S. Kovalevsky: A Mathematical Lesson Author(s): Karen D. Rappaport Reviewed work(s): Source: The American Mathematical Monthly, Vol. 88, No. 8 (Oct., 1981), pp. 564-574 Published by: Mathematical Association of America Stable URL: http://www.jstor.org/stable/2320506 . Accessed: 25/02/2013 16:08

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This content downloaded on Mon, 25 Feb 2013 16:08:52 PM All use subject to JSTOR Terms and Conditions S. KOVALEVSKY: A MATHEMATICAL LESSON

KAREN D. RAPPAPORT MathemcaticsDepartmenit, Essex CounitvCollege, Newark, NJ 07102 SofyaKovalevsky was a notedwriter whose works include both fiction and nonfiction.She was also a politicalactivist and a public advocate of feminism.In addition,she was a brilliant mathematicianwho made significant contributions despite the enormous educational and political obstaclesthat she had to overcome.Somehow her manyachievements have been forgotten.In thosefew instances where her work has not been lostit has been denigratedby such studiesas FelixKlein's history of nineteenth-centurymathematics. Klein dismisses Kovalevsky's work in the followingmanner: "Her worksare done in thestyle of Weierstrassand so one doesn'tknow how muchof herown ideas are in them."'He findssomething wrong with all herresearch and credits her withonly one positiveaccomplishment, drawing Weierstrass out of his shell throughtheir correspondence.It is timeto set thisrecord straight and to let thefacts speak for themselves. SofyaKrukovsky, known affectionately as Sonya,was bornin Moscowin 1850.Her father,a Russianarmy officer, retired in 1858and movedthe family- Sofya, her older sister, Anyuta, and heryounger brother, Fedya- to Palibino,an estatenear the Lithuanian border. Aftersettling at Palibino,the householddiscovered that theyhad not broughta sufficient amountof wallpaperwith them. Rather than travel a greatdistance to obtainnew wallpaper, they decidedto use old newspaperson thewall. Since onlythe nursery required the paper, this was deemedan adequatesolution. However, while searching the attic for newspaper, they discovered paperof a betterquality. On it werethe lecture notes from a calculuscourse taken by General Krukovsky.This is how thenursery walls came to be coveredwith the calculus notes that, in her later years,Sofya claimed to have studied.Sofya oftenrepeated this anecdoteand enjoyed reportinghow hercalculus teacher exclaimed: "You have understoodthem as thoughyou knew themin advance."2 Kovalevskyclaimed that her interestin mathematicswas arousedby her Uncle Peter,who woulddiscuss numerous abstractions and mathematicalconcepts with her. When the family tutor, JosephMalevich, read of thisin Sofya'sautobiographical work, Memories of Childhood,he was incensed.He wrotea long essayin a Russiannewspaper explaining why he shouldreceive credit forKovalevsky's mathematical development. In responseto thiscriticism Kovalevsky wrote the followingtribute in "An AutobiographicalSketch": "It is to JosephMalevich that I am indebted formy first systematic study of mathematics.It happenedso longago thatI no longerremember his lessons at all.... It was arithmeticthat Malevich taughtbest.. .I have to confessthat arithmeticheld little interest for me."3 Kovalevsky4studied mathematics against her father'swishes. When she was thirteen,she smuggledan algebratext into her room and studiedit. Whenshe was fourteenshe taughtherself trigonometryin order to studya physicsbook writtenby her neighborProfessor Tyrtov- trigonometrywas necessaryfor the opticssection, and the youngSofya taught herself without tutoror text.By constructinga chord on a circle,she was able to explainthe sine function and to developthe othertrigonometric formulas. When Professor Tyrtov saw herwork, he was struckby its similarityto the actualmathematical development. Calling her a new Pascal, Tyrtovpleaded withthe Generalto permitSofya to studymathematics. After a year of exhortation,General Krukovskyrelented and allowedSofya to go to Petersburgto studycalculus and othersubjects. Aftercompleting her studies in 1867,Sofya wanted to continueher education, but theRussian universitysystem was closedto women.The onlyoption for study was to go to Switzerland,but GeneralKrukovsky would not allowhis daughtersto go abroad.

Theauthor received her Ph.D. fromNew York University in 1975under Professor H. Weitzner.She is a tenured facultymember at EssexCounty College in NewJersey. Her interests are in thehistory of women in mathematics, exploratorydata analysis, computer applications in statistics,, and partial differential equations. -Editors 564

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Sofya'solder sister, Anyuta, felt imprisoned at Palibinoand soughta way out. She foundit throughthe radicalpolitics of the times.This was a periodof politicalferment in Russia. The nihilists,feminists, and radicalistswere all active,and theirideas werebrought to Palibinoby the local priest'sson on his vacationfrom school in Petersburg.While they scandalized the neighbor- hood, theseideas had greatinfluence on Anyuta,who in turninfluenced Sofya. Anyuta joined a radical group that advocatedhigher education for womenand promotedthe conceptof the "fictitioushusband" to enablewomen to obtainmore freedom. A marriedwoman did not need her father'ssignature for a passport,and so a fictitioushusband would enable Anyuta to travel abroad forher education. Anyuta and herfriend Zhanna founda 26-year-olduniversity student, VladimirKovalevsky, who agreedto marryone of them.Unfortunately, for Anyuta, she brought Sofyato one of theirmeetings. Vladimir became infatuated with Sofya and insistedon marrying her.After several secret meetings and muchintrigue, General Krukovsky consented, and Vladimir and Sofyawere married in September1868. Followingtheir marriage, the Kovalevskysleft for Petersburg to study,and to searchfor a husbandfor Anyuta. With little effort Sofya had won the freedomto pursueher education, the freedomand independencethat Anyuta had been fightingso hard for.Sofya's feelings of guilt aboutthis can be seenin herletters to Anyuta,who was stillconfined at Palibino.She wrote:"At timesa strangeanguish comes over me and I feelashamed that everything is comingto me so easilyand withoutany struggle."5 In Petersburg,Sofya received permission from the instructors to attendclasses unofficially. She wroteto hersister: "... Lecturesbegin tomorrow and so myreal life begins at 9 A.M. ... [Vladimir] and friendswill solemnly escort me by wayof thebackstairs so thatthere is hope of hidingfrom the administrationand fromcurious stares."6 It was at Petersburgthat Sofya decided to concentrateon mathematics.In a letterto Anyutashe said: "I have becomeconvinced that one cannotlearn everything and one lifeis barelysufficient to accomplishwhat I can in mychosen field."' The Kovalevskysand Anyuta,who was stillunmarried but chaperonedby theyoung couple, leftfor Europe in 1869. Sofyaintended to studymathematics; and Vladimir,geology. Anyuta plannedto pursueher revolutionary activities. Sofya and Vladimirsettled in Heidelberg,but Sofya was not permittedto matriculateat the university.She appealed to both the facultyand the administration.A special committeewas formed,and it was decided that each individual professorcould choosewhether to permitSofya Kovalevsky to attendhis lecturesunofficially. Kovalevskywas nowable to attendlectures, and heroutstanding mathematical ability became the talkof Heidelberg.As a firmbeliever in educationfor women, she used her reputationto assistother Russian women in theirefforts to attendthe university. One of thesewomen was her friendYulya Lermontov,who laterbecame the firstfemale chemist in Russia. For manyyears Bunsenhad describedSofya as "a dangerouswoman"8 because, according to him,Sofya had trickedhim intopermitting Yulya to use thepreviously all-male chemistry labs. In 1874,Karl Weierstrassasked Sofyafor confirmation of thestory because "he [Bunsen]writes fiction even if he doesn'tpublish it."9 Sofya,1Vladimir, and Yulya lived and studiedtogether in Heidelberguntil the fall of 1869. When Anyutaarrived for a visit,she was quite surprisedto find Sofya still livingwith her "fictitioushusband" and proceededto evictVladimir from the apartment.A shorttime later, Vladimirleft Heidelberg to studyat Jena. As hermathematics progressed, Kovalevsky felt the need to studywith ,the mostnoted mathematician of thetime, at theUniversity of Berlin.She traveledto Berlinfor the startof thefall semester of 1870,only to findthe university closed to women.Sofya wrote: "The capitalof Prussiaproved to be backward.Despite all mypleadings and effortsI had no successin obtainingpermission to attendthe University of Berlin."'0I Determinedto studymathematics, Kovalevsky personally presented herself to Weierstrassas an aspiringstudent. On the basis of recommendationsfrom Koenigsberger at Heidelberg, Weierstrasswas willingto see her.He assignedher a set of problemson hyperellipticfunctions

This content downloaded on Mon, 25 Feb 2013 16:08:52 PM All use subject to JSTOR Terms and Conditions 566 KAREN D. RAPPAPORT [October thathe hadjust givento his class.Weierstrass was so impressedwith the ability she demonstrated in her solutionsthat he personallyrequested the universityto allow her to attendhis classes unofficially.However, the university was intransigentin its decision. Not wantingto waste this mathematicaltalent, Weierstrass offered to tutorKovalevsky privately.The lessons,begun in thefall of 1870,lasted for four years. The workingrelationship lasteda lifetime.When Weierstrass made his offer, he had no idea thatKovalevsky would become theclosest of his disciplesand remainso untilher death. The lessonsbegan with twice-weekly meetings, Sundays at Weierstrass'shome and weekdaysat Kovalevsky's.The mathematicslessons are partiallydocumented in a seriesof forty-oneletters fromWeierstrass to Kovalevsky,spanning the period of March 1871to August1874. They show thatthe emphasis was on Weierstrass'sfavorite topic: Abelian functions. Kovalevsky's responses are unavailable,because Weierstrasshad burnedthem on hearingof her death.However, some idea of theimportance of thiscorrespondence to Kovalevskycan be foundin herwritings: "These studieshad thedeepest possible influence on myentire career in mathematics.They determined finallyand irrevocablythe direction I was to followin mylater scientific work: all mywork has been done preciselyin thespirit of Weierstrass."1 Theseletters also showthe development of a close personalrelationship between Kovalevsky and Weierstrass.They give a glimpseof thegrowing affection that Weierstrass felt for his pupil. Weierstrasswas unawareof Kovalevsky's marriage arrangement and did notunderstand Vladimir's appearances.It was not untiltwo yearslater (October 1872) thatWeierstrass learned the truth. The seriesof lettersat thistime indicate that the topic of a relationshipbetween Kovalevsky and Weierstrasswas raisedand thetruth about her marital status was madeknown. Both were upset at the scene,and Weierstrassmade severalattempts to reassurehis pupil thathe would thereafter onlydiscuss science. For a shortperiod the correspondence remained strictly mathematical, but it did notstay that way. Weierstrass was notable to maskhis concernand affectionfor Kovalevsky. In October of 1872, Weierstrasshad suggestedseveral possible topics for Kovalevsky's dissertation.By 1874,she had completedthree original works, any one of whichWeierstrass felt wouldbe acceptable.Now he neededto finda universitythat would award Kovalevsky a degree. In July1874, the University of Gottingenawarded Sofya Kovalevsky a Ph.D., in absentia,summa cumlaude, without either orals or defense.Needless to say,this was an unprecedentedevent. The threepapers presented for the degree were: 1. "On theTheory of PartialDifferential Equations" 2. "On the Reductionof a CertainClass of AbelianIntegrals of the ThirdRank to Elliptic " 3. "SupplementaryRemarks and Observationson Laplace's Researchon theForm of Saturn's Rings" The firstof thepapers, on partialdifferential equations, was publishedin Crelle'sjournal in 1875. This was considereda greathonor, especially for a novicemathematician, since Crelle's journal was consideredthe most serious mathematical publication in Germany. In the firstpaper, Kovalevsky had generalizeda problemthat had been posed by Cauchy. Cauchyhad examinedan existencetheorem for partial differential equations, and Kovalevsky generalizedCauchy's results to systemsof orderr containingtime derivatives of orderr. The mathematicianH. Poincaresaid that"Kovalevsky significantly simplified the proof and gave the theoremits definitiveform."'2 Today, this theorem on theexistence and uniquenessof solutions of partialdifferential equations is often,but not always,known as the Cauchy-Kovalevsky Theorem.While studying the partial problem, Kovalevsky examined the heat equation.Some of herresults were helpful to Weierstrassand he wrote:"So you see,dear Sonya, thatyour observation (which seemed so simpleto you) on the distinctiveproperty of partial differentialequations ... was for me the startingpoint for interestingand very elucidating researches."' 3 Shortlyafter Sofya received her degree from Gottingen, her friend Yulya was awardeda degree in chemistryfrom Gottingen. Vladimir had receivedhis degreein paleontologytwo years earlier.

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Althoughshe had earned her Ph.D. and had writtena highlyacclaimed paper, Sofya Kovalevskywas unableto geta job. Eventhe efforts of her mentor, Weierstrass, were fruitless. So, in 1875,together with Vladimir, Anyuta, and Anyuta'shusband, Sofya returned to the family homein Russia.Weierstrass encouraged her to relaxat homeand "enjoythe pleasures of big city life. I knowyou won't give up yourscientific work."' 4 However,Kovalevsky did not actively pursueher work that winter. She feltguilty and wrote:"I workedfar less zealouslythan I had done in Germanyand, indeed, the situation was farless propitious for scholarly work ... The only thingthat still gave me scholarlysupport was theexchange of lettersand ideas withmy beloved teacherWeierstrass." 15 As importantas thisexchange may have been,the correspondence with Weierstrassstopped in 1875.-It was resumedfor a shorttime in 1878but did notbecome regular until 1880. Weierstrasswas veryhurt by thisneglect. However, it mustbe acknowledgedthat Kovalevskywas nevera good correspondent,even while in Germany,so thatit is notsurprising to findthis lapse in writingafter her return to Russia. In Russia,Kovalevsky again tried to finda job. The onlyposition in mathematicsavailable to a womanwas teachingarithmetic in thelower grades of a girls'school, and sinceSofya admitted thatshe was "unfortunatelyweak in themultiplication table"' 6 she could not seriouslyconsider such a position.Therefore, Kovalevsky turned to otherintellectual pursuits. She wrotefiction, theaterreviews, and popular-sciencereports for a newspaper.She was instrumentalin the organizationof theBestuzhev School for Women, but because of what were considered her radical viewsshe was notpermitted to teachthere. Duringthis period in Russia, GeneralKrukovsky died and leftSofya a small inheritance. Vladimirinvested this money in businessenterprises that eventually went bankrupt. By 1878,Sofya had becomebored with her activities and wantedto returnto mathematics.She wroteto Weierstrassfor advice. Weierstrass was excitedby thisletter, the firsthe had received fromhis pupilin threeyears. However, Kovalevsky's return to mathematicswas delayedby the birthof a daughter,Sofya Vladimirovna, in Octoberof 1878. On theirreturn to Russia theKovalevskys had assumedthe obligations of a realmarriage. This was donepartly as an obligationto Sofya'sparents and partlybecause of theirnew politics. It was theirfeeling to end lyingrelationships of all kinds,and so themarriage was finallyconsummated. Kovalevsky'sreturn to mathematicswas encouragedby a scientificconference held in Petersburgin 1880.The Russianmathematician Chebyshev invited Kovalevsky to presenta paper at thisconference. She foundher unpublished dissertation on AbelianIntegrals, translated it from Germanto Russian in one night,and presentedit to the conference.Although it had lain untouchedfor six years,it was wellreceived by themathematicians. Followingher presentation,the Swedishmathematician Gosta Mittag-Leffler,who had met Kovalevskyearlier while she was a studentin Germany,offered to findher a positionin his country.Kovalevsky was veryappreciative of thisoffer and wrote,in 1881,to Mittag-Leffler:"[If I can teach]I mayin thisway open theuniversities to women,which have hitherto only been open by specialfavor, a favorwhich can be deniedat anymoment."'7 Kovalevsky'sdesire for a positionwas spurrednot onlyby herfeminism but also by herneed to do mathematics.She wished"at thesame time,to be able to live formy work, surrounded by thosewho are occupiedwith the same questions."''8 Whilewaiting for word from Sweden, Kovalevsky looked in Berlinfor research work. While Vladimirwas on a businesstrip, Sofya secretly visited Weierstrass. When she decidedthat she wouldreturn to Berlinto pursueresearch, Vladimir was quiteangry with the decision and their marriageended. For thenext two years, Kovalevsky lived a student'slife in Berlin,and thecare of herdaughter was sharedby Sofya'sfriend Yulya and herbrother-in-law Alexander. WhileSofya Kovalevsky was busyconducting her research on therefraction of light in crystals, VladimirKovalevsky again managedto gethimself into financialdifficulties. There was a stock scandal,and Vladimir,faced with ruin, committed suicide in thespring of 1883.Sofya took solace in hermathematics and hopedto finda badlyneeded job in . Mittag-Lefflerhad recentlybeen appointedthe head of the mathematicsdepartment at the

This content downloaded on Mon, 25 Feb 2013 16:08:52 PM All use subject to JSTOR Terms and Conditions 568 KAREN D. RAPPAPORT [October newlyfounded University of Stockholmand was able to offerSofya a positionthere. However, the job had conditions.In order to prove her competence,Kovalevsky was to teach for a probationaryyear, with no pay and no officialuniversity affiliation. Kovalevsky agreed to this becauseshe had no otheroptions. In the fall of 1883, Sofya Kovalevskyarrived in Stockholmto become a lecturerat the Universityof Stockholm.Her receptionwas mixed.Although hailed by many,others agreed with A. Strindberg,who wrotein thelocal paper,"A femaleprofessor is a perniciousand unpleasant phenomenon-even, one mightsay, a monstrosity."19 In thespring of 1884,Kovalevsky lectured in Germanon partialdifferential equations. These lectureswere well received, and Mittag-Lefflerwas able to obtainthe funds for her appointment as a Professorof HigherAnalysis in July1884. Word of the appointmentwas sentto Sofyain ,where she was spendingthe summer with her daughter. However, Sofya did not bring herdaughter to Stockholmin thefall because she was stillunsure of her position. She was publicly criticizedfor her child-care arrangements but chose to ignorethis. She didn'tbring her daughter to Swedenuntil 1885, when the child was eightyears old. In additionto joiningthe Stockholmfaculty in 1884,Kovalevsky became an editorfor Acta Mathematicaand publishedher firstpaper on crystals.In 1885,she receiveda secondappoint- mentto theChair of .She also publisheda secondpaper on thepropagation of lightin crystalsbut was embarrassedwhen Volterra found a seriouserror in her work.She had used a multi-valuedfunction as if it werea single-valuedone. Since thisresearch was performedafter a three-yearhiatus in hermathematical career, Kovalevsky felt that her teacher, Weierstrass, should have caughtthe errorprior to publication.Weierstrass, quite distraught,blamed illnessand overwork.(It mightbe added thatWeierstrass was 70 yearsold at thetime.) While in Stockholm,Sofya lived for a timein Mittag-Leffler'shome, where she met and developeda friendshipwith his sister,Anna Leffler.Anna Leffler,a well-knownadvocate of women'srights and a writer,encouraged Sofya's literary leanings. In 1887they collaborated on a playentitled The Struggle for Happiness. It was based on an idea thathad occurredto Sofyawhile she sat at thebedside of herdying sister. AfterAnyuta died in thefall of 1887,Sofya felt lonely and despondent.The sistershad been close,and Sofyafelt the loss deeply.However, at thistime two eventsoccurred that helped to assuageher grief. Both the announcement of a newcompetition for the Prix Bordin and thearrival in Stockholmof a Russianlawyer named Maxim Kovalevsky were to haveprofound effects on the lifeof SofyaKovalevsky. Early in 1888 the FrenchAcademy of Scienceannounced a new competitionfor the Prix Bordin.Papers on the theoryof therotation of a solid body wouldbe consideredfor the prize competitionat theend of thesummer. Gosta Mittag-Lefflerencouraged Sofya to workon a paper forthe competition. WhileSofya was engagedin her researchon thepaper, Maxim Kovalevsky arrived to givea seriesof lecturesat StockholmUniversity. He had been dismissedfrom Moscow Universityfor criticizingRussian constitutional law. Asidefrom politics, Sofya and Maximhad manyinterests in common,and theirattraction resulted in a scandalous affair.Eventually Maxim proposed marriage,onwthe condition that Sofya give up herresearch. Even if she had wishedto giveup her mathematics,Sofya was too farinto her work for the prize competition to stop.In orderto free Sofyato do herwork, Mittag-Leffler invited Maxim to his summerhome in Uppsala. Thiswas a wisemove, for, as Sofyastated, "If burlyMaxim had stayedlonger, I do notknow how I should have goton withmy work."20 WithMaxim gone, Sofya was able to finishher work and thepaper was submittedon time.Of the fifteenpapers, which were submitted anonymously, one was consideredso outstandingthat theaward was increasedfrom 3,000 francs to 5,000francs. The PrixBordin was awardedto Sofya Kovalevskyin Decemberof 1888. Sofya attendedthe awardsceremony with Maxim. Special recognitionwas givento herwork. In his congratulatoryspeech, the President of theAcademy of

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Sciencessaid: "Our co-membershave found that her work bears witness not only to profoundand broadknowledge, but to a mindof greatinventiveness.""21 Priorto SofyaKovalevsky's work the only solutions to themotion of a rigidbody about a fixed point had been developedfor the two cases wherethe body is symmetric.In the firstcase, developedby Euler,there are no externalforces, and thecenter of massis fixedwithin the body. This is thecase thatdescribes the motion of theearth. In thesecond case, derivedby Lagrange, thefixed point and thecenter of gravityboth lie on theaxis of symmetryof thebody. This case describesthe motion of thetop. Sofya Kovalevsky developed the first of thesolvable special cases foran unsymmetricaltop. In thiscase thecenter of massis no longeron an axis in thebody. She solved the problemby constructingcoordinates explicitly as ultra-ellipticfunctions of time. Kovalevskycontinued this work in twomore papers on a rigidbody motion. These both received awardsfrom the Swedish Academy of Sciencesin 1889.Her laterworks on thesubject have been lost. Kovalevsky'sprofessorship in Swedenwas due to expirein 1889.Desirous of returningto her nativecountry, she inquiredabout a positionin Russia. Her requestwas flatlydenied. Russian mathematicianswere indignant at thisslight of Kovalevskyand decided to honorher. It was suggestedthat an honorarymembership in theRussian Academy of Scienceswould provide that recognition.However, in orderto do that,the charterhad to be amendedto allow forfemale membership.In November1889, Chebyshev sent the followingtelegram to Kovalevsky:"Our Academyof Scienceshas just nowelected you a correspondingmember, having just permittedthis innovationfor whichthere has been no precedentuntil now. I am veryhappy to see this fulfillmentof one of mymost impassioned and justified desires."22 Kovalevskysought positions throughout Europe but was againunsuccessful. She thereforehad to acceptthe renewal of theprofessorship in Stockholm. Whileworking in Stockholm,Kovalevsky regularly commuted to France,where she visited Maxim at his villa. During thesevisits Maxim encouragedher literaryinterests. She wrote Memoriesof Childhood in Russian.It was translatedinto Swedish and publishedin 1889.In order not to shockSwedish society with its personal revelations, it was releasedas a novelentitled The RaevskySisters. The originalwas publishedin Russianin 1890.Kovalevsky's novel A NihilistGirl was writtenin Swedishin 1890.A Russianversion had been started,but Sofya'ssudden death left it unfinished.Maxim edited the two versions and was responsiblefor the posthumous publication of thenovel. This book was highlypraised by criticsin Russia and Scandinavia. It was Kovalevsky'sfrequent trips to Franceto visitMaxim that eventually caused her death. On returningto Stockholmfrom a visit,early in 1891,she fellill. The illnesswas misdiagnosed, and by thetime it was finallyfound to be pneumoniait was outof control.On February10, 1891, SofyaKovalevsky died. Although there was widespreadmourning and eulogieswere given around the world,the Russian Ministerof the Interior,I. N. Dumovo, was concernedthat too much attentionwas beingpaid to "a womanwho was, in thelast analysis,a Nihilist."23 The mathematicalworld was moregenerous in itspraise. Mittag-Leffler gave the official eulogy for the Universityof Stockholm.Speaking of her as a teacherhe said: "We knowwith what inspiringzeal sheexplained [her] ideas... and howwillingly she gave the riches of her knowledge."24 In his eulogy,Kronecker, of theUniversity of Berlin,spoke of Kovalevskyas "one of therarest investigators."25Karl Weierstrass,who felther loss mostdeeply, having burned all of herletters, said " 'People die,ideas endure':it wouldbe enoughfor the eminent figure of Sofyato pass into posterityon thelone virtueof hermathematical and literarywork."26 In her shortlifetime, Sofya Kovalevskyleft a notablecollection of political,literary, and mathematicalworks. Her contributions,completed in spiteof manyobstacles, certainly warrant hera place in our intellectualand mathematicalhistory. Duringher mathematicalcareer Sofya Kovalevsky published ten papersin mathematicsand mathematicalphysics. Three of thesepapers,27 28 29 werewritten during her student days under Weierstrass(1870- 1874).The articleson therefraction of light30'31, 32 werewritten years later in

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Berlin(1881-1883) afterher returnto her mathematicalresearches. The remainingresearch on rigidbody motion and Bruns'sTheorem33-36 was completedduring her tenure at theUniversity of Stockholm(1883-1891). The only completecollection of Kovalevsky'sworks is publishedin Russian.37Portions of herwork in partialdifferential equations and rigidbody motion appear in English,and sincethese are Kovalevsky'smost important works they will be discussedin some detail. The proofof thefirst general existence theorem in partialdifferential equations was presented by Cauchyin 1842,in hispublications on integratingdifferential equations with initial conditions. Here he showedthe existence of analyticsolutions for ordinary differential equations and certain linearpartial differential equations. The Cauchyproblem for the ordinarydifferential equation du/dt = f(t,u), withthe initial conditionsu = u0 and t = to states: Iff(t, u) is an analyticfunction in a neighborhoodof thepoint (t, u) thereexists a unique solutionof du/dt= f(t,u) withthe given initial conditions. For systemsof first-orderpartial differential equations of theform

___ au,l aul at! F t,Xi, Xn; Ul* Un; ax, axm ) withinitial conditions when t = 0, thatis u1(0, X1,..., xn) = w(x,..., xn) fori = 1,..., m, theCauchy problem is to finda solutionu(x, t) thatsatisfies the initial conditions.38 To solve thisproblem Cauchy assumed that Fi and wi were analytic.He obtaineda locally convergentpower series solution by usinghis "methodof majorants."The originalfunction F, is replacedby a simpleanalytic function whose power series expansion coefficients are nonnegative and greaterthan or equal to the absolutevalue of the correspondingcoefficients for F,. The derivedsystem is thenexplicitly integrated to give a solutionwhich is the majorantfor the solutionto theoriginal with t = 0. In herthesis Kovalevsky generalized Cauchy's results to systemsof an orderr containingtime derivatives,aru,/latr, of orderr. It is thisgeneralization that is knownas theCauchy-Kovalevsky Theorem. Kovalevskyused majorantsof theform M

1 [(tl + t2 + *+tr)/p] wherep and M are constants. For thehigher order system

a_n,u ak(Ua t, Xn; Un; Ul, atn, Il xI,.** U, m; atkoaXlkl ... aXkn with i, j = 1,2,..., m; ko + k1+ +kn = k < n., ko< nj, and initialvalues given at some pointt = to forthe ui and theirfirst ni - 1 derivativeswith respect to t, Kovalevskyproved the following. If all the functionsFi are analytic in a neighborhoodof the point (t, xi,.... Xn; ... Wj,ko k,.k) and all the functionswj(k) are analyticin a neighborhoodof the point (x1,..., xn), thenCauchy's problem has an analyticsolution in a certainneighborhood of the point(t, xl, X2,..., Xn), and it is theunique solution in theclass of analyticfunctions. (Note that

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The simplestform of this theoremstates that any equation of the form a u/at f(t, x, u, au/ax),wheref is analyticin its argumentsfor values near (to, xo, uo, auo/axo), possessesone and onlyone solutionu(t, x) whichis analyticnear (to, xo). The Cauchy-KovalevskyTheorem has significantlimitations. It is restrictedto analytic functions,and convergencemay fail in a regionof interest.Also the computationof the coefficientsof the seriesmay be too tediousto be practical.However, it is importantin thatit shows thatwithin a class of analyticsolutions of analyticequations the numberof arbitrary functionsneeded for a generalsolution is thesame as theorder of theequation and thesearbitrary functionsinvolve one less independentvariable than the number occurring in theequation. The workfor which Kovalevsky was bestknown in hertime was herresearch on themotion of a rigidbody about a fixedpoint. The equationsof motionof a rigidbody moving about a fixed pointwere derived by Eulerin 1750.They are as follows:

- dY =ry'-qY" Ad +(C B)qr Mg(yoyl oZy')

ddt - y dt + (AC- )rpMg( zy-xy"),

dy" Cdr - + dtqd y py Cdrdt (B-A)pq= Mg(xoy'-yoy).

HereA, B, C are theprincipal axes of theellipsoid of thebody relative to thefixed point; M is the mass; g is the accelerationdue to gravity.y, y', y" are the directioncosines of the angles whichthe three axes makeat each moment;their direction is thesame as theforce of the rigid body;p, q, r are thecomponents of theangular velocity along the principal axes; xo,YO, zo are the coordinatesof thecenter of gravityof thebody considered in a systemof coordinatesof whichthe originis the fixedpoint and whose directioncoincides with that of the principalaxes of the ellipsoidof inertia. The problemto be solvedwas theintegration of thissystem of equationsso thatthe position of themoving body at anytime could be obtained.Before 1888, the integration had been completed foronly two cases. The firstwas forthe condition xo = yo= zo-0. This case, studiedby Euler and Poisson,is theone wherethe center of gravityof themoving body coincideswith the fixed point.This is themotion of a force-freesymmetric body. There are no externalforces acting on thebody and themotion is about a fixedpoint within the body, the center of mass.If thefixed pointis thecenter of gravity,then gravity does notinfluence the motion. The axes of rotationare thusfixed in thebody. An exampleof thisforce-free motion is thefree rotation of theearth. In the case of the earth'sfree rotation, the axis of rotationdoes not coincidewith the axis of symmetry.It is veryslightly tilted, although it passes throughthe centerof mass of the earth. Whatthen happens is thatthe instantaneous axis precessesslowly about the axis of symmetry. In the secondcase, studiedby Lagrange,A = B, xo = yo= 0. Here the fixedpoint and the centerof gravitylie on thesame axis. When this axis is theaxis of symmetry,the motion is thatof thespinning top. "A top is definedto be a materialbody which is symmetricalabout an axis and terminateisin a sharppoint ... at one of theaxis."40 This top spinsabout a fixedpoint that is not thecenter of gravity,but the center of gravity and thefixed point both lie on theaxis of symmetry of thetop. The weightof thetop givesrise to a momentof forceas it spinsabout the fixed point on theplane. In bothof thesecases therigid body was symmetrical.Sofya Kovalevsky developed the first of thesoluble special cases forthe heavy or unsymmetricaltop. In thiscase, thecenter of mass no longerlies on theaxis of thebody. Instead, it is in theequatorial plane (the plane perpendicular to theaxis) and passes throughthe fixed point. In addition,two of theprincipal moments of inertia are equal and double the third.The centerof gravityis in the plane of the equal momentsof inertia. Euler'sequations containing six unknownfunctions has thefollowing first integrals:

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Ap2 + Bq2 + Cr2 - 2Mg(xoy + yoy' + zoy") = C1 Apy + Bqy' + Cry" = C2

'y2 + yi2 + r2 = c3=-

It is sufficientto findone moreintegral to obtaina completesolution in quadratures.This occursbecause the time variable appears only in theform dt and can be eliminated,leaving only fiveequations. Kovalevskyderived the fourth for the case A = B = 2C, zo = 0,41 and showedthat the onlyconditions necessary for a givenseries to be a solutionto Euler'ssystem are thatthe constantsA, B, C, x, y, z satisfyone of fourconditions:

(1) A - B = C, (2) xo yo = zo = 0 (Euler'scase), (3) A B, xo = Yo 0 (Lagrange'scase), (4) A B = 2C, zo 0 (Kovalevsky'scase).42 Kovalevskyobtained a solutionfor her case by rotatingthe coordinate axes in thexy plane and choosinga unitof length so thatyo= 0 and C 1. Withc0 Mgxothe Euler equations become:

d-y 2dp = qr qy dt d

- 2 dg , _ r dt =-p pr-c0Yco dt =PY"-ryly

dr d_y__d = _ 2 dt=CO'Y qy py. dt 'dt The threealgebraic integrals are:

2(p2 + q2) + r2 = 2cox + 61l 2(py + qy') + ry" = 21

'Y2 + yi2 + y ii2 = I where1 and 1 are constantsof integration.Kovalevsky then derived a fourthintegral:

[(p + qi)2 + Co(y + iy')][(p - qi)2 + Co(y - y'i)] = k2 where k is an arbitraryconstant. Defining xl = p + qi, X2 = p - qi, she made several transfor- mationsof thevariables. After some algebraic manipulations she obtainedthe equations:

0- ds1 + ds2 \,RI(sl) VR1(s2)

dt= s's ' + S2d2 ,RbI(l) 'R I (S2) whereRI (S) is a fifth-degreepolynomial whose zeros are uniqueand sI and s2 are polynomialsin xl and x2. This systemresults in hyperellipticintegrals which Kovalevsky solved by usingtheta functions. For thishighly praised research Kovalevsky was awardedthe Bordin Prize in 1888and a prize fromthe Swedish Academy in 1889.Historians, however, have foundthis work "not of sufficient interestfor the theory of a top to finda place here."43Today thevalue of herwork is seennot in theresults or theoriginality of themethod but in theinterest it stimulatedin theproblem of the rotationof a rigidbody. Severalresearchers have continuedthe workof findingnew cases of

This content downloaded on Mon, 25 Feb 2013 16:08:52 PM All use subject to JSTOR Terms and Conditions 1981] S. KOVALEVSKY: A MATHEMATICAL LESSON 573 special solutions.This includesChaplygin's development of the integralfor a symmetricaltop turningabout a fixedpoint, with moments of inertiaA = B = 4C and theHesse-Schiff equations of motionfor a top. Thereis also a body of workby others,for example, Bobylev, Steklov, Goryachev,and V. Kovalevsky. The remainingworks by SofyaKovalevsky are of lesserimportance and will be onlybriefly reviewed. In thepaper on Abelianintegrals, Kovalevsky showed how a certaintype of Abelianintegral could be expressedas an ellipticintegral. The paper on Saturn'srings is concernedwith the stabilityof motionof liquid ring-shaped bodies.Laplace foundthe form of thering to be a skewedcross-section of an ellipse.Kovalevsky, usinga seriesexpansion, proved that the rings were egg-shaped ovals symmetricabout a single axis. However,the subsequentproof that Saturn's rings consist of discreteparticles and not a continuousliquid made thiswork inapplicable. In thearticles on therefraction of lightin crystals,Kovalevsky applied a methoddeveloped by Weierstrassto differentiateLame's partialdifferential equations. However, Volterra discovered a basic errorin her work. She had (as had Lame) treateda multi-valuedfunction as thoughit were single-valued,and thesolution could not be appliedto theequations in herform. However, the paperdid demonstratethe previously unpublished theory of Weierstrass. Kovalevsky'slast articlederived a simplerproof of Bruns's Theorem based on a propertyof the potentialfunction of a homogeneousbody.

Notes

1. Felix Klein, VorlesungenUber die Entwicklungder MathematikIm 19. Jahrhundert,Teil I, Springer-Verlag, Berlin,1926, p. 294. 2. Sofya Kovalevskaya,"An AutobiographicalSketch," in A Russian Childhood,transl. and ed. by Beatrice Stillman,Springer-Verlag, New York, 1978,p. 216. 3. Ibid. 4. AlthoughKovalevskaya, the feminineversion of the name Kovalevsky,is used in much of the current literature,the authorwill referto Kovalevskyin thispaper, as thatwas the name underwhich Sofya Kovalevsky workedand published.However, in the notes the authorhas used the versionof the name that appeared in the reference. 5. S. V. Kovalevskaya,Vospominaniya i pis'ma, ed. S. Ya. Shtraikh,Moscow, 1951,p. 232. Cited in Beatrice Stillman,"Introduction," in Kovalevskaya,A RussianChildhood, p. 11. 6. Kovalevskaya, Vospominaniyai pis'ma, p. 225. Cited in Stillman,"Introduction," in Kovalevskaya,A Russian Childhood,p. 11. 7. P. Polubarinova-Kochina,Sophia VasilyevnaKovalevskaya, Her Life and Work,Men of Russian Science, transl.by P. Ludwick,Foreign Languages Press,Moscow, 1957,p. 25. 8. Karl Weierstrass,Briefe von Karl Weierstrassan SofieKowalevskaja 1871-1891 -Pis'ma Karla Veiershtrassa k Sofye Kovalevskoi,Nauka, Moscow, 1973,p. 51. 9. Ibid. 10. Kovalevskaya,"An AutobiographicalSketch," in A RussianChildhood, p. 218. 11. [bid. 12. P. Y. Polubarinova-Kochina,"On the ScientificWork of Sofya Kovalevskaya,"transl. by N. Koblitz, in Kovalevskaya,A Russian Childhood,p. 234. 13. Ibid., p. 235. 14. Weierstrass,Briefe, p. 52. 15. Kovalevskaya,"An AutobiographicalSketch," in A RussianChildhood, p. 220. 16. L. A. Vorontsova,Sofia Kovalevskaia,Moscow, 1957,p. 147. Cited in B. Stillman,"Sofya Kovalevskaya: GrowingUp in the Sixties,"Russian Literature Triquarterly, 9 (1974) 287. 17. Anna Carlotta Leffler,Duchess of Cajanello, Sonya Kovalevsky,transl. by A. de Furuhjelm,T. Fisher Unwin,London, 1895,p. 51 18. Ibid., p. 52. 19. Polubarinova-Kochina,Sophia V. Kovalevskaya,p. 50. 20. S. Kovalevskayato A. Leffler,1888. Cited in Anna Leffler,Sonya Kovalevsky,p. 124. 21. Polubarinova-Kochina,Sophia V. Kovalevskaya,p. 61.

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22. Kovalevskaya, Vospominaniyai pis'ma, p. 352. Cited in G. J. Tee, "Sofya Vasil'yevna Kovalevskaya," MathematicalChronicle, 5 (1977) 132- 133. 23. Kovalevskaya,Vospominaniya i pis'ma, p. 532. Cited in Tee, p. 135. 24. G. Mittag-Leffler,Commemorative Speech as Rectorof StockholmUniversity, 1891. Cited in Leffler,p. 171. 25. L. Kronecker,Journ. fur die reineund angewandte Mathematik, BD CVIII, March 1891, 1-18. 26. G. Mittag-Leffler,"Weierstrass et Sonja Kowalewsky,"Acta Mathematica,39 (1923) 170. 27. S. Kovalevsky,"Zur Theorie der partiellenDifferentialgleichungen," Journal fur die reineund angewandte Mathematik80 (1875) 1-32. 28. _ _, "Uber die Reductioneiner bestimmten Klasse von Abel'scherIntegrale 3-ten Ranges auf elliptische Integrale,"Acta Mathematica,4 (1884) 393-414. 29. _ _, "Zusatze und Bermerkungenzu Laplace's Untersuchunguber die Gestalt des Saturnringes," AstronomischeNachrichten 111 (1885) 37-48. 30. ,"Uber die Brechungdes Lichtesin cristallinischenMitteln," Acta Mathematica,6 (1883) 249-304. 31. _, "Sur la propagationde la lumieredans un milieu cristallise,"Comptes rendus des seances de I 'academie des sciences,98 (1884) 356-357. 32. _ _, "Om ljusetsfortplantning uti ettkristalliniskt medium," Ofversigt af Kongl. Vetenskaps-Akademiens Forhandlinger,41 (1884) 119-121. 33. _ _, "Sur le problemede la rotationd'un corps solide autour d'un point fixe,"Acta Mathematica,12 (1889) 177-232. 34. _ , "Memoire sur un cas particulierdu problemede la rotationd'un corps pesant autour d'un point fixe,ou I'integrations'effectue a l'aide de fonctionsultraelliptiques du temps,"Memoires presentes par diverssavants a l'academie des sciencesde l'institutnational de France,31 (1890) 1-62. 35. _ _, "Sur une proprietedu systemed'equations differentiellesqui dfinit la rotationd'un corps solide autourd'un point fixe,"Acta Mathematica,14 (1890) 81-93. 36. __, "Sur un theoreme de M. Bruns," Acta Mathematica, 15 (1891) 45-52. 37. S. V. Kovalevskaya,Nauchnye raboty (Scientific Works), ed. by P. Y. Polubarinova-Kochina,AN SSSR, Moscow, 1948. 38. G. Birkhoff,ed., A Source Book in Classical Analysis,Harvard UniversityPress, Cambridge,Mass., 1973, p. 318. 39. P. Y. Polubarinova-Kochina,"On the ScientificWork of Sofya Kovalevskaya,"transl. by N. Koblitz, in Kovalevskaya,A RussianChildhood, p. 233. 40. E. T. Whittaker,Analytical Dynamics of Particlesand Rigid Bodies, CambridgeUniversity Press, London, 1965,p. 164. 41. Kovalevsky,"Memoire sur un cas particulierdu problemede la rotationd'un corpspesant... " (seeNote 34). 42. Kovalevsky,"Sur une proprinetdu systeme..." (see Note 35). 43. A. Gray,A Treatiseon Gyrostaticsand RotationalMotion, Theory and Applications,Dover Publications,New York, 1959 (firstappeared 1918), p. 369.

THE UNIFORMIZATION THEOREM

WILLIAM ABIKOFF Departmentof Mathematics, University of Illinois, Urbana, IL 61801 Almost oile hundredyears have passed since Felix Klein discoveredthe uniformization theorem.While such theoremshad earlierbeen provedin specificcases, no one had daredeven conjecturethat everycompact Riemann surface could be parametrizedby a variablewhose

William Abikoffreceived his B.S. and M.S. in electricalengineering and his Ph.D. in mathematicsfrom the PolytechnicInstitute of Brooklyn.He workedat Bell Telephone Laboratoriesand taughtat Columbia University and theuniversities of Perugia,Firenze, and Paris. He is currentlyan associateprofessor at theUniversity of Illinois at Urbana-Champaign.He has been a researchfellow at the Mittag-LefflerInstitute and the Institutdes Hautes Etudes Scientifiques.His researchinterests have centeredon the use of geometricfunction theory in the studyof Riemann surfaces,Teichmiller theory, and Kleinian groups.Currently he is studyingthe structureat infinityof hyperbohcmanifolds.-Editors

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