Sofia Vasilyevna Kovalevskaya

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15 January 1850
Moscow, Russia
10 February 1891
Stockholm, Sweden

Sofia Kovalevskaya was a Russian born mathematician who made valuable contributions to the theory of differential equations. She finished her career in Sweden.


Sofia Kovalevskaya was the daughter of Vasily Vasilievich Korvin-Krukovsky (1801-1879), an artillery general, and Elizaveta Shubert, both well-educated members of the Russian nobility. Sofia's name is written in different forms and we should say a little about that here. Ignoring the problems of transliteration from Russian, her father's name was Krukovsky but he made applications to the Department of Heraldry to allow him to be considered a member of the nobility. This was refused until he retired in 1858 when his application was allowed and he became Korvin-Krukovsky, the name Korvin coming from Matthias Corvinus, the King of Hungary. Vasily had inherited two large estates in 1843, one at Palibino and the other at Moshino. Around the time he inherited these estates, he had married Elizaveta Fedorovna Shubert (1820-1879), the daughter of Fyodor Fyodorovich von Shubert and Sophie von Shubert. Elizaveta had been born on 15 December 1820, and was about twenty years younger than her husband. Let us note at this point that sources differ in giving Vasily Korvin-Krukovsky's year of birth, with dates ranging between 1800 and 1803. It is also worth noting that Elizaveta Shubert's grandfather was the astronomer and cartographer Theodor Friedrich Schubert (1789-1865) who has a crater on Mercury named after him. Sofia had an older sister Anyuta Vasilievna Korvin-Krukovskaya (1843-1887), and a younger brother Fyodor Vasilievich Korvin-Krukovsky (1855-1919). Fyodor went on to study in the Physics-Mathematics Faculty of St Petersburg University and then worked in a government ministry.

We need to also say a little about Sofia Kovalevskaya's name. She was given the name Sofia Vasilievna Krukovsky, only adopting the name Korvin-Krukovsky after her father's application for nobility was accepted in 1858. She is often called Sophie or Sonya, the first being an anglicised version of Sofia, the second being a familiar version by which she was known by her friends after she became an adult. Kovalevskaya is the female version of her husband's name Kovalevsky which is often transliterated as Kovalevskaia, and infrequently as Kovalevskaja. She is also known as Sonya Kovalevsky, using a masculine version of her surname, a form she sometimes used herself. We will use the form Sofia Kovalevskaya throughout this biography.

In 1855 Sofia's father, General Krukovsky, was posted to Kaluga, about 70 km south west of Moscow, and the family lived there until 1858. The family had a nanny and a governess for Anyuta who was twelve years old when they moved to Kaluga.

Sofia lived at Palibino, the Krukovsky country estate, from 1858 when he father retired, and was educated by tutors and governesses. Palibino was near the Lithuanian border and was a large estate with sheep and cattle, lakes stocked with fish, forests with game, and vegetable gardens. Sofia's father was fully occupied managing the estate and the family lived comfortably in the manor house.

It was at a very young age that Sofia was attracted to mathematics. Her uncle Pyotr Vasilievich Krukovsky, her father's elder brother, often visited Palabino. She writes about him in [62]:-
Although he was the oldest member of our family and should have been its head, the truth was that he was ordered about by anyone who felt like it, and the whole family treated him like an elderly child. He had long held the reputation of an eccentric and a dreamer. His wife had died some years before; he had handed over his entire and rather good-sized estate to his only son, leaving for himself only a very small monthly pension. Left thus without any definite business affairs to attend to, he used to come to Palibino often and stayed with us for weeks on end. His arrival was always regarded as a holiday, and the atmosphere at home became somehow livelier and cosier when he was with us.
Uncle Pyotr Vasilievich had a great respect for mathematics and often spoke about the subject. Sofia wrote in her autobiography [62]:-
The meaning of these concepts I naturally could not yet grasp, but they acted on my imagination, instilling in me a reverence for mathematics as an exalted and mysterious science which opens up to its initiates a new world of wonders, inaccessible to ordinary mortals.
When Sofia was 11 years old, the walls of her nursery were papered with pages of Ostrogradski's lecture notes on differential and integral analysis. This requires some explanation! Before the family moved from Kaluga to Palibino, they had the whole manor house redecorated. Wallpaper was ordered from St Petersburg but they had made a small error in working out how many rolls would be required and ended up one short. Rather than order one roll, they decided to paper the nursery with old sheets of paper and looked for some in their attic. Ostrogradski's lecture notes on differential and integral analysis were there because Vasily Vasilievich, Sofia's father, had attended his course when undergoing his military training. She noticed that there were certain things on the sheets she had heard mentioned by her uncle. Studying the wallpaper was Sofia's introduction to calculus.

It was under the family's tutor, Yosif Ignatievich Malevich (1813-1898), that Sofia undertook her first proper study of mathematics, beginning with arithmetic (which she found boring) and then moving on to elementary geometry and algebra. She writes that it was as his pupil that [62]:-
I began to feel an attraction for my mathematics so intense that I started to neglect my other studies.
Sofia's father decided to put a stop to her mathematics lessons but she borrowed a copy of Bourdon's Algebra Course which she read at night when the rest of the household was asleep [62]:-
Since I was under my governess's strict surveillance all day long, I was forced to practice some cunning in this matter. At bedtime I used to put the book under my pillow and then, when everyone was asleep, I would read the night through under the dim light of the icon-lamp or the night lamp. Under such circumstances, of course, I did not dare dream of continuing the systematic study of my favourite subject. My mathematical knowledge would likely have remained confined for a long time, to the contents of Bourdon's 'Algebra' if I had not been aided by the following incident, which motivated my father to reassess his views on my education to some degree.
The incident she refers to happened a year later when a neighbour, Nikolai Tyrtov, Professor of Physics at the Naval Academy, presented her family with a physics textbook which he had written, and Sofia attempted to read it. She did not understand the trigonometric formulae that she came across in the chapter on optics and attempted to explain them herself. Tyrtov realised that in her working with the concept of sine, she had used the same method by which it had been developed historically. Tyrtov argued with Sofia's father that she should be encouraged to study mathematics further but it was two years later before he permitted Sofia to take private lessons with Aleksandr Strannoliubskii (1839-1903) who had been a student of Tyrtov. These lessons on analytic geometry, and differential and integral calculus, took place when the family were in St Petersburg, where they spent some time each year visiting the Shubert aunts. There Sofia joined her family's social circle which included the author Fyodor Mikhailovich Dostoevsky.

We now need to say a little about Sofia's sister Anyuta at this point, because she was a strong influence on the young Sofia. In around 1863 Anyuta had become enthusiastic about the radical ideas that one of her friends, the son of the local priest, told her about when he was home in Palibino from his university studies during the vacation. Anyuta wanted to go to St Petersburg to live, and even proposed living in a commune where young people were living together without servants. Her father was not going to allow such behaviour and Anyuta had to remain at Palibino. She reacted by secretly writing and getting two pieces published under a male pseudonym. Well, it was done secretly from everyone except Sofia, for the two sisters kept nothing from each other. On 28 February 1865 the family went to St Petersburg. Dostoevsky came to visit and both Anyuta and Sofia would meet him. Usually they were not allowed to be alone with him but on one occasion their parents were out and the three were able to chat together. Sofia had written some poetry which was shown to Dostoevsky and he praised it. As the meetings progressed, however, Anyuta and Dostoevsky disagreed over her Nihilist views. Dostoevsky began to turn his attention more and more to Sofia and in turn she practised her piano playing to entertain him. Although Dostoevsky expressed a wish to marry Anyuta, as much as she admired him and thought he was a wonderful person, she felt they were not suited. The family returned to Palibino in April 1865.

By 1868, Anyuta had made contact with a radical group with Nihilist views when in St Petersburg. She believed her only way to travel and get an education was to marry and through the radical group, she and a friend were introduced to Vladimir Onufrievich Kovalevsky (1842-1883), with the idea that if he married one or other of them they could both "escape", one as a wife and the other chaperoned by a married couple. By this time Sofia was eighteen years old and on one occasion she came with her sister to meet Vladimir who, after that meeting insisted he would marry Sofia rather than Anyuta or her friend. What had been intended only as a nominal marriage changed when Vladimir allowed his feelings to take over. He wrote to Sofia:-
Meeting you makes me believe in the affinity of souls, so swiftly ... and genuinely did the two of us come together and, on my part at least, become friends .... Now I cannot keep from picturing much that is joyous and good in our common future. Indeed, employing the coolest possible judgment, without childish enthusiasm, one can state almost positively that Sofia Vasilievna will become a splendid doctor or scholar in some branch of the natural sciences ...
Sofia realised that by marrying Vladimir she could do what her sister had wanted and go abroad to enter higher education. If unmarried, her father would not allow her to leave home to study at a university, and women in Russia could not live apart from their families without the written permission of their father or husband. So, at the age of eighteen, she married the young palaeontologist Vladimir Kovalevsky in September 1868 having, with the greatest difficulty, received her father's permission. At first things went well for Sofia living in St Petersburg and attending the University there. She wrote to her sister back in Palibino:-
Sechenov's lectures begin tomorrow; and so my real life begins at 9 a.m. .... Vladimir Onufrievich and friends will solemnly escort me by way of the back stairs so that there is hope of hiding from the administration and from curious stares ...
Ivan Mikhailovich Sechenov (1829-1905) was Professor of Physiology at St Petersburg. Again Sofia writes to Anyuta:-
I forgot to tell you that Mechnikov promised to admit me to his lectures and get permission for me to attend the physics lectures .... I'm studying physiology and particularly anatomy; we got a skeleton from Pyotr lvanovich Bokov and brother is poking it at this moment ....
Two comments about this quote. Ilya Ilyich Mechnikov was a zoologist who at this time was a docent at St Petersburg. The "brother" she refers to is in fact her husband, who she always referred to as "brother", a clear indication of their relationship at this time.

Although her marriage was allowing Kovalevskaya to get the education she craved at this time, it soon caused her problems and, throughout its fifteen years, it was a source of intermittent sorrow, exasperation and tension and her concentration was broken by her frequent quarrels and misunderstandings with her husband. Although she was now allowed to attend university lectures, her real love was mathematics and she felt that for this she had to go to Germany.

In the spring of 1869 Kovalevskaya travelled with her husband to Heidelberg to study mathematics and the natural sciences, only to discover that women could not matriculate at the university. Eventually she persuaded the university authorities to allow her to attend lectures unofficially, provided that she obtain the permission of each of her lecturers. Sofia studied there successfully for three semesters attending lectures by Gustav Kirchhoff, Hermann Helmholtz, Leo Königsberger, and Paul Du Bois-Reymond and, according to the memoirs of a fellow student, she:-
... immediately attracted the attention of her teachers with her uncommon mathematical ability. Professor Königsberger, the eminent chemist Kirchhoff, ... and all of the other professors were ecstatic over their gifted student and spoke about her as an extraordinary phenomenon.
Following Leo Königsberger's advice, in 1871 Kovalevskaya moved to Berlin to study with Karl Weierstrass, Königsberger's former teacher. When she arrived, Weierstrass gave her some problems to test her mathematical skill and when she solved these in a week he was immediately convinced of her brilliance. Despite the efforts of Weierstrass and his colleagues the senate refused to permit her to attend courses at the university. Ironically this actually helped her since over the next four years Weierstrass tutored her privately. Soon after arriving in Berlin, she heard that Anyuta was in the Paris Commune, a radical socialist group that took over running Paris in March 1871. Sofia rushed to Paris where the Commune was being suppressed by the French army. Many were wounded and killed but the two sisters escaped uninjured. Naturally, this episode made a big impression on Sofia.

By the spring of 1874, Kovalevskaya had completed three papers. Weierstrass deemed each of these worthy of a doctorate. The three papers were on Partial differential equations, Abelian integrals and Saturn's Rings. The first of these is a remarkable contribution which was published in Crelle's Journal in 1875. The paper on the reduction of abelian integrals to simpler elliptic integrals is of less importance but it consisted of a skilled series of manipulations which showed her complete command of Weierstrass's theory.

In 1874 Kovalevskaya was granted her doctorate, summa cum laude, from Göttingen University after it received very positive reports from Karl Weierstrass, Paul Du Bois-Reymond, and Lazarus Fuchs. She was awarded the degree in absentia, writing to the Dean to explain her position:-
The very reverend Dean will graciously permit me to add something to the letter in which I present myself for admission to the degree of Doctor Phil, in the mathematical faculty ... It is only a wish to satisfy my dearest friends ... I wish to give them an incontestable proof that, in devoting myself to the study of mathematics, I am following the determined bent of my nature, and that, moreover, this study is not without result. At the same time I hope the very reverend Dean will not misconstrue me if I acknowledge openly that I do not know whether I have sufficient aplomb to undergo an examen rigorosum, and I fear that the unusual position, and having to answer, face to face, men with whom I am altogether unacquainted, would confuse me, although I know the examiners would do all they could for me. In addition to this, I speak German very badly.
Pozefsky writes in [83]:-
In 1874 the couple returned to St Petersburg, foreign doctorates in hand, and embarked on a utopian scheme. They believed that their knowledge of the sciences would allow them to engage in fail-safe investing and expected their new capital to permit them to study science in tranquillity for the rest of their lives. But the plans ended in disaster. They squandered their inheritances, fell deeply into debt ...
Vladimir never recovered from his episode and continued, unsuccessfully, with his "get-rich-quick" schemes. Sofia quickly recovered and wanted to continue her academic career but, now in St Petersburg, despite having a doctorate and letters of strong recommendation from Weierstrass, she was unable to obtain an academic position. This was for a combination of reasons, but her sex was a major handicap as was her enthusiasm for the nihilist philosophy. It is unclear how extreme her nihilist beliefs were for the biographies give somewhat different views. For example in [48] Kennedy says she was:-
... above all an aristocrat who definitely opposed revolution ... [and] an elitist who believed that a worthwhile society needs an elite intellectual class.
In [51] Koblitz paints a picture of Kovalevskaya as a radical ([28], [29]):-
Exposed to progressive thought in early adolescence, she embraced eagerly the ideas of the 'sixties, made no secret of her socialist sympathies even when they harmed her professionally, and while she never committed herself directly, she envied her radical friends their involvement in active political work, gave them her passport, accepted coded letters, and helped them out in small ways that entailed some risk.
See also Ann Hibner Koblitz's biography of Kovalevskaya at THIS LINK.

Her rejections, however, resulted in a six year period during which time she neither undertook research nor replied to Weierstrass's letters. She was bitter to discover that the best job she was offered was teaching arithmetic to elementary classes of school girls, and remarked:-
I was unfortunately weak in the multiplication table.
Unable to get a job, Kovalevskaya and her husband decided to live as man and wife rather than brother and sister. In October 1878, she gave birth to a daughter, Sofia Vladimirovna (who was called Fufa), but from 1880 increasingly returned to her study of mathematics. This was prompted by an invitation from Pafnuty Chebyshev to deliver an address to the Sixth Congress of Natural Scientists in St Petersburg in 1880. She delivered the unpublished paper on abelian integrals which she had written as part of her doctoral thesis. In 1882 she began work on the refraction of light, and wrote three articles on the topic. In 1891, Volterra discovered that Kovalevskaya had made the same mistake as Lamé, on whose work these papers were based, though she had pointed out several others which he had made in his presentation of the problem. The first of these three articles was still a valuable paper however, because it contained an exposition of Weierstrass's theory for integrating certain partial differential equations.

In the spring of 1883, Vladimir, from whom Sofia had been separated for two years, committed suicide after he was again in financial difficulties which led to a stock scandal. After the initial shock, Kovalevskaya immersed herself in mathematical work in an attempt to rid herself of feelings of guilt. Mittag-Leffler managed to overcome opposition to Kovalevskaya in Stockholm, and obtained for her a position as privatdocent. To prove her competence, she had to lecture for a year without pay and without an official position. Having no other option, she reluctantly agreed. She began to lecture there in early 1884, giving a course in German on partial differential equations. Her lectures were well received and she was appointed to a five year extraordinary professorship in Higher Analysis in June of that year on condition that she lecture in Swedish, which she did. Not everyone was happy, for example August Strindberg (1849-1912), the famous author, wrote in a local paper [76]:-
A female professor is a pernicious and unpleasant phenomenon - even, one might say, a monstrosity.
Kovalevskaya had left her daughter in Moscow and only saw her during the summer of 1884, but in 1885, when Fufa was eight years old, she brought her to Stockholm. In June 1889 she became the first woman since the physicist Laura Bassi and Maria Gaetana Agnesi to hold a chair at a European university.

During Kovalevskaya's years at Stockholm, she carried out what many consider her most important research She taught courses on the latest topics in analysis and became an editor of the new journal Acta Mathematica. She took over the task of liaison with the mathematicians of Paris and Berlin and took part in the organisation of international conferences. Her status brought her attention from society, and she began again to write reminiscences and dramas that she had enjoyed doing when young. Karen Rappaport writes [84]:-
Sofia lived for a time in Mittag-Leffler's home, where she met and developed a friendship with his sister, Anna Leffler. Anna Leffler, a well-known advocate of women's rights and a writer, encouraged Sofia's literary leanings. In 1887 they collaborated on a play entitled 'The Struggle for Happiness'. It was based on an idea that had occurred to Sofia while she sat at the bedside of her dying sister. After Anyuta died in the fall of 1887, Sofia felt lonely and despondent. The sisters had been close, and Sofia felt the loss deeply.
The topic of the Prix Bordin of the French Academy of Sciences was announced in 1886. Entries were to be significant contributions to the problem of the study of a rigid body. Kovalevskaya entered and, in 1886, was awarded the Prix Bordin for her paper Mémoire sur un cas particulier du problème de le rotation d'un corps pesant autour d'un point fixe, où l'intégration s'effectue à l'aide des fonctions ultraelliptiques du temps . There had been fifteen papers entered for the prize. In recognition of the brilliance of this work the prize money was raised from 3,000 to 5,000 francs. Entries for the prize were supposed to be anonymous and many biographies of Kovalevskaya claim that she only won because the judges did not know the entry was from a woman. This, however, is totally false for the topic of the prize was especially chosen to fit in with her research expertise. This was not done just for her, for often the prize topic was chosen to encourage an outstanding young mathematician. She accepted the prize in person and the President of the Academy of Sciences, in his speech of congratulation, said [76]:-
Our judges have found that her work bears witness not only to profound and broad knowledge, but to a mind of great inventiveness.
Kovalevskaya's further research on this subject won a prize from the Swedish Academy of Sciences in 1889, and in the same year, on the initiative of Chebyshev, Kovalevskaya was elected a corresponding member of the Imperial Academy of Sciences. Although the Tsarist government had repeatedly refused her a university position in her own country, the rules at the Imperial Academy were changed to allow the election of a woman.

Kovalevskaya's last published work was a short article Sur un théorème de M Bruns in which she gave a new, simpler proof of Bruns's theorem on a property of the potential function of a homogeneous body. In early 1891, at the height of her mathematical powers and reputation, Kovalevskaya died of influenza complicated by pneumonia. She had travelled to France and had fallen ill shortly after returning to Stockholm. The doctors made an incorrect diagnosis of her illness and by the time they realised she was suffering from pneumonia, it was too late to save her. She was buried in the Northern Cemetery of Solna, just to the north of Stockholm.

Weierstrass said that Kovalevskaya could [51]:-
... show through deeds that women have been alienated from the highest strivings of mankind because of prejudice.
Kronecker described her as "one of the rarest investigators" while Mittag-Leffler, speaking of Kovalevskaya as a teacher, said:-
We know with what inspiring zeal she explained [her] ideas ... and how willingly she gave the riches of her knowledge.
Koblitz writes [51]:-
At the time of her death, Kovalevskaia was indeed considered the equal of anyone of her generation. This included Poincaré, Picard, and Mittag-Leffler.
Let us end by showing something of the prejudice that Kovalevskaya suffered by quoting from the 1895 review of 1895 publication of A Russian childhood [3]:-
With all these adventures and successes, her life was a melancholy failure, and she knew it to be such. Even her scientific achievements were but the masterly working out of ideas derived from her teachers. It is difficult to conceive how she could have been more cruel and unregardful of her parents and of her child. Her intense passionate desire was for two things which Balzac strove for all his laborious years - to be famous, and to be loved. She attained both, as he did, to the uttermost. But her life ended, as it began, in wretchedness; while his was nobly satisfied. The man had cast out selfhood; the woman fastened the demon of self-will in her very vitals.

References (show)

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    See THIS LINK.
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Written by J J O'Connor and E F Robertson
Last Update January 2021