The Civil Calendar (Julian, Gregorian, &c.) . . . . . . . . . . 291 The Ecclesiastical Calendar (date of Easter) . . . . . . . . . . 293

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[PDF page 16] xiv TABLE OF CONTENTS. PAGE Day of the week corresponding to a given date . . . . . . . . . 296 Means of measuring Time . . . . . . . . . . . . . . . . . . . . 296 Styles, Sun-dials, Sun-rings . . . . . . . . . . . . . . . . . 296 Water-clocks, Sand-clocks, Graduated Candles . . . . . . . 300 Clocks and Watches . . . . . . . . . . . . . . . . . . . . . 300 Watches as Compasses . . . . . . . . . . . . . . . . . . . . . . 302 Chapter XIV. Matter and Ether Theories. Hypothesis of Continuous Matter . . . . . . . . . . . . . . . . 305 Atomic Theories . . . . . . . . . . . . . . . . . . . . . . . . . 305 Popular Atomic Hypothesis . . . . . . . . . . . . . . . . . 305 Boscovich’s Hypothesis . . . . . . . . . . . . . . . . . . . . 306 Hypothesis of an Elastic Solid Ether. Labile Ether . . . . . 306 Dynamical Theories . . . . . . . . . . . . . . . . . . . . . . . 307 The Vortex Ring Hypothesis . . . . . . . . . . . . . . . . . 307 The Vortex Sponge Hypothesis . . . . . . . . . . . . . . . 309 The Ether-Squirts Hypothesis . . . . . . . . . . . . . . . . 309 The Electron Hypothesis . . . . . . . . . . . . . . . . . . . 310 Speculations due to investigations on Radio-activity . . . . 310 The Bubble Hypothesis . . . . . . . . . . . . . . . . . . . . 312 Conjectures as to the cause of Gravity . . . . . . . . . . . . . 313 Conjectures to explain the finite number of species of Atoms . 317 Size of the molecules of bodies . . . . . . . . . . . . . . . . . . 319 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322 Notices of some works—chiefly historico-mathematical . . . . . . . . . . . . . . . . . 334 Project Gutenberg Licensing Information. . . . . . 353

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[PDF page 17] PART I. Mathematical Recreations. “Les hommes ne sont jamais plus ingénieux que dans l’invention des jeux; l’esprit s’y trouve à son aise.... Après les jeux qui dépendent uniquement des nombres viennent les jeux où entre la situa- tion.... Après les jeux où n’entrent que le nom- bre et la situation viendraient les jeux où entre le mouvement.... Enfin il serait a souhaiter qu’on eût un cours entier des jeux, traités mathématique- ment.” (Leibnitz: letter to De Montmort, July 29, 1715.) 1

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[PDF page 18] CHAPTER I. SOME ARITHMETICAL QUESTIONS. The interest excited by statements of the relations between num- bers of certain forms has been often remarked. The majority of works on mathematical recreations include several such problems, which are obvious to any one acquainted with the elements of algebra, but which to many who are ignorant of that subject possess the same kind of charm that some mathematicians find in the more recondite proposi- tions of higher arithmetic. I shall devote the bulk of this chapter to these elementary problems, but I append a few remarks on one or two questions in the theory of numbers. Before entering on the subject of the chapter, I may add that a large proportion of the elementary questions mentioned here and in the following two chapters are taken from one of two sources. The first of these is the classicalProblèmes plaisans et délectables, by C.G. Ba- chet, sieur de Méziriac, of which the first edition was published in 1612 and the second in 1624: it is to the edition of 1624 that the references hereafter given apply. Several of Bachet’s problems are taken from the writings of Alcuin, Pacioli di Burgo, Tartaglia, or Cardan, and possi- bly some of them are of oriental origin, but I have made no attempt to add such references. The other source to which I alluded above is Ozanam’sRécréations mathématiques et physiques. The greater por- tion of the original edition, published in two volumes at Paris in 1694, was a compilation from the works of Bachet, Leurechon, Mydorge, van Etten, and Oughtred: this part is excellent, but the same cannot be said of the additions due to Ozanam. In theBiographie Universelleal- lusion is made to subsequent editions issued in 1720, 1735, 1741, 1778, 2
