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ProjectGutenberg'sMathematical Recreationsand Essays,byW. W.Rouse Ball ThiseBookis fortheuse ofanyone anywhereatno costandwith almostnorestrictions whatsoever.Youmay copyit, giveitaway or re-useitunder thetermsof theProject GutenbergLicenseincluded withthiseBook oronlineat www.gutenberg.org

Title:MathematicalRecreations andEssays

Author:W.W. RouseBall

ReleaseDate:October 8,2008[EBook #26839]

Language:English

Charactersetencoding: ISO-8859-1

STARTOFTHIS PROJECTGUTENBERGEBOOK MATHEMATICALRECREATIONS

FirstEdition ,Feb.1892.Re printed,May,1892.

SecondEdition,1896.Re printed,1905.

MATHEMATICAL

RECREATIONSANDESSAYS

BY

W.W.ROUSE BALL

FellowandTutorofTrinityCollege,Cam bridge.

FOURTHEDITION

London:

MACMILLANANDCO.,Limited

NEWYORK:T HEMA CMILLANCOMPANY

[Allrightsreserved.] ProducedbyJosh uaHutchinson,DavidStar ner,DavidWilson and theOnlineDistributed Pro ofreadingT eamathttp://www.pgdp.net

Transcriber'snotes

Mostofthe openqu estion sdiscussedb ytheauthorwere settledduringthetw entiethcentury. Theauth or'sfootnotesarelab elledusingprinter'smarks footnotesshowingwh erecorrectionsto thetexthavebeen madeare labellednumerically 1

Minortypographicalcorrectionsare documentedintheL

A T E X source. Thisdocument isdesignedfortwo-sidedprinti ng.Conseque ntly, themany hyperlinked cross-referencesarenotvisually distinguished.Thedocument canbe recompiledformore comfortableon-screenvi ewing: seecommentsinsourceL A T E X code.

PREFACETOTHEFIRSTEDITION.

Thefollowingpagescon tain anaccountofcertainmathematical recreations,problems,and speculationsofpastand present times.I hastentoadd thatthe conclusio nsare ofnopr acticaluse,andmost oftheresults areno tnew. Iftherefo rethereaderproceeds further he isatleast forewarned. Atthesametime Ithink Imay assertthatmanyof thediversions - particularlythoseint helatterhalfo ftheb ook - areinterest ing,not afeware associate dwith thenamesofdistinguishedmathematicians, whilehitherto severaloft hememoirsquotedhavenot beeneas ilyac- cessibletoEnglishreaders. Thebo okisdividedintotwo parts,but inboth partsIhave in- cludedque stionswhichinvolveadvanced mathematics. Thefirstpartconsistsofse venc hapters,inwhichareincludedv ar- iousproblems andamusements ofthekindusuallycalle dmathematical recreations.Thequestio nsdiscussed inthefirstof thesechaptersare connectedwitharithmetic;thos einthesecondwithgeometry;and thoseinthethird relatet omechanics.Thefourthchaptercontains anaccount ofsomemiscella neousproblems whichinvolvebo thnum- berandsituatio n;thefifthchapter containsaconciseaccount ofmagic squares;andthesixthandsev enth chaptersdealwithsomeunicurs al iii ivPREFA CE problems.Severalofthe questionsmentionedinthefirstthr eec hap- tersareof asomewhat trivialc haracter,a ndhad theybeentreated in anystandardEnglish worktowhichIco uldhav ereferredthe reader,I shouldhav epointedthemout.In theabsenceofsuch awork ,Ithoug ht itbest toinsertthema ndtrustto theju diciousreadertoomitthem altogetherortos kimthemas hefeelsinclined. Thesecondpartconsistsoffiv echa pters,whicha remostlyhistori- cal.Theydea lrespectiv elywiththreeclassicalpr oblemsinge ometry - namely,theduplicatio nofthe cube,thetrisectionofanangle,a ndthe quadratureofthecircle - astrology,theh ypothes esastothenatureof spaceandmass,anda mea nso fmeasuringtime. Iha veinserteddetailedreferences,asfar asIkno w,astothe sources ofthev ariousquestio nsandsolutionsgiven;also, whereverIhave given onlytheres ultof atheorem,I have triedto indicateauthoritieswhere aproo fmaybefound.In general,unlessitis statedotherwise,I have takenthereferencesdirect fromtheo riginalworks;but,inspiteof considerabletimesp en tinverifyingthem,Idarenotsupposethat they arefreef romallerrorsormisprints. Ishallb egratef ulfornoticesofaddit ionsorcorrectionswhichmay occurtoan yofmy readers.

W.W.RO USEBALL

TrinityCollege ,Cambridge.

February,1892.

NOTETO THEFOURTH EDITION.

InthiseditionI have insert edintheearlierchaptersde scriptionsof severaladditionalRecreat ionsinvolvingelementaryma thematics,and Ihav eaddedinthesecondpa rtchapterson theHistoryoftheMathe- andCiphers . Itisw iths omehesitationthatI includeinthebookthe chapterson AstrologyandCiphers,forthes esubj ectsareonlyremo telyconnected withMathe matics,buttoaffordmyselfsomelat itudeIhave altered thetitleo fthesecondparttoMiscellaneousEssaysandProblems.

W.W.R.B.

TrinityCollege ,Cambridge.

13May,1905.

v

TABLEOFCONTENT S.

PARTI.

MathematicalRecreations.

ChapterI.SomeAr ithmeticalQuestions.

PAGE ElementaryQuestionson Numbers(Miscellaneous)......4

ArithmeticalFallacie s......................2 0

Bachet'sWeights Problem....................2 7

ProblemsinHig herArithmetic .................2 9

Fermat'sLastTheorem....................3 2

ChapterII.SomeGeome tricalQuestions.

GeometricalFallacies .......................3 5

GeometricalPar adoxes......................4 2

PhysicalGeograph y.......................46

StaticalGamesof Position....................4 8

Colour-CubeProblem....................5 1

vi

TABLEOFCONTENTS.vii

PAGE

DynamicalGamesof Position..................5 2

ShuntingProblems......................5 3

Ferry-BoatProblems.....................5 5

GeodesicProblems......................5 7

Problemswit hCountersplacedin arow..........58

Problemsona Chess-bo ardwithC ountersorPawns....60

Guarini'sProblem......................6 3

GeometricalPuzzles(rods,s trings,&c.)............64

ParadromicRings.........................6 4

ChapterIII.Some MechanicalQuestions.

Force,Inertia,Cent rifugalForce.................70

Work,Stabilityof Equilibrium,&c................72

PerpetualMotion.........................7 5

Models...............................78

Sailingquic kerthantheWind..................7 9

Boatmov edbyaropeinside theboat.............8 1

Resultsdep endentonHauksbee'sLaw.............82

Cutona tennis-ball. Spinon acricket-ball.......8 3

FlightofBirds..........................8 5

CuriosaPhysica ..........................86

ChapterIV.SomeMis cellane ousQuestions.

TheFifteenP uzzle........................8 8

TheTo werofHano¨ı.......................9 1

TheEight QueensProblem...................97

OtherProblemswithQueensand Chess-pieces... ... ... 102 TheFifteenSc hool-GirlsPro blem... ... .. ... ... ..10 3 viiiTABLEOFCO NTENTS. PAGE Problemsco nnectedwithapackofcards... ... .. ... .109 Mongeonshufflingapa cko fcards... ... .. ... ..10 9

Arrangementbyrowsand columns... ... ... ... .111

Determinationofoneoutof

1 2 n(n+1) givencouples... .113 Gergonne'sPileProblem... ... .. ... ... ... ..11 5

TheMouseT rap.T reize... ... ... ... ... ... 119

ChapterV.MagicSquares .

NotesontheHistoryof Magic Squares ... ... .. ... ..12 2 ConstructionofOdd MagicSq uares... ... ... ... ... 123 MethodofDela Loub` ere... ... .. ... ... ... ..12 4

MethodofBache t... ... ... ... ... ... .. ... 125

MethodofDela Hire... ... ... ... ... ... .. .126

ConstructionofEven MagicSqua res... ... .. ... ... 128

FirstMet hod... ... .. ... ... ... ... ... ..12 9

MethodofDela HireandLab osne... ... .. ... ..13 2

CompositeMagicSquares ... ... .. ... ... ... ... 134 BorderedMa gicSquares... ... .. ... ... ... ... .135 Hyper-MagicSquares... ... .. ... ... ... ... ... 136 Pan-diagonalorNasikSquares... ... .. ... ... ..13 6

DoublyMagicSquares ... ... ... ... ... ... .. 137

MagicPencils... ... ... .. ... ... ... ... ... .137 MagicPuzzles... ... .. ... ... ... ... ... ... .140

CardSquare ... ... .. ... ... ... ... ... ... 140

Euler'sOfficersProblem... ... ... ... ... ... .14 0

DominoSquares... ... .. ... ... ... ... ... .141

CoinSquares ... ... ... ... ... ... .. ... ..14 1

ChapterVI.Un icursalPr oblems.

Euler'sProble m... ... .. ... ... ... ... ... ... 143 Definitions... ... .. ... ... ... ... ... ... .145

Euler'sTheorems... ... .. ... ... ... ... ... 145

Examples... ... ... ... ... ... .. ... ... .148

TABLEOFCONTENTS.ix

PAGE Mazes... ... .. ... ... ... ... ... ... ... ..14 9 Rulesfo rcompletelytraversinga Maze... ... ... .. 150 NotesontheHistoryofM azes... ... .. ... ... ..15 0 GeometricalTrees... ... ... ... ... ... .. ... .154 TheHamiltonia nGame... ... ... .. ... ... ... .155 Knight'sPathon aChess-Board... ... ... ... ... ..15 8

MethodofDeMon tmort andDeMoivre... ... ... .159

MethodofEuler... ... ... .. ... ... ... ... 159

MethodofVandermonde ... ... .. ... ... ... ..16 3

MethodofWarns dorff... ... ... .. ... ... ... 164

MethodofRoget ... ... ... ... ... ... .. ... 164

MethodofMoon ... ... ... ... ... ... .. ... 167

MethodofJaenisc h... ... ... ... ... ... .. ..16 8 Numberofpossiblerout es... ... .. ... ... ... .168 PathsofotherChess-Pieces... ... ... .. ... ... ..16 8 xTABLE OFCONTENTS.

PARTII.

MiscellaneousEssaysand Problems.

ChapterVII.TheMa thematicalTripos.

PAGE MedievalCourseofStudies: Acts... ... .. ... ... ..17 1 TheRe naissanceatCambridge... ... .. ... ... ... 172 Riseofa Mathematical Schoo l... ... .. ... ... ..17 2 Subject-MatterofActsatdifferentperiods... ... .. ... 172 DegreeLists... ... ... ... ... ... .. ... ... ..17 4 OralExaminations alwayspossible... ... ... ... ... 174 PublicOral Examinationsbecome customary,1710- 30... .175 Additionalworkthro wnonModerators. Stipendsraised.175 Facilitatesorderofmerit... ... ... ... ... ... .17 6 SchemeofExaminatio nin1750... ... ... ... ... ..17 6 RightofM.A.stota kepart init... ... .. ... ... .176 SchemeofExaminatio nin1763... ... ... ... ... ..17 7 FoundationsofSmith'sPrizes ,1768... ... ... ... ... 178 IntroductionofaWrittenExamination,circ.17 70... ... .179 Descriptionofthe Examinat ionin 1772... ... ... ... .179 SchemeofExaminatio nin1779... ... .. ... ... ... 182 SystemofBra ckets ... ... ... ... ... ... .. ..18 2 ProblemPap ersin1785and1786 ... ... .. ... ... ..18 3 Descriptionofthe Examinat ionin1 791... ... .. ... ..18 4 ThePo llPartoftheExaminatio n... ... ... ... ..18 5 APass Standardintroduced ... ... ... .. ... ... ..18 6 ProblemPap ersfrom1802onwa rds... ... .. ... ... .186 Descriptionofthe Ex aminationin 1802... ... ... ... .187 SchemeofRe adingin1806 ... ... .. ... ... ... ... 189 Introductionofmodernanalyticalnotation... ... .. ... 192 AlterationsinSchemeso fStudy, 1824... ... .. ... ..19 5 SchemeofExaminatio nin1827... ... ... ... ... ..19 5 SchemeofExaminatio nin1833... ... ... ... ... ..19 7 Allthepa pers marked... ... .. ... ... ... ... 197 SchemeofExaminatio nin1839... ... ... ... ... ..19 7

TABLEOFCONTENTS.xi

PAGE SchemeofExaminatio nin1848... ... .. ... ... ... 198 CreationofaBoa rdofMathematicalStudies ... ... .. .198 SchemeofExaminatio nin1873... ... .. ... ... ... 199 SchemeofExaminatio nin1882... ... .. ... ... ... 200 Fallinnumb ero fstudentsreadingmathematics... ... 201 OriginoftermT ripos... ... ... ... ... ... .. ... 201

TriposVerses... ... ... ... ... ... .. ... ..20 2

ChapterVIII.Thr eeGeometricalProblems.

TheThreeP roblems ... ... .. ... ... ... ... ... 204 TheDuplication oftheCube... ... .. ... ... ... .205 Legendaryoriginof theproblem... ... .. ... ... .205 LemmaofHipp ocrat es... ... .. ... ... ... ... ..20 6 SolutionsofArch ytas,Plato, Menaechmus,Apollonius,and

Sporus.... ... ... ... ... .. ... ... ... ..207

SolutionsofVieta, Descartes, GregoryofStVincent, and

Newton.. ... ... ... ... ... .. ... ... ... 209

TheTrise ctionofanAngle... ... .. ... ... ... ... 210 Solutionsquotedby Pappus(three)... ... ... ... ... 210 SolutionsofDescartes ,Newton,Clairaut,andC hasles... .211 TheQuadra tureoftheCircle... ... ... ... ... ... 212 Incommensurabilityofπ... ... .. ... ... ... ... .212 Definitionsofπ... ... ... .. ... ... ... ... ... 213 Originofsymbol π... ... ... .. ... ... ... ... .214 Methodsofappro ximatingtothen umericalvalueof π... .214 Geometricalmethodsofapproximat ion... ... ... ... .214 ResultsofEg yptians,Ba bylonians,Jews... ... .. ..21 5 ResultsofAr chimedes andotherGreekwriters... ... .215 ResultsofR omansurv eyorsandGerbert ... ... .. ..21 6 ResultsofIndia nand Easternwriters... ... .. ... .216 ResultsofEuro pe anwriters,1200-1630... ... .. ... 217 Theoremsof WallisandBr ouncker... ... ... ... ... 220 Analyticalmethods ofapproximation.Gregory'sseries ... .220 ResultsofEuro pe anwriters,1699-1873... ... .. ... 220 Geometricalapproximations... ... .. ... ... ... ..22 2 Approximationsbythetheoryof probability... ... .. ..22 2 xiiTABLEOFCO NTENTS.

ChapterIX.Mersenn e'sNum bers.

PAGE Mersenne'sEnunc iationoftheTheorem... ... .. ... .224 Listofkno wnresults... ... .. ... ... ... ... ... 225 Casesaw aitingverification... ... .. ... ... ... ... 225 HistoryofInv estigations... ... .. ... ... ... ... .226 Methodsusedinatta cking theproblem ... ... ... ... .230 Bytrial ofdivisorsofknownforms ... ... .. ... ..23 1 Byindeterminate equations... ... .. ... ... ... 233 Byprop ertiesofquadraticforms... ... .. ... ... 234

Bythe useofaCanonAr ithmeticus... ... .. ... .234

Byprop ertiesofbinarypow ers... ... ... ... ... 235 Bythe useofthebinaryscale ... ... .. ... ... ..23 5 Bythe useofFermat'sThe orem... ... .. ... ... .236 Mechanicalmethodsof FactorizingNumbers ... ... ... .236

ChapterX.Astrolo gy.

Astrology.Twobranche s:natalandhorary astrology... ..23 8 Rulesfo rcastingandreadingah oroscope... ... .. ... 238 Housesand theirsignific ations... ... .. ... ... ..23 8 Planetsandtheir significatio ns... ... ... ... ... 240 Zodiacalsignsandtheirs ignifications ... ... .. ... .242 Knowledgethatr uleswerewo rthless... ... ... ... ..24 3 Notableinstancesofhoro scopy... ... .. ... ... ... 246 Lilly'spredictionof theGreat Fireand Plague... ... .24 6 Flamsteed'sgues s... ... .. ... ... ... ... ... 246 Cardan'shoroscopeof EdwardVI... ... ... ... ..24 7

ChapterXI.Cryptograph sand Ciphers.

ACr yptograph.Definition.Illustration... ... .. ... ..25 1 ACipher. Definition.Illustratio n... ... .. ... ... ..25 2 EssentialFeaturesofCryptographsa ndCiphers... ... ..25 2 CryptographsofThreeTy pes.Illustrations ... ... .. ..25 3 Orderoflettersre-arra nged... ... .. ... ... ... 253 Useof non-significantsym bols.TheGrille... ... ... 256 Useof brokensymb ols.TheScytale... ... ... ... .258 Ciphers.Useof arbitrary symbols unnecessary... ... .. .259

TABLEOFCONTENTS.xiii

PAGE Ciphersof FourTypes ... ... .. ... ... ... ... ..25 9 Ciphersof theFirstType .Illustrations ... ... ... ..26 0 Ciphersof theSecondT ype. Illustrations... ... .. ..26 3 Ciphersof theThirdType. Illustrations... ... .. ..26 5 Ciphersof theFourthTyp e.Illustratio ns... ... .. ..26 7 Requisitesina goo dCipher ... ... .. ... ... ... ..26 8 CipherMac hines... ... .. ... ... ... ... ... ..26 9 HistoricalCiphers... ... .. ... ... ... ... ... ..26 9 JuliusCae sar,Augustus... ... ... ... ... ... .26 9

Bacon... ... .. ... ... ... ... ... ... ... 269

CharlesI... ... ... ... ... ... .. ... ... ..26 9

Pepys... ... ... ... ... ... .. ... ... ... 271

DeRohan ... ... .. ... ... ... ... ... ... .272

MarieAnto inette... ... ... ... ... ... .. ... 272

TheCode Dictionary... ... ... ... ... ... .. .274

Poe'sWritings... ... ... ... ... ... .. ... ..27 5

ChapterXII.Hyper-space.

Twosubjectsof speculationon Hyper-space... ... .. ..27 8quotesdbs_dbs22.pdfusesText_28

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