The history of mathematics PDF
Preview The history of mathematics
Revised Con rming Pages The History of Mathematics AN INTRODUCTION Seventh Edition David M. Burton University of New Hampshire bur83155 fmi-xii.tex i 01/13/2010 16:12 Revised Con rming Pages THEHISTORYOFMATHEMATICS:ANINTRODUCTION,SEVENTHEDITION PublishedbyMcGraw-Hill,abusinessunitofTheMcGraw-HillCompanies,Inc.,1221Avenueofthe Americas,NewYork,NY10020.Copyright(cid:2)c 2011byTheMcGraw-HillCompanies,Inc.Allrights reserved.Previouseditions(cid:2)c 2007,2003,and1999.Nopartofthispublicationmaybereproducedor distributedinanyformorbyanymeans,orstoredinadatabaseorretrievalsystem,withoutthepriorwritten consentofTheMcGraw-HillCompanies,Inc.,including,butnotlimitedto,inanynetworkorother electronicstorageortransmission,orbroadcastfordistancelearning. Someancillaries,includingelectronicandprintcomponents,maynotbeavailabletocustomersoutsidethe UnitedStates. Thisbookisprintedonacid-freepaper. 1234567890DOC/DOC109876543210 ISBN 978–0–07–338315–6 MHID 0–07–338315–5 EditorialDirector:StewartK.Mattson SponsoringEditor:JohnR.Osgood DirectorofDevelopment:KristineTibbetts DevelopmentalEditor:EveL.Lipton MarketingCoordinator:SabinaNavsariwala-Horrocks ProjectManager:MelissaM.Leick SeniorProductionSupervisor:KaraKudronowicz DesignCoordinator:BrendaA.Rolwes CoverDesigner:StudioMontage,St.Louis,Missouri (USE)CoverImage:Royalty-Free/CORBIS SeniorPhotoResearchCoordinator:JohnC.Leland Compositor:LaserwordsPrivateLimited Typeface:10/12TimesRoman Printer:R.R.Donnelley Allcreditsappearingonpageorattheendofthebookareconsideredtobeanextensionofthecopyright page. LibraryofCongressCataloging-in-PublicationData Burton,DavidM. Thehistoryofmathematics:anintroduction/DavidM.Burton.—7thed. p.cm. Includesbibliographicalreferencesandindex. ISBN978-0-07-338315-6 (alk.paper) 1.Mathematics–History.I.Title. QA21.B962011 510.9–dc22 2009049164 www.mhhe.com bur83155 fmi-xii.tex ii 01/13/2010 16:15 Revised Con rming Pages A llthesewerehonoredintheirgenerations,and werethegloryoftheirtimes. T herebeofthem,thathaveleftanamebehind them,thattheirpraisesmightbereported. A ndsometherebe,whichhavenomemorial;who areperished,asthoughtheyhadneverbeen;andare becomeasthoughtheyhadneverbeenborn;and theirchildrenafterthem. ECCLESIASTICUS 44:7–9 bur83155 fmi-xii.tex iii 01/13/2010 16:14 This page intentionally left blank Revised Con rming Pages C o n t e n t s EarlyEgyptianMultiplication 37 TheUnitFractionTable 40 RepresentingRationalNumbers 43 2.3 FourProblemsfromtheRhindPapyrus 46 TheMethodofFalsePosition 46 ACuriousProblem 49 Preface x–xii EgyptianMathematicsasAppliedArithmetic 50 2.4 EgyptianGeometry 53 ApproximatingtheAreaofaCircle 53 Chapter1 TheVolumeofaTruncatedPyramid 56 Early Number Systems and SpeculationsAbouttheGreatPyramid 57 Symbols 1 2.5 BabylonianMathematics 62 ATabletofReciprocals 62 1.1 PrimitiveCounting 1 TheBabylonianTreatmentofQuadraticEquations 64 ASenseofNumber 1 TwoCharacteristicBabylonianProblems 69 NotchesasTallyMarks 2 2.6 Plimpton322 72 ThePeruvianQuipus:KnotsasNumbers 6 ATabletConcerningNumberTriples 72 1.2 NumberRecordingoftheEgyptiansandGreeks 9 BabylonianUseofthePythagoreanTheorem 76 TheHistoryofHerodotus 9 TheCairoMathematicalPapyrus 77 HieroglyphicRepresentationofNumbers 11 Chapter3 EgyptianHieraticNumeration 15 TheGreekAlphabeticNumeralSystem 16 The Beginnings of Greek 1.3 NumberRecordingoftheBabylonians 20 Mathematics 83 BabylonianCuneiformScript 20 DecipheringCuneiform:GrotefendandRawlinson 21 3.1 TheGeometricalDiscoveriesofThales 83 TheBabylonianPositionalNumberSystem 23 GreeceandtheAegeanArea 83 WritinginAncientChina 26 TheDawnofDemonstrativeGeometry: ThalesofMiletos 86 Chapter2 MeasurementsUsingGeometry 87 Mathematics in Early 3.2 PythagoreanMathematics 90 Civilizations 33 PythagorasandHisFollowers 90 Nicomachus’sIntroductioArithmeticae 94 2.1 TheRhindPapyrus 33 TheTheoryofFigurativeNumbers 97 EgyptianMathematicalPapyri 33 Zeno’sParadox 101 AKeytoDeciphering:TheRosettaStone 35 3.3 ThePythagoreanProblem 105 2.2 EgyptianArithmetic 37 GeometricProofsofthePythagoreanTheorem 105 v bur83155 fmi-xii.tex v 01/13/2010 16:43 Revised Con rming Pages vi Contents EarlySolutionsofthePythagoreanEquation 107 TheAlmagestofClaudiusPtolemy 188 TheCrisisofIncommensurableQuantities 109 Ptolemy’sGeographicalDictionary 190 Theon’sSideandDiagonalNumbers 111 4.5 Archimedes 193 EudoxusofCnidos 116 TheAncientWorld’sGenius 193 3.4 ThreeConstructionProblemsofAntiquity 120 EstimatingtheValueof³ 197 HippocratesandtheQuadratureoftheCircle 120 TheSand-Reckoner 202 TheDuplicationoftheCube 124 QuadratureofaParabolicSegment 205 TheTrisectionofanAngle 126 ApolloniusofPerga:TheConics 206 3.5 TheQuadratrixofHippias 130 Chapter5 RiseoftheSophists 130 HippiasofElis 131 The Twilight of Greek TheGroveofAcademia:Plato’sAcademy 134 Mathematics: Diophantus 213 Chapter4 5.1 TheDeclineofAlexandrianMathematics 213 The Alexandrian School: TheWaningoftheGoldenAge 213 Euclid 141 TheSpreadofChristianity 215 Constantinople,ARefugeforGreekLearning 217 4.1 EuclidandtheElements 141 5.2 TheArithmetica 217 ACenterofLearning:TheMuseum 141 Diophantus’sNumberTheory 217 Euclid’sLifeandWritings 143 ProblemsfromtheArithmetica 220 4.2 EuclideanGeometry 144 5.3 DiophantineEquationsinGreece,India, Euclid’sFoundationforGeometry 144 andChina 223 Postulates 146 TheCattleProblemofArchimedes 223 CommonNotions 146 EarlyMathematicsinIndia 225 BookIoftheElements 148 TheChineseHundredFowlsProblem 228 Euclid’sProofofthePythagoreanTheorem 156 5.4 TheLaterCommentators 232 BookIIonGeometricAlgebra 159 TheMathematicalCollectionofPappus 232 ConstructionoftheRegularPentagon 165 Hypatia,theFirstWomanMathematician 233 4.3 Euclid’sNumberTheory 170 RomanMathematics:BoethiusandCassiodorus 235 EuclideanDivisibilityProperties 170 5.5 MathematicsintheNearandFarEast 238 TheAlgorithmofEuclid 173 TheAlgebraofal-Khowaˆrizmˆı 238 TheFundamentalTheoremofArithmetic 177 AbuˆKaˆmilandThaˆbitibnQurra 242 AnInfinityofPrimes 180 OmarKhayyam 247 4.4 Eratosthenes,theWiseManofAlexandria 183 TheAstronomersal-Tuˆsˆıandal-Kashˆı 249 TheSieveofEratosthenes 183 TheAncientChineseNineChapters 251 MeasurementoftheEarth 186 LaterChineseMathematicalWorks 259 bur83155 fmi-xii.tex vi 01/13/2010 16:43 Revised Con rming Pages Contents vii Chapter6 Cardan’sSolutionoftheCubicEquation 320 BombelliandImaginaryRootsoftheCubic 324 The First Awakening: 7.4 Ferrari’sSolutionoftheQuarticEquation 328 Fibonacci 269 TheResolvantCubic 328 TheStoryoftheQuinticEquation: 6.1 TheDeclineandRevivalofLearning 269 Ruffini,Abel,andGalois 331 TheCarolingianPre-Renaissance 269 TransmissionofArabicLearningtotheWest 272 Chapter8 ThePioneerTranslators:GerardandAdelard 274 The Mechanical World: 6.2 TheLiberAbaciandLiberQuadratorum 277 Descartes and Newton 337 TheHindu-ArabicNumerals 277 Fibonacci’sLiberQuadratorum 280 8.1 TheDawnofModernMathematics 337 TheWorksofJordanusdeNemore 283 TheSeventeenthCenturySpreadofKnowledge 337 6.3 TheFibonacciSequence 287 Galileo’sTelescopicObservations 339 TheLiberAbaci’sRabbitProblem 287 TheBeginningofModernNotation: SomePropertiesofFibonacciNumbers 289 Franc¸oisVie`ta 345 6.4 FibonacciandthePythagoreanProblem 293 TheDecimalFractionsofSimonStevin 348 PythagoreanNumberTriples 293 Napier’sInventionofLogarithms 350 Fibonacci’sTournamentProblem 297 TheAstronomicalDiscoveriesofBraheand Kepler 355 Chapter7 8.2 Descartes:TheDiscoursdelaMe´thode 362 The Renaissance of Mathematics: TheWritingsofDescartes 362 Cardan and Tartaglia 301 InventingCartesianGeometry 367 TheAlgebraicAspectofLaGe´ome´trie 372 7.1 EuropeintheFourteenthandFifteenth Descartes’sPrincipiaPhilosophiae 375 Centuries 301 PerspectiveGeometry:DesarguesandPoncelet 377 TheItalianRenaissance 301 8.3 Newton:ThePrincipiaMathematica 381 ArtificialWriting:TheInventionofPrinting 303 TheTextbooksofOughtredandHarriot 381 FoundingoftheGreatUniversities 306 Wallis’sArithmeticaInfinitorum 383 AThirstforClassicalLearning 310 TheLucasianProfessorship:BarrowandNewton 386 7.2 TheBattleoftheScholars 312 Newton’sGoldenYears 392 RestoringtheAlgebraicTradition:RobertRecorde 312 TheLawsofMotion 398 TheItalianAlgebraists:Pacioli,delFerro,and LaterYears:AppointmenttotheMint 404 Tartaglia 315 8.4 GottfriedLeibniz:TheCalculusControversy 409 Cardan,AScoundrelMathematician 319 TheEarlyWorkofLeibniz 409 7.3 Cardan’sArsMagna 320 Leibniz’sCreationoftheCalculus 413 bur83155 fmi-xii.tex vii 01/13/2010 16:43 Revised Con rming Pages viii Contents Newton’sFluxionalCalculus 416 ScientificSocieties 497 TheDisputeoverPriority 424 MarinMersenne’sMathematicalGathering 499 MariaAgnesiandEmilieduChaˆtelet 430 Numbers,PerfectandNotSoPerfect 502 10.2 FromFermattoEuler 511 Chapter9 Fermat’sArithmetica 511 The Development of Probability TheFamousLastTheoremofFermat 516 Theory: Pascal, Bernoulli, TheEighteenth-CenturyEnlightenment 520 and Laplace 439 Maclaurin’sTreatiseonFluxions 524 Euler’sLifeandContributions 527 9.1 TheOriginsofProbabilityTheory 439 10.3 ThePrinceofMathematicians:Carl Graunt’sBillsofMortality 439 FriedrichGauss 539 GamesofChance:DiceandCards 443 ThePeriodoftheFrenchRevolution: ThePrecocityoftheYoungPascal 446 Lagrange,Monge,andCarnot 539 PascalandtheCycloid 452 Gauss’sDisquisitionesArithmeticae 546 DeMe´re´’sProblemofPoints 454 TheLegacyofGauss:CongruenceTheory 551 9.2 Pascal’sArithmeticTriangle 456 DirichletandJacobi 558 TheTraite´duTriangleArithme´tique 456 MathematicalInduction 461 Chapter11 FrancescoMaurolico’sUseofInduction 463 Nineteenth-Century 9.3 TheBernoullisandLaplace 468 Contributions: Lobachevsky to ChristiaanHuygens’sPamphletonProbability 468 Hilbert 563 TheBernoulliBrothers:JohnandJames 471 DeMoivre’sDoctrineofChances 477 11.1 AttemptstoProvetheParallelPostulate 563 TheMathematicsofCelestialPhenomena: TheEffortsofProclus,Playfair,andWallis 563 Laplace 478 SaccheriQuadrilaterals 566 MaryFairfaxSomerville 482 TheAccomplishmentsofLegendre 571 Laplace’sResearchinProbabilityTheory 483 Legendre’sEle´mentsdege´ome´trie 574 DanielBernoulli,Poisson,andChebyshev 489 11.2 TheFoundersofNon-EuclideanGeometry 584 Gauss’sAttemptataNewGeometry 584 Chapter10 TheStruggleofJohnBolyai 588 The Revival of Number Theory: CreationofNon-EuclideanGeometry:Lobachevsky 592 Fermat, Euler, and Gauss 497 ModelsoftheNewGeometry:Riemann, Beltrami,andKlein 598 10.1 MarinMersenneandtheSearch GraceChisholmYoung 603 forPerfectNumbers 497 11.3 TheAgeofRigor 604 bur83155 fmi-xii.tex viii 01/13/2010 16:43 Revised Con rming Pages Contents ix D’AlembertandCauchyonLimits 604 ZermeloandtheAxiomofChoice 701 Fourier’sSeries 610 TheLogisticSchool:Frege,Peano,andRussell 704 TheFatherofModernAnalysis,Weierstrass 614 Hilbert’sFormalisticApproach 708 SonyaKovalevsky 616 Brouwer’sInstitutionism 711 TheAxiomaticMovement:PaschandHilbert 619 11.4 ArithmeticGeneralized 626 Chapter13 BabbageandtheAnalyticalEngine 626 Extensions and Generalizations: Peacock’sTreatiseonAlgebra 629 Hardy, Hausdorff, TheRepresentationofComplexNumbers 630 and Noether 721 Hamilton’sDiscoveryofQuaternions 633 MatrixAlgebra:CayleyandSylvester 639 13.1 HardyandRamanujan 721 Boole’sAlgebraofLogic 646 TheTriposExamination 721 TheRejuvenationofEnglishMathematics 722 Chapter12 AUniqueCollaboration:HardyandLittlewood 725 Transition to the Twentieth India’sProdigy,Ramanujan 726 Century: Cantor and 13.2 TheBeginningsofPoint-SetTopology 729 Frechet’sMetricSpaces 729 Kronecker 657 TheNeighborhoodSpacesofHausdorff 731 12.1 TheEmergenceofAmericanMathematics 657 BanachandNormedLinearSpaces 733 AscendencyoftheGermanUniversities 657 13.3 SomeTwentieth-CenturyDevelopments 735 AmericanMathematicsTakesRoot:1800–1900 659 EmmyNoether’sTheoryofRings 735 TheTwentieth-CenturyConsolidation 669 VonNeumannandtheComputer 741 12.2 CountingtheInfinite 673 WomeninModernMathematics 744 TheLastUniversalist:Poincare´ 673 AFewRecentAdvances 747 Cantor’sTheoryofInfiniteSets 676 Kronecker’sViewofSetTheory 681 CountableandUncountableSets 684 GeneralBibliography 755 TranscendentalNumbers 689 AdditionalReading 759 TheContinuumHypothesis 694 TheGreekAlphabet 761 12.3 TheParadoxesofSetTheory 698 SolutionstoSelectedProblems 762 TheEarlyParadoxes 698 Index 777 bur83155 fmi-xii.tex 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