The History of Mathematics: A Source-Based Approach, Volume 2 (AMS/MAA Textbooks, 61)

704 Pages·2022·90.41 MB·English

by  June Barrow-Green, Jeremy Gray, Robin Wilson

AMS / MAA TEXTBOOKS VOL 61
The History of Mathematics:
A Source-Based Approach
Volume 2
June Barrow-Green, Jeremy Gray, and Robin Wilson

The History of
Mathematics:
A Source-Based
Approach
Volume 2
FrontispiecetoEuler’sIntroductioinAnalysinInfinitorum,1748
AMS/MAA TEXTBOOKS
VOL 61
The History of
Mathematics:
A Source-Based
Approach
Volume 2
June Barrow-Green
Jeremy Gray
Robin Wilson
MAATextbooksEditorialBoard
StanleyE.Seltzer,Editor
MatthiasBeck SuzanneLynneLarson JeffreyL.Stuart
DebraSusanCarney MichaelJ.McAsey RonD.Taylor,Jr.
HeatherAnnDye VirginiaA.Noonburg ElizabethThoren
WilliamRobertGreen ThomasC.Ratliff RuthVanderpool
2020MathematicsSubjectClassification.Primary01-01,01A05;
Secondary01A45,01A50,01A55.
Foradditionalinformationandupdatesonthisbook,visit
www.ams.org/bookpages/text-61
TheISBNnumbersforthisseriesofbooksincludes
ISBN978-1-4704-4382-5(number2)
ISBN978-1-4704-4352-8(number1)
LibraryofCongressCataloging-in-PublicationData
Thefirstvolumewascataloguedasfollows:
Names:Barrow-Green,June,1953–author.|Gray,Jeremy,1947–author.|Wilson,RobinJ.,author.
Title: Thehistoryofmathematics: asource-basedapproach/JuneBarrow-Green, JeremyGray, Robin
Wilson.
Description: Providence,RhodeIsland: MAAPress,animprintoftheAmericanMathematicalSociety,
[2018]-|Series:AMS/MAAtextbooks;volume45|Includesbibliographicalreferencesandindex.
Identifiers:LCCN2018034323|ISBN9781470443825(paperback)|9781470456931(ebook)
Subjects: LCSH:Mathematics–History. |Mathematics–Studyandteaching. |AMS:Historyandbiography
–Instructionalexposition(textbooks,tutorialpapers,etc.). |Historyandbiography–Historyofmath-
ematicsandmathematicians–Generalhistories,sourcebooks. |Historyandbiography–Historyof
mathematicsandmathematicians–IndigenousEuropeancultures(pre-Greek,etc.).|Historyandbiog-
raphy–Historyofmathematicsandmathematicians–Egyptian. |Historyandbiography–Historyof
mathematicsandmathematicians–Babylonian.|Historyandbiography–Historyofmathematicsand
mathematicians–Greek,Roman.|Historyandbiography–Historyofmathematicsandmathematicians
–China. |Historyandbiography–Historyofmathematicsandmathematicians–India. |Historyand
biography–Historyofmathematicsandmathematicians–Medieval.|Historyandbiography–History
ofmathematicsandmathematicians–17thcentury.
Classification:LCCQA21.B242018|DDC510.9–dc23
LCrecordavailableathttps://lccn.loc.gov/2018034323
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Contents
Acknowledgments ix
Permissions&Acknowledgments x
Introduction 1
PartI The17thand18thcenturies 5
1 Introduction: The17thand18thcenturies 7
2 TheInventionoftheCalculus 13
Introduction 13
2.1 Tangents,maxima,andminima 14
2.2 Areaandvolumeproblems 29
2.3 Thesituationmid-century 44
2.4 Furtherreading 50
3 NewtonandLeibniz 51
Introduction 51
3.1 Newton 51
3.2 Newtonandhiscalculus 59
3.3 Leibnizandhiscalculus 67
3.4 Furtherreading 75
4 TheDevelopmentoftheCalculus 77
Introduction 77
4.1 Inversetangentproblems 78
4.2 Newton’scalculusandinversetangentproblems 86
4.3 Newton’smaturecalculus 97
4.4 Leibniz’smaturecalculus 105
4.5 Acomparison 118
4.6 Furtherreading 121
5 Newton’sPrincipiaMathematica 123
Introduction 123
5.1 ThecreationofNewton’sPrincipia 124
5.2 ThecontentofthePrincipia 132
5.3 ResponsestothePrincipia 148
5.4 Newton’sfinalyears 151
5.5 Furtherreading 153
v
vi Contents
6 TheSpreadoftheCalculus 155
Introduction 155
6.1 Thenextgeneration 155
6.2 Thecalculus,1690–1730 158
6.3 TheContinentalreceptionofthePrincipia 172
6.4 Furtherreading 189
7 The18thcentury 191
Introduction 191
7.1 Euler 192
7.2 D’AlembertandLagrange 204
7.3 Algebra 207
7.4 Furtherreading 217
8 18th-centuryNumberTheoryandGeometry 219
Introduction 219
8.1 Numbertheory 219
8.2 Infiniteseries 225
8.3 Eulerandgeometry 229
8.4 Thestudyofcurves 232
8.5 Furtherreading 244
9 Euler,Lagrange,and18th-centuryCalculus 247
Introduction 247
9.1 Earlycritiquesofthecalculus 247
9.2 Euler’scalculus 255
9.3 Differentialequations 265
9.4 Thefoundationsofthecalculus 274
9.5 Furtherreading 281
10 18th-centuryAppliedMathematics 283
Introduction 283
10.1 Thevibratingstring 286
10.2 Euler’svisionofmechanics 298
10.3 Furtherreading 307
11 18th-centuryCelestialMechanics 309
Introduction 309
11.1 TestingthePrincipia 309
11.2 Academyprizes 310
11.3 Laplace 311
11.4 Thestabilityofthesolarsystem 314
11.5 JupiterandSaturn 322
11.6 Furtherreading 329
PartII The19thCentury 331
12 Introduction: The19thCentury 333
Contents vii
13 TheProfessionofMathematics 335
Introduction 335
13.1 Thesocialcontext 335
13.2 MathematicsinFrance 336
13.3 MathematicsinGermany 350
13.4 Journalsandpublishing 360
13.5 Thelater19thcentury 368
13.6 Furtherreading 371
14 Non-EuclideanGeometry 373
Introduction 373
14.1 ThefirstWesternattempts 377
14.2 LobachevskiiandBolyai 390
14.3 Thereformulationofmetricalgeometry 401
14.4 Furtherreading 412
15 ProjectiveGeometryandtheAxiomatisationofMathematics 413
Introduction 413
15.1 TherediscoveryofprojectivegeometryinFrance 413
15.2 ProjectivegeometryinGermany 426
15.3 Theestablishmentofprojectivegeometry 432
15.4 There-unificationofgeometry 434
15.5 Theaxiomatisationofgeometry 442
15.6 Furtherreading 449
16 TheRigorisationofAnalysis 451
Introduction 451
16.1 Bolzano,Cauchy,andcontinuity 451
16.2 Cauchy’smistake 463
16.3 Cauchyondifferentiationandintegration 469
16.4 Conclusion 472
16.5 Furtherreading 473
17 TheFoundationsofMathematics 475
Introduction 475
17.1 Dedekind’sdefinitionoftherealnumbers 475
17.2 Cantor,sets,andtheinfinite 481
17.3 Foundationalquestions 490
17.4 Thephilosophyofmathematics 494
17.5 Settheoryandlogic 502
17.6 Furtherreading 508
18 AlgebraandNumberTheory 511
Introduction 511
18.1 Numbertheory 511
18.2 Primenumbers 520
18.3 Complexnumbersandquaternions 528
18.4 Vectors 538
18.5 Furtherreading 543
viii Contents
19 GroupTheory 545
Introduction 545
19.1 Solvingpolynomialequations 545
19.2 GaloisandGaloistheory 552
19.3 Impossibilitytheorems 560
19.4 Galois’stheoryofgroupsandequations 563
19.5 Grouptheory 565
19.6 Furtherreading 566
20 AppliedMathematics 569
Introduction 569
20.1 TheusesofFourierseries 570
20.2 Potentialtheory 581
20.3 Transatlanticcables 590
20.4 Furtherreading 593
21 PoincaréandCelestialMechanics 595
Introduction 595
21.1 Late19th-centurycelestialmechanics 595
21.2 HenriPoincaré 599
21.3 Poincaréanddifferentialequations 601
21.4 Poincaréandcelestialmechanics 603
21.5 Poincaré’smemoir 610
21.6 Poincaré’slaterworkincelestialmechanics 619
21.7 Conclusion 621
21.8 Furtherreading 622
22 Coda 623
Introduction 623
22.1 Theinternationalcommunityofmathematicians 623
22.2 Furtherreading 633
23 Exercises 635
Adviceontacklingtheexercises 635
Exercises: PartA 641
Exercises: PartB 657
Exercises: PartC 660
Bibliography 663
Index 683