The History of Mathematics

643 Pages·2014·11.48 MB·English

by  Roger L. Cooke

THE HISTORY OF
MATHEMATICS

THE HISTORY OF
MATHEMATICS
A BRIEF COURSE
THIRD EDITION
Roger L. Cooke
DepartmentofMathematicsandStatistics
UniversityofVermont
Burlington,VT
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LibraryofCongressCataloging-in-PublicationData:
Cooke,Roger,1942-
Thehistoryofmathematics:abriefcourse/RogerL.Cooke.–3rded.
p.cm.
Includesbibliographicalreferencesandindex.
ISBN978-1-118-21756-6(cloth)
1. Mathematics–History. I. Title.
QA21.C6492013
510’.9–dc23
2012020963
PrintedintheUnitedStatesofAmerica
10 9 8 7 6 5 4 3 2 1
CONTENTS
PREFACE xxiii
ChangesfromtheSecondEdition xxiii
ElementaryTextsontheHistoryofMathematics xxiv
PARTI. WHATISMATHEMATICS?
ContentsofPartI 1
1. MathematicsanditsHistory 3
1.1. TwoWaystoLookattheHistoryofMathematics 3
1.1.1. History,butnotHeritage 4
1.1.2. OurMathematicalHeritage 4
1.2. TheOriginofMathematics 5
1.2.1. Number 5
1.2.2. Space 5
1.2.3. AreMathematicalIdeasInnate? 7
1.2.4. SymbolicNotation 7
1.2.5. LogicalRelations 7
1.2.6. TheComponentsofMathematics 8
1.3. ThePhilosophyofMathematics 8
1.3.1. MathematicalAnalysisofaReal-WorldProblem 9
1.4. OurApproachtotheHistoryofMathematics 11
QuestionsforReflection 12
2. Proto-mathematics 14
2.1. Number 14
2.1.1. Animals’UseofNumbers 14
2.1.2. YoungChildren’sUseofNumbers 15
2.1.3. ArchaeologicalEvidenceofCounting 15
2.2. Shape 16
2.2.1. PerceptionofShapebyAnimals 16
2.2.2. Children’sConceptsofSpace 16
2.2.3. GeometryinArtsandCrafts 17
2.3. Symbols 18
v
vi CONTENTS
2.4. MathematicalReasoning 20
2.4.1. AnimalReasoning 20
2.4.2. VisualReasoning 21
ProblemsandQuestions 22
MathematicalProblems 22
QuestionsforReflection 24
PARTII. THEMIDDLEEAST,2000–1500BCE
ContentsofPartII 25
3. OverviewofMesopotamianMathematics 27
3.1. ASketchofTwoMillenniaofMesopotamianHistory 27
3.2. MathematicalCuneiformTablets 29
3.3. SystemsofMeasuringandCounting 30
3.3.1. Counting 31
3.4. TheMesopotamianNumberingSystem 31
3.4.1. Place-ValueSystems 32
3.4.2. TheSexagesimalPlace-ValueSystem 33
3.4.3. ConvertingaDecimalNumbertoSexagesimal 33
3.4.4. IrrationalSquareRoots 36
ProblemsandQuestions 36
MathematicalProblems 36
HistoricalQuestions 36
QuestionsforReflection 37
4. ComputationsinAncientMesopotamia 38
4.1. Arithmetic 38
4.1.1. SquareRoots 39
4.2. Algebra 40
4.2.1. LinearandQuadraticProblems 41
4.2.2. Higher-DegreeProblems 43
ProblemsandQuestions 44
MathematicalProblems 44
HistoricalQuestions 44
QuestionsforReflection 44
5. GeometryinMesopotamia 46
5.1. ThePythagoreanTheorem 46
5.2. PlaneFigures 48
5.2.1. MesopotamianAstronomy 48
5.3. Volumes 49
5.4. Plimpton322 49
5.4.1. ThePurposeofPlimpton322:SomeConjectures 53
CONTENTS vii
ProblemsandQuestions 54
MathematicalProblems 54
HistoricalQuestions 55
QuestionsforReflection 55
6. EgyptianNumeralsandArithmetic 56
6.1. Sources 56
6.1.1. MathematicsinHieroglyphicsandHieratic 57
6.2. TheRhindPapyrus 58
6.3. EgyptianArithmetic 58
6.4. Computation 59
6.4.1. MultiplicationandDivision 61
6.4.2. “Parts” 62
ProblemsandQuestions 65
MathematicalProblems 65
HistoricalQuestions 65
QuestionsforReflection 65
7. AlgebraandGeometryinAncientEgypt 66
7.1. AlgebraProblemsintheRhindPapyrus 66
7.1.1. AppliedProblems:ThePesu 67
7.2. Geometry 68
7.3. Areas 69
7.3.1. Rectangles,Triangles,andTrapezoids 69
7.3.2. Slopes 69
7.3.3. Circles 70
7.3.4. ThePythagoreanTheorem 71
7.3.5. SpheresorCylinders? 72
7.3.6. Volumes 73
ProblemsandQuestions 76
MathematicalProblems 76
HistoricalQuestions 76
QuestionsforReflection 76
PARTIII. GREEKMATHEMATICSFROM500BCETO500CE
ContentsofPartIII 77
8. AnOverviewofAncientGreekMathematics 79
8.1. Sources 80
8.1.1. LossandRecovery 81
8.2. GeneralFeaturesofGreekMathematics 82
8.2.1. Pythagoras 83
8.2.2. MathematicalAspectsofPlato’sPhilosophy 85
viii CONTENTS
8.3. WorksandAuthors 87
8.3.1. Euclid 87
8.3.2. Archimedes 87
8.3.3. Apollonius 88
8.3.4. Zenodorus 88
8.3.5. Heron 88
8.3.6. Ptolemy 89
8.3.7. Diophantus 89
8.3.8. Pappus 89
8.3.9. TheonandHypatia 89
Questions 90
HistoricalQuestions 90
QuestionsforReflection 90
9. GreekNumberTheory 91
9.1. TheEuclideanAlgorithm 92
9.2. TheArithmeticaofNicomachus 93
9.2.1. Factorsvs.Parts.PerfectNumbers 94
9.2.2. FigurateNumbers 95
9.3. Euclid’sNumberTheory 97
9.4. TheArithmeticaofDiophantus 97
9.4.1. AlgebraicSymbolism 98
9.4.2. ContentsoftheArithmetica 99
9.4.3. Fermat’sLastTheorem 100
ProblemsandQuestions 101
MathematicalProblems 101
HistoricalQuestions 102
QuestionsforReflection 102
10. Fifth-CenturyGreekGeometry 103
10.1. “Pythagorean”Geometry 103
10.1.1. TransformationandApplicationofAreas 103
10.2. ChallengeNo.1:UnsolvedProblems 106
10.3. ChallengeNo.2:TheParadoxesofZenoofElea 107
10.4. ChallengeNo.3:IrrationalNumbersand
IncommensurableLines 108
10.4.1. TheArithmeticalOriginofIrrationals 110
10.4.2. TheGeometricOriginofIrrationals 110
10.4.3. ConsequencesoftheDiscovery 111
ProblemsandQuestions 113
MathematicalProblems 113
HistoricalQuestions 113
QuestionsforReflection 114