Sunday, September 9, 2007
Sofia Kovalevskaya (1850-1891)
Many who have had an opportunity of knowing any more about mathematics confuse it with arithmetic, and consider it an arid science. In reality, however, it is a science which requires a great amount of imagination.
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Sofia Kovalevskaya was the middle child of Vasily Korvin-Krukovsky, an artillery general, and Velizaveta Shubert, both well-educated members of the Russian nobility. Sofia was educated by tutors and governess's, lived first at Palabino, the Krukovsky country estate, then in St. Petersburg, and joined her family's social circle which included the author Dostoevsky. Sofia was attracted to mathematics at a very young age. Her uncle Pyotr Vasilievich Krukovsky, who had a great respect for mathematics, spoke about the subject. Sofia wrote in her autobiography:
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. She noticed that 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, Y I Malevich, that Sofia undertook her first proper study of mathematics, and she says that it was as his pupil that 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 Bourdeu's Algebra which she read at night when the rest of the household was asleep. A year later a neighbour, Professor Tyrtov, presented her family with a physics textbook which he had written, and Sofia attempted to read it. She did not understand the trigonometric formulas 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 developed historically. Tyrtov argued with Sofia's father that she should be encouraged to study mathematics further but it was several years later that he permitted Sofia to take private lessons. Sofia was forced to marry so that she could go abroad to enter higher education. 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. At the age of eighteen, she entered a nominal marriage with Vladimir Kovalevski, a young palaeontologist. This marriage caused problems for Sofia 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. In 1869 Sofia travelled 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 and, according to the memoirs of a fellow student, she
immediately attracted the attention of her teachers with her uncommon mathematical ability. Professor Konigsberger, the eminent chemist Kirchhoff, .... and all of the other professors were ecstatic over their gifted student and spoke about her as an extraordinary phenomenon.
In 1871 Kovalevskaya moved to Berlin to study with Weierstrass, Konigsberger's teacher. 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. 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 Gottingen University. Despite this doctorate and letters of strong recommendation from Weierstrass, Kovalevskaya was unable to obtain an academic position. This was for a combination of reasons, but her sex was a major handicap. Her rejections 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.
In 1878, Kovalevskaya gave birth to a daughter, but from 1880 increasingly returned to her study of mathematics. In 1882 she began work on the refraction of light, and wrote three articles on the topic. In 1916, Volterra discovered that Kovalevskaya had made the same mistake as Lame, 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 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 privat docent. She began to lecture there in early 1884, was appointed to a five year extraordinary professorship in June of that year, and in June 1889 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. 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 Memoire sur un cas particulier du probleme de le rotation d'un corps pesant autour d'un point fixe, ou l'integration s'effectue a l'aide des fonctions ultraelliptiques du temps. In recognition of the brilliance of this work the prize money was raised from 3,000 to 5,000 francs. 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 theoreme de M. Bruns in which she gave a new, simpler proof of Bruns' 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.
An Alternate Biography Based on a project by Tom Burslem, University of St. Andrews
Born in Moscow on January 15th, 1850, Sofia was the middle child of Vasily Korvin-Krukoskii and Velizaveta Schubert. Vasily was an artillery general from the Russian Nobility and Velizaveta was from a family of German scholars who had settled in Russia in the 18th Century, so both had benefited from a good education. When Sofia was born she was a disappointment to her parents as they had waited six years from the birth of her sister Aniuta for the arrival of a son and heir. Her elder sister had attention lavished on her as the first born and her mother never considered Sofia as attractive or as socially acceptable as her sister. So when the family had guests at the house, it was always Aniuta who was brought out to meet them. The family nurse, Praskovia, recognised the rejection of Sofia by her parents and consequently took the girl under her wing. Sofia was a strong willed youngster and there are stories of her resorting to her fists when she did not get her own way, this strong character would help her later in life as she faced many obstacles, not least the prejudices against women in society. Five years after Sofia's birth a son arrived and he received even more attention than his sister Aniuta. When Sofia was eight the family moved to Palibino. Upon moving there, Vasily had more time to spend at home and decided that Sofia should be educated before she turned into the spoiled and ignorant girl her sister had become. So Sofia was taken out of the nurse's charge and put into the care of the family governess, Margarita Frantsevna Smith (see right). The governess was a strict English woman who was raised in Russia. She was forbidden to inflict punishment in the household, so would instead humiliate Sofia before the family and servants. Sofia would be given a yellow ticket that spelt out her misdemeanour and made to wear it to dinner. Sometimes Sofia was caught reading books from the family library that weren't assigned to her and on hearing this news her father would have to reprimand her to protect the governess's authority.
In the same year that the family moved Vasily decided to get Sofia a tutor to give her a broader education. At the time her sister was too much of a headstrong young lady for a return to the classroom and her brother was not old enough. Vasily's decision was not the norm in all families at the time, most noblewomen were educated in the arts of music, needlecraft and painting and also in French, but it was very rare for women to be educated in the sciences. An arrangement such as that was not satisfactory for a family with such a broad range of academic interests as the Korvin Krukovskys. Something else that may have influenced Vasily's decision to educate Sofia was his acquaintance with Dr Nicholas Pirogov, a surgeon during the Crimean War. Pirogov, now rector of Kiev University helped start the debate over education for women. He was a man fed up with education that turned upper class girls into dolls, dressed up and put on display. He thought that women should also be challenged academically as intelligent beings, though his reason was partly so that they could become better wives and mothers. So Vasily enlisted the help of Joseph Ignatvich Malevich as Sofia's tutor. Malevich's previous experience had been almost exclusively in preparing boys for military, civil or academic careers. However he did not vary his teaching methods greatly for his female pupils and it may well have benefited them to be treated as equal to their male counterparts. Malevich was to be Sofia's tutor from the age of eight to seventeen. During this time he taught her a broad range of subjects in such diverse fields as penmanship and mineralogy. Often new topics would be discussed at length with other members of the family, and encouraging Sofia to participate in these discussions helped her to develop an admirable skill in being able to argue her views clearly, concisely and convincingly.
It was also during this time of education that there was another great influence in her life, that of Pyotr Vasilievich Krukovsky. Pyotr was her father's elder brother and had escaped a life in the military due to his feeble constitution. He was only educated to secondary level and had no formal training in mathematics, but was enthusiastic and loved to question the things he read. Pyotr was rather like a grandfather figure to Sofia, he had time to lavish on this middle child, envious of the attention and privileges that her siblings received, playing games with her and telling her stories. He would play chess with the young girl and discuss at length a wide range of subjects, many scientific, treating her as if she were an adult. Although the young Sofia could hardly grasp many of the ideas that he talked about, he sparked an interest in mathematics that would stay with her throughout her life. In her words:
Although he had never studied mathematics, he cherished the most profound respect for that science. He had gathered a certain amount of mathematical knowledge from various books, and loved to philosophise about them, on which occasions he often thought aloud in my presence. I heard from him for the first time, for example, about the quadrature of the circle, about asymptotes which the curve always approaches without ever attaining, and about many other things of the same sort- the sense of which I could of course not understand as yet, but which acted on my mind, imbuing me with a reverence for mathematics, as for a very lofty and mysterious science, which opened out to those who consecrated themselves to it a new and wonderful world not attained by simple mortals.
Not just concerned with science, Pyotr read with equal enthusiasm books on travel, history and politics. He took a special interest in politics and was full of many social and economic ideas that he dreamt might benefit mankind. His passion for politics no doubt played some part in the political beliefs that Sofia came to hold as an adult. Another of her earliest memories of a curiosity for mathematics was also due, partly to her uncle Pyotr's influence. Upon moving to Palibino the family redecorated their new home, and as there were many rooms in their massive house, there was not enough paper to cover one of them, the children's nursery. It was considered too much effort to order paper all the way from Petersburg just for one room and so eventually a few years later, in an effort to improve it Vasily covered the walls with sheets of Ostrogradski's lithographed lectures that he had used in his youth. The lectures were on differential and integral analysis and Sofia upon reading them recognised things that her uncle had talked about. This could be described as Sofia's first encounter with calculus. Sofia attributed much to the time with her uncle, and in her Recollections placed little importance on the tutoring provided by Malevich, much to his annoyance. As Sofia progressed in her tutoring, her enthusiasm for maths grew and grew, though it took a while for Malevich to notice her growing interest. It grew such that she started to neglect her other studies, and on hearing this, according to Sofia, her father decided that she had progressed far enough in her mathematical studies and put an end to them (though in Malevich's recollections Vasily was pleased with this news). This was not enough to stop the headstrong young girl and she would sit up late at night reading books on mathematics while the rest of her family slept.
It was after her father put an end to her studies that a neighbour, Professor Tyrtov brought around to the house a copy of his new book on Physics as a gift to the family. Sofia reading the book didn't understand the trigonometric functions as neither she nor her tutor had ever encountered them. She attempted to reason through the functions herself replacing values of sine with the chord of an arc. With small angles these values almost coincide and luckily Tyrtov had used only small angles. When she told Tyrtov that she had read his book with great interest he thought that she was boasting, but when she discussed with him the methods she had used of arrive at the solutions of the formulae, Tyrtov was astounded. The steps she had followed were similar to those used in the first explanations of trigonometric functions. He immediately went to Sofia's father to convince him of the need for Sofia to be taught mathematics to a higher level. He went as far a to compare her to Pascal. This pressure eventually resulted in the hiring of Alexander Nikolayevich Strannoliubsky as a tutor for mathematics when the family visited Petersburg. Strannoliubsky was an accomplished teacher who came to be considered the foremost Russian teacher of mathematics at the end of the nineteenth century. It is thought that Malevich was not made aware of this appointment in order to spare him of the acknowledgement of his own inadequacies in mathematics. When Sofia was eighteen the family moved to Petersburg to further educate her and her brother. Here she continued to study under Strannoliubsky. He was just twenty-seven when he first started tutoring Sofia. In their time together they covered topics rapidly and her unusual aptitude for the subject spurred him into attempting to further the movement for women's education. Convinced by this young lady's abilities he went on take an active role in organising gymnasium schools for women and helping to raise the money necessary to maintain them. During this time Sofia's elder sister Aniuta was growing and maturing. The friendships she had developed with a number of young men during this time helped to bring her into contact with new political ideas and she took an interest in many of these, ordering books so that she could understand them further. Aniuta developed what would be a lifelong friendship with the great novelist Fedor Dostoevsky (see right). She sent many letters to him and impressed by her writing he replied. Dostoevsky met Aniuta and would talk to her at length, while Sofia watched them taking in all that he said. He was very much older than Aniuta but he fell in love with her and proposed to her, only to be rejected. However, he recognised Aniuta's literary talents and encouraged her to pursue them further. This no doubt helped her to mature and also develop a friendship with her younger sister, in turn helping to plant new ideas into the young girl's mind.
Having spent time studying with Strannoliubsky, it became clear that Sofia was ready for a wider education. Sofia wanted to take her study further, looking to enrol in university to study mathematics, but as a young woman she needed her father's permission to travel. Unfortunately Russian universities were closed to women at the time, as indeed were most universities around the world. The general argument against the academic training of women was that a free choice of education would produce competitive and aggressive women, lacking the skills that would enable them to be refined in the home and in society. It wasn't that there hadn't been any debate previously. As early as 1697, Queen Anne had tried to open English universities to women, but was opposed by her advisors.
While Sofia's learning was blossoming her sister was becoming acquainted with many young women in nihilist circles and was beginning to entertain ideas of a university education herself, even though she had not received the formal training necessary to embark on one. With her tales of women entering nominal marriages in order to travel abroad and study, she gave Sofia some hope that she might be able to get the education she wanted. The plan formed that Aniuta would marry and then Sofia would be allowed to travel around chaperoned by her and her husband. The first attempt failed, with Aniuta's proposal being politely declined. As the girls became more deeply involved with nihilism, they met more people sympathetic to their cause. One such person was Maria Bokova, whom Sofia enlisted to help find a partner for Aniuta. Maria found Vladimir Onufrievich Kovalevsky, who knew the standing of the Korvin-Krukovskys, and on this basis agreed to help Aniuta, even though they had never met. All did not go exactly as planned however and on their first meeting neither party was impressed with the other. Aniuta usually energetic and talkative was rather quiet and withdrawn, Sofia on the other hand was lively and conversational and it was also apparent that she was serious in her intent to learn. A few days later the news came that Vladimir would not marry Aniuta, due partly to the fact that she was so hesitant in accepting his offer and also that she lacked the courage to tell her father of her intentions to marry. Nevertheless, so impressed by her sister Vladimir expressed his wish to marry Sofia. This must have been a fantastic boost for Sofia's confidence after years of living in the shadow of Aniuta's beauty, she had actually been chosen above her.
This was not the complete solution to the problem though. It was still customary at the time for the older daughter to be married first, and they would need to persuade their father to permit Sofia's marriage. Vasily was diplomatic with Vladimir when he revealed his intentions, but made it clear that he would have to wait. Thinking that this would be enough to deter him Vasily considered the situation to be dealt with and in doing so underestimated his strong willed daughter. So intent was Sofia on leaving home and continuing with her studies that she decided to take firm action. She chose the evening of a dinner party at the house to make her move, she did not arrive at the table, leaving the message that she was busy preparing her wedding plans with Vladimir. In the company of his guests Vasily had no choice but to feign knowledge of this to prevent a scene, and by doing so he made a public blessing of the union that could not easily be retracted. Sofia was at last free to pursue her further education. The couple were eventually married on September 27th 1869 and had a blessing in a Russian Orthodox church. Three days after the wedding the Kovalevskys arrived in Petersburg.
Soon after arriving in Petersburg Sofia made acquaintance with a woman who had her own gynaecological practice and by this was inspired to take up medical studies herself, with the view to becoming a physician. She met with members of St Petersburg University to discuss attending lectures and it was agreed that she would be permitted, though it would be unofficial and she would not be able to obtain a degree from the university. She was heartened by her time at Petersburg and by the reaction of other students in the classes, there was no bad feeling towards the female students, their attendance not provoking any hostilities. Nevertheless, she was to grow tired of her studies, longing to concentrate on mathematics, so from Petersburg she headed to Vienna to investigate the possibilities of study there.
The feedback in Vienna was very positive and she was offered a place on the mathematics courses. However Sofia felt that the quality of the courses on offer could not match the quality of those at Heidelberg. So she went to Germany to investigate this further, leaving Vladimir behind in Austria in case she was unsuccessful. Arriving in Heidelberg, Sofia was disappointed to find that the University did not admit women and would not allow her to attend lectures, so from her she travelled to Berlin.
The initial reaction that she received on approaching the faculty in Berlin was very unwelcoming and it was made very clear that women no matter how gifted, were under no circumstances to be admitted. If anything this frosty reception gave the impression that Berlin was more backward in its attitudes than Heidelberg. However all was not lost, as while in Berlin Sofia was fortunate enough to meet Weierstrauss, who was so impressed at how prepared the young woman was that he decided he would take her under his wing and give her personal tuition. Together they worked hard and she progressed very well, but Weierstrauss did not consider the possibility of Sofia obtaining a doctorate, because as far as he knew she did not need to pursue a career because she was a married woman. When he discovered the truth behind Sofia's marriage he encouraged her to try and obtain a doctorate and helped her in her preparations.
As Berlin was effectively closed to her Sofia approached Gottingen University instead and they allowed her to try for a doctorate. However instead of requiring one dissertation, as was usual for men, they required three. Sofia duly obliged and produced three dissertations for them of a standard so high that she was not required to defend them or to sit an exam, as was the usual procedure. In fact her first thesis was to be published in Crulle's Journal fur die reine und angewante Mathematik that same year. The journal was perhaps the most serious mathematical publication in Germany at the time, and for a novice to have an article published was no mean feat.
In 1874 Sofia and her husband returned to Russia to search for academic posts, despite their excellent degrees and strong references neither of them were successful. Disheartened by their rejection from the academic world they turned their back on it and followed a quite different path. Vladimir embarked on a financial enterprise, while Sofia turned to writing. She found a job on a newspaper and wrote on a wide range of subjects, from theatre performances to the workings of mechanical objects. She also helped to set up higher education courses for women, with the intention that they would form the basis for a women's university. However despite having helped to set these up, when she offered to teach the courses unsalaried, her offer was rejected.
It was during this time in Russia that the couple decided to consummate their marriage. The reasons are unclear as to why they chose to validate a relationship that was based on convenience and many reports suggest that Vladimir had hoped the marriage would develop into a loving one. It is clear that they were going through many problems and would persistently row and argue when in each other's company. Perhaps they thought a child would make things better, or perhaps they felt guilty at deceiving their parents and thought they ought to try to make the marriage work. Whatever the reasons were, the result was the birth their first daughter Sofya Vladimirovna in October of 1878. For two years, all of Sofia's efforts were put into raising the infant and this was frustrating to a woman with a desire to dedicate herself completely to mathematics.
The turning point would come in the January of 1880 with the Sixth Congress of Natural Scientists in Petersburg. P I Cheyshev, was a friend of the Kovavlevskys and invited Sofia to give a paper at the congress. Six years had elapsed since Sofia had done any serious mathematical work and it was with reluctance that she finally agreed. The paper she decided to present was a dissertation on Abelian integrals that was amongst the three she had presented for her doctorate in 1874. She took a night to translate the paper into Russian and presented it the following morning. The response was fantastic, although six years old the paper was still fresh and introduced a new way of approaching the problems. Among those at the listening to her presentation was Gosta Mittag-Leffler a fellow student of Weierstrauss. He talked positively to her about the chances of finding her a place in the faculty of mathematics at Helsingfors in Helsinki. Suddenly Sofia had been reinstated as a serious figure in the academic world.
The change in Sofia's fortunes came at an awful time for Vladimir, whose financial enterprises had collapsed. He had ploughed all of Sofia's inheritance into failed plans and had now reached rock bottom. People around him were insensitive and unsympathetic, not realising the full extent of his depression and that it was a symptom of the medical illness from which he was suffering. He searched for a university post in an attempt to try and restore his academic career, but was initially unsuccessful. When he was offered a job with a petroleum firm, his dreams of financial success got the better of him and he accepted the position. Shortly after this he was offered a palaeontology post at the University of Moscow. Foolishly he took on the academic post whilst also holding his position with the oil company.
While her husband set about juggling two careers Sofia visited Weierstrauss in Berlin to discuss how to make up for lost time. She immersed herself in work, inspired by the thought of Mittag-Leffler's promising words. However, she was to discover that her chance of teaching at Helsingfors had fallen through and she was in the same position as before. Still she continued with her efforts in mathematics, working in Berlin and Paris on research into the refraction of light in crystals. Unfortunately, there was to be another disturbing event in her life. Frustrated at his inability to create any original work and with a scandal emerging at the oil firm where he worked, Vladimir took his own life by inhaling a bottle of chloroform. On hearing the news Sofia blamed herself and locked herself inside for days in attempt to starve herself to death. Luckily she was unsuccessful and on recovering was to immerse herself in her work.
Soon after her recovery Sofia was contacted by Mittag-Leffler who was now the rector of Stockholm University, with the news that the way was now open for her to come and teach there, with the one catch being that she could not become a member of the faculty. Stockholm was a university that had been set up out of necessity and with private money. The aim was free education without discrimination and the freedom to attend the lectures you wanted rather than study a narrow core syllabus. Sofia arrived apprehensively in Sweden in 1883, worried at her lack of preparation to become a lecturer. These apprehensions were proven to be completely unfounded when her lectures were a huge success and by the end of her first year she had been appointed as a full professor, published her work on the refraction of light, and had even become fluent enough to lecture in Swedish. The post was fantastic for her, giving her the chance to work on mathematical problems and use her gift of teaching and in 1885 she was appointed Chair of Mechanics.
While in Stockholm she developed a close friendship with Mittag-Leffler's sister Anne-Charlotte. Anne-Charlotte encouraged Sofia to follow a literary path and in 1887 they worked together on the play A Struggle for Happiness, and Sofia became completely absorbed in it, much to the annoyance of Mittag-Leffler who wanted Sofia to concentrate on her mathematics. However, in the autumn of 1887 Sofia's sister Aniuta died quite suddenly, leaving Sofia feeling desperately lonely stuck in a world where she could not express her true feelings as she did with her sister. In December of the same year the play was published and received very negative reviews. So when the French Academy of Science announced their new competition for the Prix Bordin in early 1888 it gave Sofia something new to immerse herself in, with the encouragement of Mittag-Leffler. The papers were to be on the problems of the rotation of a solid body around a fixed point under the influence of a gravitational force. This was a topic on which Sofia had done a lot of work on and it is thought that the problem may have been set with Sofia in mind.
It was as Sofia was making a start on her paper that Maxim Maximovich Kovavlesky appeared in her life. He was a distant relative of Vladimir and had arrived in Stockholm to deliver a series of lectures on sociology. He impressed Sofia tremendously and she wrote of him:
He's a real Russian from head to foot. And it's true that he has more intelligence and originality in his little finger than you could extract from both Messrs. X together, even if you put them under a hydraulic press.
Unfortunately his arrival came at the wrong time for Sofia's work and Mittag-Leffler encouraged him to retreat to Uppasala so that Sofia could complete her work. Nevertheless he proposed to Sofia, but with the condition that she would have to give up her work. Sofia felt that she had to finish her paper and turned him down, but she remained torn up by her feelings towards him and her panic to complete her work on time. She managed to successfully complete her paper and it was one of fifteen entries, all submitted anonymously. When the entries were read one was judged to be such an outstanding contribution towards the problem that the prize money was increased from 3000 to 5000 francs. When the contributions were matched to the contributors the outstanding paper was that of Sofia's and she was propelled even higher in the mathematical world. She attended the award ceremony with Maxim, but after the ceremony he left for Beaulieu by himself. Her contract at the University of Stockholm was due to expire in 1889, so Sofia sought to find a post in closer proximity to Maxim. She tried for posts in France and had high hopes of a position in Russia after being elected as a corresponding member of the Russian Academy of Sciences. These hopes became truly dashed when she received a letter form the Academy explaining that there were no places for women no matter how great their ability and learning that would be as distinguished and remunerative as the one she [Sofia] now occupies in Stockholm. The response from France was similar and so with regret at their separation, Sofia accepted a lifetime appointment in Stockholm. In the time after her appointment at Stockholm she took began to spend a lot of time writing, working feverishly on what had previously been more of a hobby. She wrote some radical political works including A Nihilist Girl and accounts of her own life. Some works were never published in Russia or had chapters removed due to their political content.
Sofia was never to receive the Russian academic post that she deserved. Over the Christmas of 1890 she met with Maxim at his Riviera Villa and arranged to meet on New Year's Day with Anne-Charlotte whom had never been introduced to Maxim. However due to a misaddressed telegram the meeting never took place, leaving Sofia and Maxim to spend the day together in Genoa, getting increasingly aggravated with each other. Sofia left for Stockholm and the resumption of lectures, stopping off in Denmark on the way. Arriving late at night, with no Danish money she could not pay a porter to take her luggage to her room, so in the rain she carried it all herself. On arriving in Sweden she took her classes even though she was quite ill. It was only when she was completely exhausted and fit to collapse that she sent for a doctor who misdiagnosed her as having kidney colic. By the time she was properly diagnosed with having pneumonia it was too late and the disease claimed her life within six days. She was taken from the world while at the peak of her academic powers and social standing, but her life would go on to influence many others all around the globe. [Adapted from MacTutor]
Books from Alibris: Sofia Kovalevskaya