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 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 not-for-profit 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 [email protected]. Mathematical Association of America is collaborating with JSTOR to digitize, preserve and extend access to The American Mathematical Monthly. http://www.jstor.org 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,perturbation theory, and partial differential equations. -Editors 564 This content downloaded on Mon, 25 Feb 2013 16:08:52 PM All use subject to JSTOR Terms and Conditions S. KOVALEVSKY: A MATHEMATICALLESSON 565 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 Karl Weierstrass,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