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The Mathematical Gazette Felix Klein and Sophus Lie: Evolution of the Idea of Symmetry in the Nineteenth Centuryby I. M....
Felix Klein and Sophus Lie: Evolution of the Idea of Symmetry in the Nineteenth Centuryby I. M. Yaglom; Sergei Sossinsky; Hardy Grant
Review by: Douglas QuadlingBu kitabı ne kadar beğendiniz?
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Cilt:
72
Dil:
english
Dergi:
The Mathematical Gazette
DOI:
10.2307/3619976
Date:
December, 1988
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PDF, 149 KB
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Review Author(s): Douglas Quadling Review by: Douglas Quadling Source: The Mathematical Gazette, Vol. 72, No. 462 (Dec., 1988), pp. 341342 Published by: Mathematical Association Stable URL: http://www.jstor.org/stable/3619976 Accessed: 31122015 04:56 UTC Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at http://www.jstor.org/page/ info/about/policies/terms.jsp JSTOR is a notforprofit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. Mathematical Association is collaborating with JSTOR to digitize, preserve and extend access to The Mathematical Gazette. http://www.jstor.org This content downloaded from 134.129.120.3 on Thu, 31 Dec 2015 04:56:25 UTC All use subject to JSTOR Terms and Conditions REVIEWS 341 FelixKleinandSophusLie:Evolutionof the ideaof symmetryin thenineteenthcentury,by I. M. Yaglom. Translatedby Sergei Sossinsky and edited by Hardy Grant and Abe Shenitzer.Pp 237. SFr68. 1988.ISBN 0817633162(Birkhiiuser) In the 1940sthe undergraduate mathematicscourseat Cambridgewas embeddedfirmlyin the nineteenthcentury.In my lecture notes can be found Abel's test for convergence, Chasles'theorem,Cliffordnumbers,the CauchyRiemannequations,the CayleyHamilton theorem,Dedekindsections,Dirichlet series, Gauss'slaw of reciprocity,Jacobians,the Jordancurve theorem,Pluckercoordinates,Poncelet'sporism,Weierstrass'sMtestand muchelse besidesfromthat vastlyproductiveera. Significantly,I can findin themneither the nameof Klein nor that of Lie: for this was still a time when 'groups'wereconsidered dangerousgroundfor studentspriorto takingtheir bachelor'sdegree.But more seriously of mathematicswas an appreciationof how(orevenwhere) lackingfrommy understanding theseindividualsworked,of the wa; ysin whichthey influencedone another,or of the steps and the motivationwhich led them to make their own distinctivecontributionsto the subject.It hasbeena considerablepleasure,fourdecadeslater,to havethe opportunityto fill some of these gaps with the help of ProfessorYaglom'sstimulatingbook. The prefacetells us that 'the book is based on the author'slecturesto graduatelevel studentsmajoringin puremathematicsat YaroslavlUniversity,mostof whomsubsequently go on to teachin secondaryschools'.Onecan only say, 'luckyteachers'and 'luckypupils'. For this is no catalogueof namesand dates, but a sharplyfocused (thoughappropriately discursive)studyof the evolutionof a rangeof mathematicalideaswhichcametogetherin the work of Lie and Klein. It is a work of considerablescholarship,thougheminently readableby a studentwitha reasonablecompetencein mathematicswho is preparedto do a bit of skippingwhenthegoinggetstough.It couldbe recommended as a sequelto E. T. Bell's more lightweight Men of mathematics. To be moreprecise,whatwe havewithinthesecoversis twobooksforthepriceof one:the notes transcriptof the originallectures,anda collectionof no fewerthan312supplementary whichoccupyalmost75%as muchspaceas the maintext. Apartfromthe usualsupportive references(whichareremarkably uptodateandinternationalin theircoverage),thesenotes filloutthetextwithfurtherdetailof themoreadvancedmathematicaltopics,withadditional biographicalinformationabout some of the minor contributorsto the action, and occasionallywithdiscussionof deeperissuesin the philosophyof mathematics.Note 260,for instance,raisessome fundamentalquestionsas to what is meantby 'completeproof',and questionswhethera problemforwhichtheonlyproofavailableis one producedwithsupport from a computer(such asat the time of writingthe fourcolourproblem, or the classificationof sporadicsimplegroups)can be said to be truly'solved'. The title of the book is somewhatmisleading,thoughit is none the worsefor that. An opening chapterintroducesthe pivotal characterin the storyCamille Jordan,whose discoveryof an unansweredletterfromGaloisto Cauchyamongstthelatter'spapersled him to become the principal advocate and publicist of the theory of symmetrygroups promulgatedby thaterstwhileneglectedmathematician.It was a shareddesireto meetand workwith Jordanthatpromptedtwo studentfriendsin Berlin,the GermanFelix Kleinand the NorwegianSophusLie, to travelto Parisin 1870a visit sooncut shortby the outbreak of the FrancoPrussianwar. But it was long enough to direct them both towardsthe applicationsof grouptheorywith which their names are respectivelyassociated. However,the centralhalf of the book traces the developmentof three themes from nineteenthcenturygeometryprojectivegeometry,nonEuclideangeometriesandmultidimensionalspacesin whichthereis littlementioneitherof symmetryor of Klein and Lie. Thesechaptersarenonethelesscrucialin demonstrating the profoundchangein thenatureof mathematicalthoughtwhichdistinguished1820from1870;yearsin which(as MorrisKline has cogentlyargued)the functionof mathematicsceasedto be seen as the revelationof the truthof God'screation,andturnedto the investigationof thecreationsof man.Formuchof this periodthe scene was dominatedby Gaussborn, incredibly,only 50 yearsafter the deathof Newton,buta pioneerthroughouthislonglife. He attendeda lectureof Riemannon the foundationsof geometryat the age of 77, in which the ideas put forwardwere 'so far This content downloaded from 134.129.120.3 on Thu, 31 Dec 2015 04:56:25 UTC All use subject to JSTOR Terms and Conditions 342 THE MATHEMATICAL GAZETTE ahead of their time that only Gauss could have understoodthem'. (But he gets a bad characterreferencefromYaglomforhis lackof generosity:'Supportof youngtalentwasnot to be expectedfromthe world'sforemostmathematician.') Othernamesprominentin these chaptersare M6bius,Steiner,Bolyai,Lobachevsky,Cayley,Grassmannand Hamilton. Sucha list remindsus thatthedevelopmentof mathematicshasto be consideredagainsta backgroundof history,geography,economicsandsociology.The rise andfallof Napoleon, the turmoilsof 1848and Prussianexpansionismall had theireffect.The inventionof the betweencentreseasier,butanyoneworking railwayandthesteamshipmadecommunication outsidethe LeipzigBerlinGottingen triangleor the environsof Pariswas 'provincial'.It is not surprisingthat some of the wayoutideas aroseaway fromthese centres,but often in unconventionalforms:nonEuclideangeometryin Hungaryand Russia,formalalgebrasin the BritishIslesandthe UnitedStates.Nor wasall this researchcarriedoutin the congenial environmentof a university:Grassmannwas a schoolteacher(as, for a time, wereSteiner andvonStaudt),PonceletandBolyaiweresoldiers,Cayley(fromchoice)practisedat thebar. LieandKleinweregoodfriends,andtheircareerswereintertwined:whenKleinmovedto Gottingenin 1886,Lie succeededhimas professorof geometryat Leipzig.Butin everyother respectthey were very differentfrom each other. Lie appearsto have sharedwith his Ibsenand Griegthe problemsof combininga passionate compatriotsand contemporaries love of his native Norwaywith the need to travelsouthto protecthimselffrom cultural isolation. As a mathematicianhe devoted himself entirelyto the study of continuous transformation groups,and he remainedcreativelyactive until his deathat the age of 56. in geometryand Klein,by contrast,aftertwelveyearsof intenseproductivityparticularly the theoryof functionsestablisheda greatreputationas a teacher,writerand scientific He directedthe publicationof the worksof Gauss,was for nearlyfiftyyears administrator. editorof Mathematische Annalen,andhis influenceon the Germanschoolcurriculumis felt to thepresentday.He wasa leadingmemberof ICMIin its earlyyears,andhe playeda large partin establishingthe internationalreputationwhichthe Universityof Gottingenenjoyed duringthe firstthreedecadesof the presentcentury. This bookis not totallyfreefromblemishes:thereis a confusingmismatchbetweentext and diagramsin the accountof projectivegeometry,Isaac Newton is promotedfromthe officeof Masterof the Mintto Chancellorof the Exchequer,and(Sylvesternotwithstanding) it seemsexcessiveto describethe royalmilitaryschoolat Woolwichas the principalcentre formathematicsin Britainduringthe nineteenthcentury.Mostseriously,the bookis flawed by the absenceof an index. But it tells a fascinatingstory in a style which consistently commandsattention,and I commendit warmlyto all teachersand studentsof advanced mathematics. I hadintendedto endthe reviewat this point;butbetweenfinishingthe bookandputting fingersto keyboardI had the good fortuneto hearIan Stewart'slectureat the Association conferencein Birminghamon 'Symmetrybreakingin mathematicsandnature',basedround a descriptionof the experimentsin fluid mechanics known as the TaylorCouette phenomenon.Withthis text freshin mind,one couldnot fail to be struckby the immediate relevanceof the work of Klein and Lie to the understandingand analysis of these observations.It was one more powerfulreminderhow the abstractmathematicsof one generationcan providethe frameworkneededto interpretthe physicalworldof the next. Whetherin teachingor research,we neglectthe historicalperspectiveof the subjectat our peril. DOUGLASQUADLING 12 Archway Court, CambridgeCB3 9LW A new regularpolygon "At the top of the borough's long thin equilateral triangle lie Vauxhall and the Oval cricket ground." From the Sunday Times of 7 June, sent in by Hamish Sloan. This content downloaded from 134.129.120.3 on Thu, 31 Dec 2015 04:56:25 UTC All use subject to JSTOR Terms and Conditions