A History of Mathematics

996 Pages·2008·9.76 MB·English

by  Victor J. Katz

A History of
MATHEMATICS
An Introduction
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A History of
MATHEMATICS
An Introduction
Third Edition
Victor J. Katz
University of the District of Columbia
Addison-Wesley
Boston SanFrancisco NewYork
London Toronto Sydney Tokyo Singapore Madrid
MexicoCity Munich Paris CapeTown HongKong Montreal
To Phyllis, for her patience, encouragement, and love
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Katz,VictorJ.
Ahistoryofmathematics/VictorKatz.—3rded.
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1.Mathematics—History. I.Title.
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Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . xi
PART ONE Ancient Mathematics
Chapter 1 EgyptandMesopotamia 1
1.1 Egypt . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Mesopotamia . . . . . . . . . . . . . . . . . . . . . . 10
1.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . 27
Exercises . . . . . . . . . . . . . . . . . . . . . . . 28
ReferencesandNotes . . . . . . . . . . . . . . . . . . 30
Chapter 2 TheBeginningsofMathematicsinGreece 32
2.1 TheEarliestGreekMathematics . . . . . . . . . . . . . . 33
2.2 TheTimeofPlato . . . . . . . . . . . . . . . . . . . . 41
2.3 Aristotle . . . . . . . . . . . . . . . . . . . . . . . . 43
Exercises . . . . . . . . . . . . . . . . . . . . . . . 47
ReferencesandNotes . . . . . . . . . . . . . . . . . . 48
Chapter 3 Euclid 50
3.1 IntroductiontotheElements . . . . . . . . . . . . . . . . 51
3.2 BookIandthePythagoreanTheorem . . . . . . . . . . . . 53
3.3 BookIIandGeometricAlgebra . . . . . . . . . . . . . . 60
3.4 CirclesandthePentagonConstruction . . . . . . . . . . . . 66
3.5 RatioandProportion . . . . . . . . . . . . . . . . . . . 71
3.6 NumberTheory . . . . . . . . . . . . . . . . . . . . . 77
3.7 IrrationalMagnitudes . . . . . . . . . . . . . . . . . . 81
3.8 SolidGeometryandtheMethodofExhaustion . . . . . . . . 83
3.9 Euclid’sData . . . . . . . . . . . . . . . . . . . . . . 88
Exercises . . . . . . . . . . . . . . . . . . . . . . . 90
ReferencesandNotes . . . . . . . . . . . . . . . . . . 92
vi Contents
Chapter 4 ArchimedesandApollonius 94
4.1 ArchimedesandPhysics . . . . . . . . . . . . . . . . . 96
4.2 ArchimedesandNumericalCalculations . . . . . . . . . . . 101
4.3 ArchimedesandGeometry . . . . . . . . . . . . . . . . 103
4.4 ConicSectionsbeforeApollonius . . . . . . . . . . . . . . 112
4.5 TheConicsofApollonius . . . . . . . . . . . . . . . . . 115
Exercises . . . . . . . . . . . . . . . . . . . . . . . 127
ReferencesandNotes . . . . . . . . . . . . . . . . . . 131
Chapter 5 MathematicalMethodsinHellenisticTimes 133
5.1 AstronomybeforePtolemy . . . . . . . . . . . . . . . . 134
5.2 PtolemyandtheAlmagest . . . . . . . . . . . . . . . . . 145
5.3 PracticalMathematics . . . . . . . . . . . . . . . . . . 157
Exercises . . . . . . . . . . . . . . . . . . . . . . . 168
ReferencesandNotes . . . . . . . . . . . . . . . . . . 170
Chapter 6 TheFinalChaptersofGreekMathematics 172
6.1 NicomachusandElementaryNumberTheory . . . . . . . . . 173
6.2 DiophantusandGreekAlgebra . . . . . . . . . . . . . . . 176
6.3 PappusandAnalysis . . . . . . . . . . . . . . . . . . . 185
6.4 HypatiaandtheEndofGreekMathematics . . . . . . . . . . 189
Exercises . . . . . . . . . . . . . . . . . . . . . . . 191
ReferencesandNotes . . . . . . . . . . . . . . . . . . 192
PART TWO Medieval Mathematics
Chapter 7 AncientandMedievalChina 195
7.1 IntroductiontoMathematicsinChina . . . . . . . . . . . . 196
7.2 Calculations . . . . . . . . . . . . . . . . . . . . . . 197
7.3 Geometry . . . . . . . . . . . . . . . . . . . . . . . 201
7.4 SolvingEquations . . . . . . . . . . . . . . . . . . . . 209
7.5 IndeterminateAnalysis . . . . . . . . . . . . . . . . . . 222
7.6 TransmissionToandFromChina . . . . . . . . . . . . . . 225
Exercises . . . . . . . . . . . . . . . . . . . . . . . 226
ReferencesandNotes . . . . . . . . . . . . . . . . . . 228
Chapter 8 AncientandMedievalIndia 230
8.1 IntroductiontoMathematicsinIndia . . . . . . . . . . . . 231
8.2 Calculations . . . . . . . . . . . . . . . . . . . . . . 233
8.3 Geometry . . . . . . . . . . . . . . . . . . . . . . . 237
Contents vii
8.4 EquationSolving . . . . . . . . . . . . . . . . . . . . 242
8.5 IndeterminateAnalysis . . . . . . . . . . . . . . . . . . 244
8.6 Combinatorics . . . . . . . . . . . . . . . . . . . . . 250
8.7 Trigonometry . . . . . . . . . . . . . . . . . . . . . . 252
8.8 TransmissionToandFromIndia . . . . . . . . . . . . . . 259
Exercises . . . . . . . . . . . . . . . . . . . . . . . 260
ReferencesandNotes . . . . . . . . . . . . . . . . . . 263
Chapter 9 TheMathematicsofIslam 265
9.1 IntroductiontoMathematicsinIslam . . . . . . . . . . . . 266
9.2 DecimalArithmetic . . . . . . . . . . . . . . . . . . . 267
9.3 Algebra . . . . . . . . . . . . . . . . . . . . . . . . 271
9.4 Combinatorics . . . . . . . . . . . . . . . . . . . . . 292
9.5 Geometry . . . . . . . . . . . . . . . . . . . . . . . 296
9.6 Trigonometry . . . . . . . . . . . . . . . . . . . . . . 306
9.7 TransmissionofIslamicMathematics . . . . . . . . . . . . 317
Exercises . . . . . . . . . . . . . . . . . . . . . . . 318
ReferencesandNotes . . . . . . . . . . . . . . . . . . 321
Chapter 10 MathematicsinMedievalEurope 324
10.1 IntroductiontotheMathematicsofMedievalEurope . . . . . . 325
10.2 GeometryandTrigonometry . . . . . . . . . . . . . . . . 328
10.3 Combinatorics . . . . . . . . . . . . . . . . . . . . . 337
10.4 MedievalAlgebra . . . . . . . . . . . . . . . . . . . . 342
10.5 TheMathematicsofKinematics . . . . . . . . . . . . . . 351
Exercises . . . . . . . . . . . . . . . . . . . . . . . 359
ReferencesandNotes . . . . . . . . . . . . . . . . . . 362
Chapter 11 MathematicsaroundtheWorld 364
11.1 MathematicsattheTurnoftheFourteenthCentury . . . . . . . 365
11.2 MathematicsinAmerica,Africa,andthePacific . . . . . . . . 370
Exercises . . . . . . . . . . . . . . . . . . . . . . . 379
ReferencesandNotes . . . . . . . . . . . . . . . . . . 380
PART THREE Early Modern Mathematics
Chapter 12 AlgebraintheRenaissance 383
12.1 TheItalianAbacists . . . . . . . . . . . . . . . . . . . 385
12.2 AlgebrainFrance,Germany,England,andPortugal . . . . . . 389
12.3 TheSolutionoftheCubicEquation . . . . . . . . . . . . . 399
viii Contents
12.4 Vie`te,AlgebraicSymbolism,andAnalysis . . . . . . . . . . 407
12.5 SimonStevinandDecimalFractions . . . . . . . . . . . . 414
Exercises . . . . . . . . . . . . . . . . . . . . . . . 418
References . . . . . . . . . . . . . . . . . . . . . . 420
Chapter 13 MathematicalMethodsintheRenaissance 423
13.1 Perspective . . . . . . . . . . . . . . . . . . . . . . . 427
13.2 NavigationandGeography . . . . . . . . . . . . . . . . 432
13.3 AstronomyandTrigonometry . . . . . . . . . . . . . . . 435
13.4 Logarithms . . . . . . . . . . . . . . . . . . . . . . 453
13.5 Kinematics . . . . . . . . . . . . . . . . . . . . . . . 457
Exercises . . . . . . . . . . . . . . . . . . . . . . . 462
ReferencesandNotes . . . . . . . . . . . . . . . . . . 464
Chapter 14 Algebra,Geometry,andProbabilityintheSeventeenthCentury 467
14.1 TheTheoryofEquations . . . . . . . . . . . . . . . . . 468
14.2 AnalyticGeometry . . . . . . . . . . . . . . . . . . . 473
14.3 ElementaryProbability . . . . . . . . . . . . . . . . . . 487
14.4 NumberTheory . . . . . . . . . . . . . . . . . . . . . 497
14.5 ProjectiveGeometry . . . . . . . . . . . . . . . . . . . 499
Exercises . . . . . . . . . . . . . . . . . . . . . . . 501
ReferencesandNotes . . . . . . . . . . . . . . . . . . 504
Chapter 15 TheBeginningsofCalculus 507
15.1 TangentsandExtrema . . . . . . . . . . . . . . . . . . 509
15.2 AreasandVolumes . . . . . . . . . . . . . . . . . . . 514
15.3 RectificationofCurvesandtheFundamentalTheorem . . . . . 532
Exercises . . . . . . . . . . . . . . . . . . . . . . . 539
ReferencesandNotes . . . . . . . . . . . . . . . . . . 541
Chapter 16 NewtonandLeibniz 543
16.1 IsaacNewton . . . . . . . . . . . . . . . . . . . . . . 544
16.2 GottfriedWilhelmLeibniz . . . . . . . . . . . . . . . . 565
16.3 FirstCalculusTexts . . . . . . . . . . . . . . . . . . . 575
Exercises . . . . . . . . . . . . . . . . . . . . . . . 579
ReferencesandNotes . . . . . . . . . . . . . . . . . . 580
PART FOUR Modern Mathematics
Chapter 17 AnalysisintheEighteenthCentury 583
17.1 DifferentialEquations . . . . . . . . . . . . . . . . . . 584
17.2 TheCalculusofSeveralVariables . . . . . . . . . . . . . . 601
Contents ix
17.3 CalculusTexts . . . . . . . . . . . . . . . . . . . . . 611
17.4 TheFoundationsofCalculus . . . . . . . . . . . . . . . . 628
Exercises . . . . . . . . . . . . . . . . . . . . . . . 636
ReferencesandNotes . . . . . . . . . . . . . . . . . . 639
Chapter 18 ProbabilityandStatisticsintheEighteenthCentury 642
18.1 TheoreticalProbability . . . . . . . . . . . . . . . . . . 643
18.2 StatisticalInference . . . . . . . . . . . . . . . . . . . 651
18.3 ApplicationsofProbability . . . . . . . . . . . . . . . . 655
Exercises . . . . . . . . . . . . . . . . . . . . . . . 661
ReferencesandNotes . . . . . . . . . . . . . . . . . . 663
Chapter 19 AlgebraandNumberTheoryintheEighteenthCentury 665
19.1 AlgebraTexts . . . . . . . . . . . . . . . . . . . . . 666
19.2 AdvancesintheTheoryofEquations . . . . . . . . . . . . 671
19.3 NumberTheory . . . . . . . . . . . . . . . . . . . . . 677
19.4 MathematicsintheAmericas . . . . . . . . . . . . . . . 680
Exercises . . . . . . . . . . . . . . . . . . . . . . . 683
ReferencesandNotes . . . . . . . . . . . . . . . . . . 684
Chapter 20 GeometryintheEighteenthCentury 686
20.1 ClairautandtheElementsofGeometry . . . . . . . . . . . . 687
20.2 TheParallelPostulate . . . . . . . . . . . . . . . . . . 689
20.3 AnalyticandDifferentialGeometry . . . . . . . . . . . . . 695
20.4 TheBeginningsofTopology . . . . . . . . . . . . . . . . 701
20.5 TheFrenchRevolutionandMathematicsEducation . . . . . . 702
Exercises . . . . . . . . . . . . . . . . . . . . . . . 706
ReferencesandNotes . . . . . . . . . . . . . . . . . . 707
Chapter 21 AlgebraandNumberTheoryintheNineteenthCentury 709
21.1 NumberTheory . . . . . . . . . . . . . . . . . . . . . 711
21.2 SolvingAlgebraicEquations . . . . . . . . . . . . . . . . 721
21.3 SymbolicAlgebra . . . . . . . . . . . . . . . . . . . . 730
21.4 MatricesandSystemsofLinearEquations . . . . . . . . . . 740
21.5 GroupsandFields—TheBeginningofStructure . . . . . . . . 750
Exercises . . . . . . . . . . . . . . . . . . . . . . . 759
ReferencesandNotes . . . . . . . . . . . . . . . . . . 761
Chapter 22 AnalysisintheNineteenthCentury 764
22.1 RigorinAnalysis . . . . . . . . . . . . . . . . . . . . 766
22.2 TheArithmetizationofAnalysis . . . . . . . . . . . . . . 788
22.3 ComplexAnalysis . . . . . . . . . . . . . . . . . . . . 795