A Historian Looks Back PDF
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P1:JZP MABK013-FM MABK013/Grabiner TrimSize:7in×10in July28,2010 19:19 A Historian Looks Back The Calculus as Algebra and Selected Writings Part I was formerly published as The Calculus as Algebra: J.-L. Lagrange, 1736–1813 by Garland Publishing, New York, 1990. ©2010 by The Mathematical Association of America (Incorporated) Library of Congress Catalog Card Number 2010927261 Print ISBN 978-0-88385-572-0 Electronic ISBN 978-1-61444-506-7 Printed in the United States of America P1:JZP MABK013-FM MABK013/Grabiner TrimSize:7in×10in July28,2010 19:19 A Historian Looks Back The Calculus as Algebra and Selected Writings By Judith V. Grabiner Pitzer College PublishedandDistributedby TheMathematicalAssociationofAmerica P1:JZP MABK013-FM MABK013/Grabiner TrimSize:7in×10in July28,2010 19:19 SPECTRUM SERIES The Spectrum Series of the Mathematical Association of America was so named to reflect its purpose: to publish a broad range of books including biographies, accessible expositions of old or new mathematical ideas, reprints and revisions of excellent out-of-print books, popular works, and other monographs of high interest that will appeal to a broad range of readers, including students and teachers of mathematics, mathematical amateurs, and researchers. Council on Publications and Communications Frank Farris, Chair Spectrum Editorial Board Gerald L. Alexanderson, Editor Robert F. Bradley Kenneth A. Ross Susanna S. Epp Sanford Segal Richard K. Guy Franklin F. Sheehan Michael A. Jones Robin Wilson Keith M. Kendig Robert S. Wolf Jeffrey L. Nunemacher 777 Mathematical Conversation Starters, by John de Pillis 99 Points of Intersection: Examples—Pictures—Proofs, by Hans Walser. Translated from the original German by Peter Hilton and Jean Pedersen. Aha Gotcha and Aha Insight, by Martin Gardner All the Math That’s Fit to Print, by Keith Devlin Calculus Gems: Brief Lives and Memorable Mathematics, by George F. Simmons Carl Friedrich Gauss: Titan of Science, by G. Waldo Dunnington, with additional material by Jeremy Gray and Fritz-Egbert Dohse The Changing Space of Geometry, edited by Chris Pritchard Circles: A Mathematical View, by Dan Pedoe Complex Numbers and Geometry, by Liang-shin Hahn Cryptology, by Albrecht Beutelspacher The Early Mathematics of Leonhard Euler, by C. Edward Sandifer The Edge of the Universe: Celebrating 10 Years of Math Horizons, edited by Deanna Haunsperger and Stephen Kennedy Euler and Modern Science, edited by N. N. Bogolyubov, G. K. Mikhailov, and A. P. Yushkevich. Translated from Russian by Robert Burns. Euler at 300: An Appreciation, edited by Robert E. Bradley, Lawrence A. D’Antonio, and C. Edward Sandifer Five Hundred Mathematical Challenges, Edward J. Barbeau, Murray S. Klamkin, and William O. J. Moser The Genius of Euler: Reflections on his Life and Work, edited by William Dunham The Golden Section, by Hans Walser. Translated from the original German by Peter Hilton, with the assistance of Jean Pedersen. The Harmony of the World: 75 Years of Mathematics Magazine, edited by Gerald L. Alexanderson with the assistance of Peter Ross A Historian Looks Back: The Calculus as Algebra and Selected Writings, by Judith Grabiner P1:JZP MABK013-FM MABK013/Grabiner TrimSize:7in×10in July28,2010 19:19 HistoryofMathematics:HighwaysandByways,AmyDahan-DalmedicoandJeannePeiffer.Translatedby SanfordSegal. HowEulerDidIt,byC.EdwardSandifer IsMathematicsInevitable?AMiscellany,editedbyUnderwoodDudley IWanttoBeaMathematician,byPaulR.Halmos JourneyintoGeometries,byMartaSved JULIA:alifeinmathematics,byConstanceReid TheLighterSideofMathematics:ProceedingsoftheEuge`neStrensMemorialConferenceonRecreational Mathematics&ItsHistory,editedbyRichardK.GuyandRobertE.Woodrow LureoftheIntegers,byJoeRoberts MagicNumbersoftheProfessor,byOwenO’SheaandUnderwoodDudley MagicTricks,CardShuffling,andDynamicComputerMemories:TheMathematicsofthePerfectShuffle, byS.BrentMorris MartinGardner’sMathematicalGames:TheentirecollectionofhisScientificAmericancolumns TheMathChatBook,byFrankMorgan MathematicalAdventuresforStudentsandAmateurs,editedbyDavidHayesandTatianaShubin.Withthe assistanceofGeraldL.AlexandersonandPeterRoss. MathematicalApocrypha,byStevenG.Krantz MathematicalApocryphaRedux,byStevenG.Krantz MathematicalCarnival,byMartinGardner MathematicalCirclesVolI:InMathematicalCirclesQuadrantsI,II,III,IV,byHowardW.Eves MathematicalCirclesVolII:MathematicalCirclesRevisitedandMathematicalCirclesSquared,byHoward W.Eves MathematicalCirclesVolIII:MathematicalCirclesAdieuandReturntoMathematicalCircles,byHoward W.Eves MathematicalCircus,byMartinGardner MathematicalCranks,byUnderwoodDudley MathematicalEvolutions,editedbyAbeShenitzerandJohnStillwell MathematicalFallacies,Flaws,andFlimflam,byEdwardJ.Barbeau MathematicalMagicShow,byMartinGardner MathematicalReminiscences,byHowardEves MathematicalTreks:FromSurrealNumberstoMagicCircles,byIvarsPeterson MathematicsinHistoricalContext,byJeffSuzuki Mathematics:QueenandServantofScience,byE.T.Bell MemorabiliaMathematica,byRobertEdouardMoritz MusingsoftheMasters:AnAnthologyofMathematicalReflections,editedbyRaymondG.Ayoub NewMathematicalDiversions,byMartinGardner Non-EuclideanGeometry,byH.S.M.Coxeter NumericalMethodsThatWork,byFormanActon NumerologyorWhatPythagorasWrought,byUnderwoodDudley OutoftheMouthsofMathematicians,byRosemarySchmalz PenroseTilestoTrapdoorCiphers...andtheReturnofDr.Matrix,byMartinGardner Polyominoes,byGeorgeMartin PowerPlay,byEdwardJ.Barbeau P1:JZP MABK013-FM MABK013/Grabiner TrimSize:7in×10in July28,2010 19:19 ProofandOtherDilemmas:MathematicsandPhilosophy,editedbyBonnieGoldandRogerSimons TheRandomWalksofGeorgePo´lya,byGeraldL.Alexanderson RemarkableMathematicians,fromEulertovonNeumann,IoanJames TheSearchforE.T.Bell,alsoknownasJohnTaine,byConstanceReid ShapingSpace,editedbyMarjorieSenechalandGeorgeFleck SherlockHolmesinBabylonandOtherTalesofMathematicalHistory,editedbyMarlowAnderson,Victor Katz,andRobinWilson StudentResearchProjectsinCalculus,byMarcusCohen,ArthurKnoebel,EdwardD.Gaughan,Douglas S.Kurtz,andDavidPengelley Symmetry,byHansWalser.TranslatedfromtheoriginalGermanbyPeterHilton,withtheassistanceof JeanPedersen. TheTrisectors,byUnderwoodDudley TwentyYearsBeforetheBlackboard,byMichaelStuebenwithDianeSandford WhoGaveYoutheEpsilon?andOtherTalesofMathematicalHistory,editedbyMarlowAnderson,Victor Katz,andRobinWilson TheWordsofMathematics,byStevenSchwartzman MAAServiceCenter P.O.Box91112 Washington,DC20090-1112 800-331-1622 FAX:301-206-9789 P1:JZP MABK013-FM MABK013/Grabiner TrimSize:7in×10in July28,2010 19:19 P1:JZP MABK013-FM MABK013/Grabiner TrimSize:7in×10in July28,2010 19:19 Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi PartI. TheCalculusasAlgebra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1 PrefacetotheGarlandEdition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3 Acknowledgements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9 1. TheDevelopmentofLagrange’sIdeasontheCalculus:1754–1797 . . . . . . . . . . .17 2. TheAlgebraicBackgroundoftheTheoryofAnalyticFunctions . . . . . . . . . . . . .37 3. TheContentsoftheFonctionsAnalytiques . . . . . . . . . . . . . . . . . . . . . . . . . . .63 4. FromProof-TechniquetoDefinition:ThePre-HistoryofDelta-EpsilonMethods . . 81 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .101 Appendix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .103 Bibliography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .105 PartII. SelectedWritings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .125 1. TheMathematician,theHistorian,andtheHistoryofMathematics. . . . . . . . . . .127 2. WhoGaveYoutheEpsilon?CauchyandtheOriginsofRigorousCalculus∗ . . . . 135 3. TheChangingConceptofChange:TheDerivativefromFermattoWeierstrass† . .147 4. TheCentralityofMathematicsintheHistoryofWesternThought† . . . . . . . . . . 163 5. DescartesandProblem-Solving† . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .175 6. TheCalculusasAlgebra,theCalculusasGeometry:Lagrange,Maclaurin, andTheirLegacy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .191 7. Was Newton’s Calculus a Dead End? The Continental Influence of Maclaurin’s TreatiseofFluxions∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 8. Newton,Maclaurin,andtheAuthorityofMathematics∗ . . . . . . . . . . . . . . . . . .229 9. WhyShouldHistoricalTruthMattertoMathematicians?DispellingMythswhile PromotingMaths. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .243 10. WhyDidLagrange“Prove”theParallelPostulate?∗ . . . . . . . . . . . . . . . . . . . . 257 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 AbouttheAuthor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .287 ∗RecipientoftheLesterR.FordAward †RecipientoftheCarlB.AllendoerferAward ix
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