Maria Gaetana Agnesi – The Middle Ages/Renaissance Mathematics
For 1500 years Hypatia was considered the only woman scientist in history. Her death was followed by a general decline in women’s education until 1453 when the Turks took Constantinople.
Christianity engulfed most of Europe which, as Hypatia found, frowned upon education for females, even the most basic skills such as reading and writing were sometimes forbidden. It was claimed such skills would provoke temptation and sin within females. Small intellectual centres did develop in Italy, Gaul, Britain, Ireland and Germany, but even these enlightened centres disapproved of higher education for women to some extent. It was only the start of the Renaissance that changed this viewpoint.
The only way females could access education was in monasteries and nunneries, so once again religion played a dominant part in the access to mathematics. Knowledge was guarded by great secrecy in such institutes, there was no way it could be accessed outside of the religion. Therefore to study mathematics and other such topics you had to subscribe to Christianity, quite a commitment!
Among the nuns educated at this time there were two in particular that displayed skills in mathematics. These were Hroswitha, a tenth century nun from the Benedictine Abbey in Saxony, and Saint Hildegard, Abbess of Bingen on the Rhine in the twelfth century. Hroswitha’s writings have become important references to the monastic mathematics of this era. She demonstrated knowledge of Greek/Boethian arithmetic, and the use of perfect numbers. Hildegard wrote many treatises on science, and she was granted recognition of these and her mathematical capabilities. She was supposed to have foreseen Copernicus by centuries when she claimed that the sun was the centre of the heavens.  
Both nuns would not take credit for their own writings instead claiming the works came directly from God. Hroswitha would humbly apologise for her texts making excuses for their ‘clownishness’. This demonstrates how ridiculed and unimportant women felt in this era. They could not speak with authority, but had to justify their thoughts by the authority of God. It is believed many medieval women adopted this tactic.
As mentioned in chapter one, the clergy had chastity enforced among them at this point. This meant any priests that were/or had been involved with women were de-robed, imprisoned, sometimes even killed, and their land was taken by noblemen. As the first universities (Bologna 1190, Paris 1200, and Oxford 1210) were solely training institutes for clergy, it was decided that academics teaching within these should also be celibate. This caused a great hatred of women from the clergy and academia. The archdeacon of Oxford, Walter Map, wrote a very popular anti-matrimonial treatise that highlighted the extreme negative attitude towards females. The treatise was in the form of a letter to a friend trying to persuade him not to marry, in it he wrote –
“Women, journey by widely different ways, but whatever windings they may wander, and through however many trackless regions they may travel, there is only one outlet, one goal of all their trails, one crown and common ground of all their differences – wickedness.” 
This enforcement of chastity on academia also meant that no daughters transpired, and professors could not share their knowledge and nurture potential women academics like Hypatia of Alexandria. Even this access to education was cut off.
Universities were the only training places for mathematics, so many women were excluded for centuries in the development of it.
After the fall of Constantinople in 1453 many great scholars flocked to the Italian Peninsula. Their ideas and knowledge began the Renaissance. At this point Italian women began a revolt against the male dominance in academia. The universities accepted them and gladly awarded them doctorates, professorships, and even teaching roles. As a result men never ridiculed them, they would only respect them for their motivation to advance their already obtained knowledge. Any man that did express an opinion against their achievements/activities would ensue a backlash of women writers defending their capabilities . Italian women had opinions and certain attitudes, and they weren’t afraid to openly air them!
In Italy women were treated completely as equals – a far cry from the recent years and the rest of the World. The status of women in the rest of Europe did eventually change, but unfortunately it took a lot longer. France and Germany saw a revival of anti-feminism that was indicative of the ancient Greeks and Romans. In England the standard education system for both sexes were adapted to a male only policy by Henry VII and Elizabeth I did not do much to advance this. Therefore women were left without any standard education for a long time.
In the early seventeenth century certain revelations looked to aid the case for women’s equality. For instance the famous mathematician Descartes set upon a mission to prove Aristotle incorrect, i.e. mathematics was the key to unlocking the secrets of nature. His thorough approach proved him much recognition, and his theory that mind and matter were completely separate was a great feat in the argument for women’s equality. If mind and matter were distinct then this challenged the Aristotelian view that women were mentally inferior because their bodies were less perfect than men’s, i.e. nothing about the female body could influence how great their minds were – that was a separate entity. Although Descartes supported several independent women he never declared that his theory defended the female intelligence . Many philosophers demonstrated enlightenment towards women, but none of them, like Descartes, would commit to an opinion on the subject of equality. Cleverly keeping a very neutral position on such a hot political topic, although all the same it did question the stale viewpoints of Aristotelians. One man that did commit himself was François Poullain de la Barre. After being taught women were ‘monsters’ he embraced Cartesianism and furthered Descartes’ theory to the social domain, and declared that the brain was sexless and therefore women were very capable of academics. Unfortunately Cartesianism did not prove popular, as such opinions were rare in male academia.
During the second half of the seventeenth century Newton discovered the Universal Law of Gravity – the force that radiates out from every massive body and draws things toward it. He related this to how the moon is held in orbit around the Earth, and consequently it explained the orbits of the planets around the Sun. This equation was one of the most powerful equations in the history of science, uniting heaven and earth, and providing a secure mathematical grounding for heliocentrism. Newton then went on to discover the three laws of motion, which describe the action of all material bodies, not just celestial ones. Newton was a strong follower of the Anglican Church and he wanted his theory to be compatible with it, but also to reinforce it. He believed his discoveries would fill the bleak philosophies of Descartes, whose theorems were seen by the Public as a recipe for atheism (this was not the original intention!). To publicise his natural philosophy he looked to the clergy to help him, his text called the ‘Principia’ became an argument for God. This strong link between Newtonianism and religion helped to promote a ‘priestly’ view of the scientist – returning the social attitude that mathematical science is a sacred activity and therefore should be male only. Newtonianism was even associated with gender to enhance the male status, it was believed men could be thought of as planets fixed in their orbits, then women would be viewed as moons ‘naturally’ compelled to stay in orbit around their men . Therefore what Descartes did to enhance the intellectual status of women, Newton did the opposite!
Although Newtonianism left a bleak outlook for women wanting to enter academia, during the Renaissance period some women did manage to access such education and went on to achieve great accomplishments. Such ladies were Tarquina Molza, Maria Angela Ardinghelli of Naples, Clelia Borromeo of Genoa, Elena Cornaro Piscopia, Laura Bassi and Diamente Medaglia. One of the most remarkable ladies came from the intellectual centre of Italy where, as mentioned earlier in this chapter, women’s intellectual status was equal in comparison to their male counterparts. This lady was Maria Gaetana Agnesi, she is classed as one of the most extraordinary women scholars of all time.
Maria was born in Milan on 16 May 1718, she was the eldest of twenty-one children! Her father, Don Pietro Agnesi, had three wives and came from a wealthy family. Some sources say that he was a professor of mathematics at the University of Bologna , but other sources declare that the family made their money from silk, and Pietro was not a professor at all .
Maria was recognised as a child prodigy from an early age by Pietro and her mother Anna Brivia. They employed the best tutors available (young men of learning from the church) to teach her. She spoke French by the age of five, and Latin, Greek, Hebrew and other modern languages by the age of nine. At this tender age she also translated and delivered a discourse in Latin defending the higher education of women to a group of academics who were invited to the family home. Some sources claim that Maria actually composed this text , but it is believed by others that the article was written by one of her tutors in Italian and Maria just translated and presented it . This topic did have an impact on her though, and it became an important feature of her later life.
During her teenage years Maria continued her education, specifically mathematics, where she mastered the current day theories of Newton, Leibniz, Fermat, Descartes, Euler and the Bernoulli brothers. She also taught her younger brothers, and contributed to her father’s intellectual gatherings where scholars would visit the house to discuss current views/opinions. In 1738 she published a series of essays on philosophy and natural science called ‘Propositiones Philosophicae’. It contained one hundred and ninety-one philosophical theses, which Maria had, and would continue to, dispute with the specially invited guests visiting her father. Indeed this sounds a strange situation, with Pietro showing off his daughter like a circus act, but it was quite common in this era. Monsieur Charles De Brosses, the president of the parliament of Burgundy, attended one of these meetings with his nephew. Apparently at this particular meeting there were approximately thirty people from several different nations of Europe, seated in a circle, questioning Maria. De Brosses said of Maria’s abilities –
“ She is much attached to the philosophy of Newton, and it is marvellous to see a person of her age so conversant with such abstract subjects. Yet however much I was amazed at her learning, I was perhaps more amazed to hear her speak Latin with such purity, ease and accuracy…” 
Although Pietro was understandably proud of his daughter’s performance at such gatherings, Maria did not enjoy participating in these. She was naturally shy, and although always obedient to her father’s wishes, she did ask to be excused from these duties when she was twenty years of age. Then she proposed an even bigger request, she wanted to enter a convent and become a nun, so that she may continue her study in seclusion and continue her work with the poor. Her father was horrified that his dearest daughter would want to leave him, and he denied the request. Maria stayed at home on three conditions, the first was that she could attend church whenever she wished, the second was that she could dress simply and humbly, and the third was that she would not be forced to attend leisure pursuits such as balls, theatre visits etc.
After her mother’s death Maria concentrated her efforts on studying religious books, learning mathematics, and caring for her younger brothers and sisters. She wrote a commentary on de L’Hôpital’s ‘Traite analytique des section coniques’ but it was never published. To advance her mathematical learning she was fortunate enough to meet Ramiro Rampinelli, a monk, mathematician and a professor at Rome and Bologna. He helped her study Reyneau’s calculus text ‘Analyse démontrée’. After encouragement from Rampinelli she began to write her own book on differential and integral calculus, initially as a teaching text for her siblings, but it grew to a more serious large two-volume text that took ten years to create. The book was written in Italian and called ‘Analytical Institutions’ (or ‘Istituzioni analitiche ad uso della gioventù italiana’). She controlled and monitored every process of its production, with her father’s wealth she even paid for it to be privately printed in her home so she could observe every step. In 1745 she wrote to Rampinelli’s teacher, Riccati, for the final draft to be checked and suggestions to be made. Riccati also got his two sons to check it and he sent Maria some of his earlier work on integration to be included in the book. Maria sent the book in parts to Riccati, who took a while to respond, resulting in the extreme delay in it’s production. In 1748 the first volume was published, and then the following year the second volume was also published.
During the books production it came to light that Maria was a somnambulist. On several occasions after working all day on a difficult problem that she was unable to solve, she would go to sleep to then arise in a somnambulist state, make a light, go to her study, and solve the problem. When she awoke in the morning she was surprised to find the solution carefully worked out on the paper. This demonstrated her extreme natural talent for mathematics.
The final production of her book brought her much credit and acclaim. It was documented as one of the most important mathematical publications produced by a woman up until that time. It was the first comprehensive textbook on calculus since L’Hôpital’s book, and it was also one of the first and most complete works on finite and infinitesimal analysis. The book was not superseded until Euler produced his great texts later on in the century. The volumes were also translated into French and English and were widely used as textbooks.
The success of Maria’s book was due to her acquired knowledge of languages. With them she was able to collate various works from different sources and by different mathematicians, translate them, and allow the reader to learn of all developments and methods in just one book.
The first section of the book discusses the analysis of finite quantities, construction of loci, conic sections, and basic maxima, minima, tangent and inflection problems. The second section discusses ‘infinitely small quantities’. The third section discusses integral calculus and recent developments of it at that time, covering rules for integration, and power series. The fourth section discusses ‘inverse method of tangents’ and basic differential equations. This latter section of the book has become the most famous part. Maria discusses here a verses sine curve originally studied by Fermat and Guido Grandi.
The plane cubic curve has the cartesian equation xy² = a²(a–x). An approximate outline of it’s derivation is –
“Agnesi begins with the geometrical principle that if the abscissa of corresponding points on a curve is equal to that of a given semicircle, then the square of the abscissa is to the square of the radius of the semicircle in the same ratio as that in which the abscissa would divide the diameter of the semicircle.” 
This curve was known as ‘versiera’, derived from the Latin word vertere, which means ‘to turn’, but it is also an abbreviation for the Italian word avversiera which means ‘wife of the devil’ i.e. a witch. In 1801 Maria’s book was translated into English by John Colson, the professor of mathematics at Cambridge. Colson translated versiera as ‘witch’. The curve then came to be known as the ‘witch of Agnesi’, and subsequently this title is now associated with Maria whenever she is documented, whether the curve is mentioned or not!
Her book was received with much praise, the French academy of sciences assessed the book, and a deputy wrote to Maria commenting –
“ I do not know of any work of this kind that is clearer, more methodical or more comprehensive… There is none in mathematical sciences. I admire particularly the art with which you bring under uniform methods the diverse conclusions scattered among the works of geometers and reached by methods entirely different.” 
Even with such an excellent review she was still not allowed to become a member of the academy, due to a strict male only policy. (It is interesting to note though that the academy was actually founded in a woman’s salon – Madame de Rambouillet.) Maria was allowed to become a member of the Bologna academy of sciences though, as at this time Italians were more liberal about women’s equal rights. Pope Benedict XIV also wrote to her expressing how her work would be of great benefit to Italy and the academy, he had studied mathematics at a younger age and he recognised the exceptional ability of Maria. This recognition was the most rewarding for Maria, being such a religious follower. Initially the Pope assigned her the position of honorary lecturer at the University of Bologna, but soon after he approached her again, along with three professors from the academy to invite her to accept the chair of mathematics at the University of Bologna. There is much discussion as to whether or not Maria accepted this invitation. On the 5th October 1750 she was granted her diploma from the University, and was added to the faculty roll until 1795 (45 years!). Some sources say that she occupied the chair of mathematics and natural philosophy at Bologna from 1750 to 1752 when her father died, she then returned to a quieter life of study and religious solitude. Other sources say she only occupied the chair to fill in for her father during his last illness (that is if her father was a professor at the University, as mentioned earlier in the chapter there is also some debate about this issue). Other sources say she avoided the invitation altogether, never went to Bologna, and devoted herself to a holy and retired life.  
Maria also received recognition from the Empress Maria Theresa, as she had dedicated the book to her. The Empress showed her appreciation by sending Maria a lovely diamond ring and a small crystal casket set with diamonds and precious stones.
One of Maria’s brothers friends, Frisi, said that during visits to the Agnesi family home at this time he noticed that Maria chose to inhabit rooms of the house that were secluded from the rest of the family. This isolation enabled her to help old women who were ill. She still had to comply with severe constraints made by her father though, which she obediently did. This indicated the type of life Maria longed for.
After her father’s death in 1752 she was finally allowed to fulfil her own aspirations, and she devoted the rest of her life to charitable projects with the sick and poor, at the hospital of Maggiore and her parish, San Nazaro. Like many religious figures she never married – for one reason such time consuming activities would not allow it!
In 1762 she was asked by the University of Turin to comment on the young Lagrange’s articles on calculus of variations but she denied the request, claiming she was not interested in such pursuits anymore. Her life was dominated with caring for other people. Initially she turned her house into a shelter for the helpless, aged, sick and poor, she even gave up her own room for neglected women if there was no space elsewhere. It is believed that she even economised on her dresses, meals and beloved books to help her patients. She even sold the imperial gifts she received in recognition of her academic work.
In 1771 the archbishop asked Maria to become the director of ‘Pio Istituto Trivulzio’ – a newly opened home for the ill and infirm. She gladly accepted this position whilst still maintaining the shelter at her home. In 1783, when the maintenance of both of these facilities became too much she moved into the Institute. Endeavouring not to deny the patients of anything she insisted on paying rent – although she was an elderly lady herself!
The records maintained by the Institute described her as -
“an angel of consolation to the sick and dying women until her death at the age of eighty-one years on January 9, 1799.” 
Maria was therefore highly honoured not only for her exceptional achievements in mathematics but also for her extreme charitable works. Even her grave situated outside the Roman gate of the city walls, is shared with fifteen old people of the Luogo Pio. There is no monument to mark the burial place of such a selfless lady, but Maria will never be forgotten. On the one-hundredth anniversary of her death, streets in Milan, Monza and Masciago were given her name. A school in Milan is also dedicated to her, with scholarships for poor girls donated in her honour. On the outside of the Luogo Pio a cornerstone dedicated to her bears the inscription ‘erudite in mathematics glory of Italy and of her century’.
Considering Maria’s life of selfless service and devotion to God, the irony is laughable that her academic achievements link her name to a curve called the ‘Witch of Agnesi’, nothing could be more further from the truth!
Sonya Kovalevskaya – The French Revolution and 18/19thCentury Mathematics
In France in the late seventeenth century a society of salons was constructed, these were social societies for male intellectuals. Unusually, they also became an arena where women could access academia. Within the salons, social, political and cultural issues were discussed, of which mathematical science was a key topic. They became testing grounds for potential members of the Académie Française. Interestingly women ran these salons, hence they provided an excellent opportunity for females to gain insight into such topics. The famous sallonnières included Madame de Lambert and Madame de Tencin. It has been suggested that if a scientist wanted to gain entry to the academy they had to first pass through the salon of Madame de Lambert. In an unexpected switch of roles these women were central and powerful figures, and could make or break a male career! Consequently budding male scientists had to present their work in an accessible and non-specialist way so that they could convince the salon women and the general public. This meant literature was readily available for educating women in the new sciences, which was something that had been denied for centuries. Bernard de Fontenelle and Francesco Algarotti both penned books that were targeted at a female audience, including the title ‘Newtonianism for Ladies’. Although such advances in women’s access to education had been made, the academy was still strictly male only, it was thought of as beyond the grasp of the feminine intellect. Therefore the female status was still classed as inferior to the males.
This enlightenment for women in France did not last long, in the middle of the eighteenth century the salon scene was increasingly attacked, especially by Jean-Jacques Rousseau. He believed the company of women lowered the level of discourse between men, as their presence forced men to be more attentive towards them, whilst if the salons were male only a more serious discourse could commence. He also believed that women lacked the strength of body and therefore the strength of mind to participate in science, unlike males (a reflection of Aristotle’s opinion). Rousseau also attacked the poetic style of scientific writing – linking it to feminism. Many males supported this style of writing, especially Denis Diderot who once said –
“Women, accustom us to discuss with charm and clearness the driest and thorniest subjects”. 
Unfortunately opinions such as Rousseau’s meant that the poetic style was eventually eliminated from academic writings, and it became even more abstract, mathematical and technical – a more masculine style (?). This disadvantaged women greatly who had no access to higher education and self teaching was impossible with such complicated texts. Once again, women were segregated from academia, much like they were in the dark ages when education was only taught in Latin, a language which women had no knowledge of. In the wake of the French Revolution the salons disappeared entirely, along with the opportunities for French women.
During this period the general attitude towards French women was poor, this was emphasised/reflected through bad publicity, which came from books and plays. J. Molière wrote two destroying plays ‘Les Précieuses ridicules’ and ‘Les Femmes savantes’, and N.Boileau wrote the work titled ‘Contre les Femmes’. All of these ridiculed women, especially learned women. In his defence Molière claimed that his works were not aimed at the genuinely educated woman, but ridiculed instead the woman that perceived herself as intelligent. The content supported the Aristotelian/Rousseau viewpoint, so as a result it did heighten already prejudice views on their inferior intelligence. With the current climate, and such social awareness concerning the education of women, a lot of learned women would actually hide their education as it was classed as socially unacceptable.
Along with Rousseau, Molière and Boileau, popular males such as Voltaire and Baron Montequieu agreed that women should only deal with practical and domestic matters as abstract thinking such as mathematics were beyond their capabilities. They all were against education for women, whether it be for the daughters of noblemen or the lower class, but interestingly they all chose highly intelligent women as their companions. In Voltaire’s case Émilie de Breteuil’s intelligence was a valued substitute when his was lacking during the production of his book. Quite a turnaround from dismissing the intelligence of women!
Émilie de Breteuil (Marquis du Châtelet) was another extraordinary female mathematician, whose work did not get the recognition it deserved due to male interception. Émilie was fascinated by Newtonian philosophy and in addition to aiding Voltaire with the mathematical expertise he lacked, she wrote her own book on Newtonian physics. Unfortunately before the book was completed her male tutor discovered parts of it and was horrified to think he would be considered as just a lady’s tutor, so to save his pride he began a rumour that he actually wrote the upcoming book. Considering the general attitude at the time, male academia were only too keen to believe this story. Therefore Émilie had to anonymously publish the book. Still undeterred, Émilie translated into French and produced a commentary on Newton’s work the ‘Principia’, she included explanatory work within this to aid her readers. This introduced France to Newtonianism and it still remains the only translation. Unfortunately before it was published Émilie died due to a complicated childbirth, once again she could not enjoy her achievements . Her only chance to achieve any academic respect in her lifetime was spoilt by the pride of a man and the social views on gender equality in France at that time.
The close of the eighteenth century saw a few lady mathematicians like Émilie striving to be recognised for their achievements. In Germany Caroline Herschel became a great astronomer, with no formal mathematical training she accomplished numerous calculations to aid her catalogue keeping. She was the first woman to detect a comet, and along with Mary Somerville became one of the first women in England to be honoured by the Royal Astronomical Society for her achievements. 
Mary Somerville also taught herself mathematics, her parents were so distressed at this activity that they confiscated her candles so she could not study her brother’s textbooks, in retaliation Mary would just memorise the problems and work on them in her head! Her contribution to Science reflected Émilie’s – she translated Laplace’s monumental book on celestial mechanics from French to English, including additional explanatory notes and calculations. Her book became the standard textbook for Cambridge University students. She was never allowed to enter the said University though (as it was strictly male only), as a compromise a bust of her was displayed in its Grand Hall.  
Another French lady, Sophie Germain was also determined to tackle the male prejudices of this period. Once again she taught herself mathematics from her father’s library, which her family became very opposed to, confiscating from her bedroom clothing, light and heat in an attempt to stop her studying. Her family eventually relented, and after much study she began to correspond with the great mathematicians of that era – Gauss and Lagrange, but continuously signing her letters under the male name M.le Blanc, to give her opinions a fair (!) chance. Sophie became interested in the mathematical laws of elastic surfaces, and after much study she produced a treatise on it that provided her with great acclaim. This included winning the Grand Prix of the French academy of sciences, and being invited to attend sessions at the ‘Institut’ (the highest honour given to a woman by this famous body). When the Eiffel tower was erected knowledge of elasticity of materials was an important factor, and seventy-two names were inscribed on the tower in honour of their contribution to the advancement of this topic. Sophie’s name was not included although her contribution was vast, maybe because she was a woman?  
All of these ladies appear to follow the same pattern, they were all denied an education in mathematics, they were all stopped from individually studying it, and after admirable determination and hard work against such prejudices they accomplished great things with little recognition. In the majority of these cases the females were thwarted by either male or social views supporting male-only academia.
There was some enlightenment to the cause of women’s education – in the early nineteenth century the first female colleges were opened in America. These colleges took science and mathematics seriously, unlike the college’s before the civil war that opened to teach women to be missionaries and teachers. Unfortunately after actually receiving the education employment was scarce, it was only the female colleges that would employ female teachers, and jobs in industry were not even open to female candidates. Therefore another huge obstacle had to be overcome before any true enlightenment for women’s equal status could be demonstrated.
The nineteenth century did see the mathematical genius of another great woman though. Her name was Sonya Corvin-Krukovsky Kovalevskaya. Although her first name is sometimes documented as Sonja, Sophie, Sof’ya or Sofia, and her surname has been spelt Kovalevsky, Kowalewski, Kovalevski or Kryakovskaya. She was born in Moscow, Russia on 15 January 1850.
Her father, Vasilii Kovalevsky was a general in the Russian army, which was reflected in her disciplined family life. His displeasure could throw the entire household into sheer terror. Her mother, Elizaveta Schubert was an excellent pianist and socialite who took a keen interest in the arts. The family was classed as minor gentry and Sonya lived in plush surroundings. She had an older sister Anya, and a younger brother (the only male heir) Fedya. As the middle child she felt that her parents neglected her in favour of her siblings. This inferiority complex shadowed most of her life, and as a result she grew to become fairly nervous and withdrawn. She was prone to giving extravagant affection and had an astonishing jealousy. The attention and devotion she often required from her friends was sometimes beyond the capabilities of even her loved ones.  
Both her mother and father dismissed the responsibility of bringing up their own children, a nanny was employed to do this. In 1858 Sonya’s father retired from the army and to his horror found that his two daughters were very ignorant, so he hired an English governess, Margaret Smith, to replace the nanny. The strict governess saw it as her duty to turn Sonya into a young lady. This, along with her insecurities that she had already developed, resulted in a reasonably unhappy childhood for Sonya.
If you look at Sonya’s family tree it might be obvious that she was destined for mathematics, as her grandfather, Feodor Feodorovitch Schubert, was a fine mathematician, and her great-grandfather had even more recognition as a mathematician as well as an astronomer.
After Sonya’s father retired from the army, when Sonya was about six years old, the family moved to their prosperous estate at Palibino, which was in a remote part of Russia. Her initial interest in mathematics came strangely from the wallpaper in her bedroom at Palibino. Apparently not enough wallpaper had been sent from St.Petersburg to decorate the house, and whilst awaiting the arrival of new supplies, old lecture notes from a course her father had attended were used instead. This course was actually differential and integral calculus taught by the great mathematician Ostrogradlskii. These notes fascinated Sonya at only eight years of age she would spend many hours trying to decipher phrases and the order in which the notes should have gone. Although the notes were fairly incomprehensible the formulas and text stayed engraved in her mind until she encountered them again later on in her education.
Sonya also got a lot of her mathematical inspiration from her Uncles. Pytor Vasilievich Krukovsky (her father’s brother) had a great respect for mathematics, and he discussed with Sonya topics such as the quadrature of the circle, asymptotes of a curve etc. Obviously she could not fully understand all of these concepts but it sparked her interest. Her mother’s brother was also well educated and discussed with Sonya the intricacies of infusoria and algae.
Her father hired a Polish tutor, Joseph Malevich, and Sonya’s education began. It was at this point that Sonya is quoted as saying –
“I began to feel an attraction for my mathematics so intense that I started to neglect my other studies”. 
She was a very bright, committed student and after a few years of lessons she began to show her deeper understanding and talent for mathematics, this is demonstrated by a particular comment made by Malevich –
“But when our study of geometry reached the ratio of the circumference of a circle to its diameter…My pupil, when explaining this topic in the next lesson, astonished me by arriving at the same result in a completely different way by using her own reasoning.” 
When Sonya was fourteen Margaret Smith (the governess) resigned her position after numerous attempts to shield both the strong-minded Sonya and Anya from ‘too much’ education. Sonya’s father also tried to put a stop to Sonya’s mathematics lessons, but defiantly Sonya borrowed a copy of Bourdeu’s ‘Algebra’ which she would read at night in secret to continue her studies in mathematics.
During this time Sonya’s neighbour, who was the physicist N.P.Tyrtov brought her a copy of his physics textbook. She eagerly read it, but as she had not been taught trigonometry she had to decipher by herself what the sine function was. She guessed that the sine of an angle is proportional to the chord in a circle subtended by a central angle. In fact it is proportional to the chord subtended by an inscribed angle, but for small angles the difference is negligible. Tyrtov was amazed initially that the young girl was even reading the book, but to be able to derive so accurately the sine function, like it had been derived historically, was astonishing. Tyrtov then portrayed to Sonya’s father the extreme capabilities of his daughter and how she needed a more advanced tutor. Therefore her father allowed her to attend a Naval school in St.Petersburg under the tuition of Alexander Nikolaevitch Strannolyubskii, an excellent teacher. During her first lesson of differential calculus from Strannolyubskii he was amazed at her easy grasp of the terms and derivatives, just as if she had known them before, but this was nearer to the truth than he had imagined as Sonya expressed that –
“…in truth the fact was that at the moment when he began to explain to me these conceptions, I immediately and vividly remembered that all this had stood on the pages of Ostrogradsky [on her bedroom walls in Palibino], so memorable to me, and the conception of space seemed to have been familiar to me for a long time.” 
Sonya was also gifted in her literary talents as well as mathematics, this side of her education was encouraged by the influence of her older sister who was primarily interested in literature, not science like Sonya. Anya wrote short stories, and at twenty years of age had one published by Dostoevsky in his journal ‘Epoch’. Anya was also passionate about radical causes, which she expressed to Sonya. Dostoevsky introduced both sisters to an elite circle of European intellectuals in Moscow. Consequently during the Polish uprising of 1863 both sisters sided with the Polish rebels. To add to Sonya’s political fervour she grew very fond of a Polish rebel named Bujnicki who fled the country, increasing her interest in the current situation. Such political awareness was very unusual for middle-class girls as, along with education, it was actively discouraged. Sonya felt a great sense of guilt for not contributing more actively to the cause and this guilt became the motivation for a lot of her poetry and playwriting later on in life.
After completing her secondary education in St.Petersburg Sonya wished to continue her education, but she came across many obstacles. Her first obstacle was her father, he was reluctant to let Sonya study mathematics with Strannolyubskii, so he was even more reluctant to allow her to follow this uncommon pursuit to university and then onto a serious career. Sonya received stern lectures from her father concerning such improper behaviour from young girls. Her second obstacle was that Russian universities were closed to women. Her final obstacle being that even though the nearby Swiss universities permitted women to enrol, females were not allowed to travel alone, they had to be escorted by their father or husband (women became just an amendment to their male’s passport). Therefore unless they were married, or had an understanding father, they were not able to actually get to the university.
This last obstacle was common to many young Russian girls, but it was the only one that they had a chance of overcoming. They did this by entering into a ‘fictitious marriage’ – this involved finding a politically conscious man that was willing to rebel against the conventions of society, but also a gentleman that would treat his lady with respect and keep the marriage of convenience purely platonic. This solution meant that fathers had no authority over their daughters, and they were free to travel with their husband to seek out education. They could also escort sisters and girlfriends with complete respectability. Although this solution provided a loophole for Russian young ladies wanting to further their education, it was a rather drastic measure!
Therefore Sonya and Anya (who also wanted to travel to seek education) set about finding a man that either one could marry, so they could then escort the other sister and numerous girlfriends abroad. Finding a man was not a problem, it was convincing their father to agree to the arrangement that was the tricky part. Sonya’s first candidate was rejected outright by her father, so with her second candidate she was more careful. Vladimir Kovalevsky was a student of parentology (examining theories and research relating to parenting and parenthood across the lifespan) at the University of Moscow. At the time he was corresponding with Darwin concerning the Russian translation of the second volume of Darwin’s work on domesticated animals. Sonya helped Vladimir to edit this translation. Vladimir was very impressed with Sonya’s talents in literacy, languages and mathematics, and her remarkable beauty aided the situation somewhat! The pair were both nihilists – people who thought the current state of society was hopeless and wanted to start again on a foundation of science and humanitarian values. As a result Vladimir was very agreeable to the proposal. To get Sonya’s father to agree, it is alleged that Sonya visited Vladimir at his home and promptly stayed there until her father agreed to the marriage, knowing full well that the reputation of the family was at stake if he didn’t comply with her wishes! Subsequently Sonya and Vladimir were married in September 1868.
Initially the couple lived in St.Petersburg, where they both attended lectures on anatomy and physiology. Entering the lectures with her husband meant that Sonya avoided any unwanted attention, (although it was legal for her to attend these lectures it obviously wasn’t yet socially acceptable). Sonya missed out on a great opportunity at this point. The great mathematician Pafnutii L’vovich Chebyshev held weekly ‘open’ lectures at the university, which presented the opportunity for any one to consult with him on mathematical problems. Unfortunately, as mentioned before, the universities were not open to females so Sonya could not attend these lectures and demonstrate her talent, if she could have attended she most definitely would have become his student. Although Sonya did make her acquaintance with Chebyshev it went no further.
Sonya was keen to move to Europe to begin her higher education so in April 1869 the couple went to Vienna. Unfortunately the cost of living was high and they could not find a mathematician for Sonya to work with, so they moved again in May 1869 to Heidelberg, Germany.
Although the local university at Heidelberg did not strictly accept female admissions, they left it up to the individual lecturers to decide if they would allow women to attend their lectures. Fortunately Sonya’s reputation was already strong enough to prompt the Professor’s to welcome her into their lectures. Therefore she was able to study with famed professors such as Helmholtz, Kirchhoff, Bunsen, Leo Königsberger and Emil Du Bois-Reymond.
Right from the beginning Sonya’s lecturers were very impressed with her abilities and demeanour, the memoirs of a fellow student state 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.” 
A particular incident with Bunsen did spoil his attitude towards her though. Bunsen – a chemist, was very proud to have a male-only laboratory, but one of Sonya’s fellow Russian female students desperately wanted to study chemistry with him, and after being turned down by Bunsen she turned to Sonya for help. It is believed that Sonya utilised her persuasive powers so successfully on the Professor that he actually agreed to the request, and only after she had left did he realise how manipulated he had been. This was an embittering experience for Bunsen as he was an established woman hater, especially Russian women, so to be outwitted by a Russian woman and consequently having to permit one into his precious male laboratory was disastrous. From then on Bunsen branded Sonya as a ‘dangerous woman’, even portraying this a few years later to what would be her future tutor, adding “And now that woman has made me eat my own words”  – a very bitter man!
Although Sonya was seemingly confident she still had a natural shyness that never faltered however much attention she got. Her insecurities from her childhood still haunted her. This quality was most appealing to the Germans though, and eventually her nature and incredible capabilities meant that local people would even point her out in the street!
After two years of study at Heidelberg Sonya realised that she would never be able to obtain a degree, she would need the support of more influential people to achieve that. Whilst at Heidelberg she studied under Leo Königsberger, who was the former student of Karl.T.Weierstrass. Königsberger rated Weierstrass highly, he introduced Sonya to his teachings on elliptic functions, and as a result Sonya also contracted Königsberger’s enthusiasm for Weierstrass.
Weierstrass was famously known as the ‘father of mathematical analysis’, he had a good reputation and great influence on the European scholars of that time, so Sonya couldn’t have picked a better mentor. Therefore in 1870 she set off for the University of Berlin where he lectured.
Once again Sonya was not allowed to enrol at the university because she was a woman, despite having impressive references from her previous university. Not to be deterred she appealed to Weierstrass himself. Fortunately current World events were to her advantage, in August of that year the Franco-Prussian War began and the attendance rates for Weierstrass’ lectures had dwindled from 50 students the year before to only 20 students. Considering this, the fact that Sonya came with good references from his associates at Heidelberg, and that Weierstrass was always grateful to his teacher (Christoph Gudermann) for the chance he gave him, he agreed to see her. Sonya was in awe of the famous mathematician but she appeared very eager and determined with her mathematics, and Weierstrass was a sympathetic and understanding man, especially to the ambitions of the young, so he gave her a set of problems that he had designed for his more advanced pupils. Sonya was well prepared for such a task and she easily solved the questions. Weierstrass was amazed at the speed, clarity and originality of the solutions that she had produced, so he wrote to Königsberger to enquire about her abilities and personal qualities. Obviously Königsberger reassured him that Sonya had the traits of a great mathematician.
In some sources it’s suggested that during this first meeting it was Sonya’s feminine charm that won Weierstrass’ attention not her abilities. Most sources refute this declaration though, claiming prejudices on behalf of the male authors who may have believed a female mathematician couldn’t gain the role entirely as a result of her intelligence but instead by her beauty? Unfortunately Sonya was supposed to be very attractive and obviously young, and Weierstrass was thirty-five years her senior and a renowned bachelor, therefore accusations could be viable, but it must be remembered that Sonya was a married woman!
Weierstrass went to the university to request that Sonya could attend his lectures, but after a second rejection for Sonya, Weierstrass committed himself to giving her private lessons instead. Therefore for the next four years he let her have lecture notes, showed her unpublished works, and taught her the latest scientific developments and theories. This relationship actually continued until Sonya’s death, up to one hundred and fifty letters have been found from Weierstrass to her, (Weierstrass burnt all of her letters upon hearing of her death). The correspondence between Weierstrass and Sonya does indicate that they were great friends, which is evident in their continued communication. It has been commented that Weierstrass particularly had a deep affection for Sonya. Sonya once said when reflecting on this period –
“These studies had the deepest possible influence on my entire career in mathematics. They determined finally and irrevocably the direction I was to follow in my later scientific work: all my work has been done precisely in the spirit of Weierstrass”. 
In 1872 (during her period of tutelage) Sonya’s life got somewhat erratic. It is believed she would invite fellow Russian females to stay in her and Vladimir’s small apartment, supporting them all on the minimal allowance she received from her father. This and the exhaustion of a heavy work load meant quarrels were rife between her and Vladimir. Sonya became so miserable as a result of this that she made a full explanation to Weierstrass about the situation of her marriage and background (this arrangement was something she had not explained to him at the beginning which she always regretted). After Weierstrass realised the full extent of the situation he proposed that Sonya should write a dissertation, and admit it to the more liberal University of Göttingen. Therefore over the next year and a half Sonya, with guidance from Weierstrass, composed several works, three of which became potential dissertations.
Her chosen doctoral dissertation was the paper entitled ‘Theory of Partial Differential Equations’. This detailed a general system of first order differential equations in any number of variables. It was an extension to Weierstrass’ paper on an analogous structure for total equations. Her paper was even published in Crelle’s journal, which was a great honour for an unknown mathematician. The other two papers were concerned with abelian integrals and Saturn’s rings. The former of these papers was entitled ‘On the Reduction of a Definite Class of Abelian Integrals of the Third Range’. This dealt with the reduction of abelian integrals to simpler elliptic integrals, consisting of a skilled series of manipulations that demonstrated her complete command of Weierstrass’ theory of abelian integrals. The latter of these two papers was entitled ‘Supplementary Research and Observations on Laplace’s Research on the Form of the Saturn Ring’. Another one of her works was ‘On the Property of a System of Equations’.
Her dissertation granted her a doctorate of Philosophy from the University of Göttingen in 1874. She was awarded exemption from the usual aural examination, as Weierstrass believed that her nerves and poor German would disable her from demonstrating her true genius. (This was authorised by the university as her outstanding dissertation and glowing references from fellow scientists confirmed her ability without examination). With such an achievement it is claimed that Sonya was the first woman outside of Renaissance Italy to receive a doctorate in such a field.
After such severe studying, homesickness and in search of employment, Sonya and Vladimir returned to Russia in the autumn of 1874. Unfortunately both Sonya and Weierstrass’ attempts at finding employment worthy of her talents were in vein, leaving Weierstrass shocked at the misogynist viewpoints of academics. Sonya was most annoyed to discover that the best job she could find was teaching arithmetic to elementary classes of schoolgirls, her response to this was –
“I was unfortunately weak in the multiplication table”. 
As often seen with women in the nineteenth century, obstacles were rife in a woman’s career path, each one being tougher than the next – after accomplishing a doctorate against all odds it was even tougher for Sonya to find a job.
Vladimir was just as unsuccessful at finding employment, he could have found a position teaching, but after criticising the work of one of his examiners whilst sitting the exam for his teaching certificate, his career in such a role was cut short! 
Sonya easily fell into the role of her mother – a society wife. As there was no call for her mathematical abilities she returned to her other talents, such as literature, writing articles, poetry, theatrical criticisms and even a small novel entitled ‘The Privat-Docent’. Much of her work was based on her continual struggle – the equal rights of women.
Soon after Sonya’s return to Russia her father died leaving her a small legacy. She and Vladimir re-invested this into real estate hoping to become wealthy as a result, unfortunately it was not as simple as they had hoped and ultimately they became bankrupt.
Concurrently Sonya and Vladimir became close and decided to consummate their fictitious marriage, which in 1878 produced their only daughter, Sonya Vladimirovna Kovalevskaya, or as they affectionately called her ‘Fufa’.
The social views of academia were still very influential and even a woman as liberated, educated and intelligent as Sonya still believed that women did not have the same strength to engage in education as men did. (Reflecting Rousseau’s and Aristotle’s theory that women did not have the strength of body, therefore neither the strength of mind to participate in academia.) In the nineteenth century it was believed that women could not possibly withstand the rigors of University life, and Sonya – who had had such a life, once made the passing comment to her friend –
“Thank heaven I did not completely waste my strength studying mathematics; now at least my little girl will inherit some intellectual ability”. 
Fufa’s first year was not a successful one for Sonya, as well as the bankruptcy her mother died in February 1879. This put a lot of strain on her marriage and the quarrelling began again. These events did make Sonya reassess her life though, and after only one letter to Weierstrass in three years (often referred to as ‘the wilderness years’) to notify him of her father’s death she initiated their correspondence again. She had become impatient with the social whirl of St.Petersburg and was keen to get back to her true origins - mathematics.
Weierstrass was obviously concerned for Sonya after hearing of her father’s death, and he forwarded his condolences. For two years he never heard from Sonya and he grew worried that she had never received the letter, so he wrote to her in 1878 enquiring about this and the rumour that she had ‘gone social’, begging her to deny it. He still heard nothing, even though he had told her he was in bad health. When Sonya eventually did answer his letter begging for his advice, Weierstrass ungrudgingly encouraged her even after such neglect on Sonya’s behalf.
In 1879 Sonya attended a scientific conference at St.Petersburg, she gave a talk on her unpublished paper concerning abelian integrals which she had produced potentially for her dissertation before her wilderness years. (This paper was later published in 1880, and was highly accredited by the great French mathematician Henri Poincaré). At this conference Sonya was re-introduced to another of Weierstrass’ students, a Swedish mathematician Gösta Mittag-Leffler who was at the University of Helsinki. They had already become acquainted in February 1876. Mittag-Leffler was already aware of her reputation as a theoretical mathematician despite her absence, he describes her in one of his diary entries –
“As a scholar she is distinguished by a rare clarity and a precision of expression, as well as an extraordinary quick perception. It is also easy to see the degree of profundity to which she has pursued her studies, and I understand perfectly why Weierstrass regards her as the most talented of his students.” 
Meanwhile Sonya had been guided by Weierstrass to apply a mathematical technique he had developed to solve a problem in mathematical physics concerning the refraction of light in a crystalline medium. Working conditions were tough for Sonya though, her marriage was strained, she was poor and in debt after taking a 65,000 rubles loan, and she had a baby daughter to care for.
In 1880 Sonya returned to Berlin to consult with Weierstrass on the problem he had set her. She left Fufa in Russia with her Godmother, Yuliya Lermontova. Two months later she returned to Russia to collect Fufa, then in 1881 she went back to Berlin with both Fufa and a nanny (even considering her poverty) so that she would be free of the responsibility of childcare.
During this time Vladimir decided to try and earn some money again by entering into business with unscrupulous partners who set him up in an illegal stock scandal, so along with bankruptcy he also faced prosecution.
Sonya continued her friendship with Mittag-Leffler who had developed a great respect for her. He tried in vein to secure a position for Sonya at the University of Helsinki. Unfortunately World events disadvantaged her at this time as the assassination of the liberal Tsar Alexander II in March 1881 provoked a crackdown on radicals. Sonya, as mentioned earlier, was a nihilist (a radical) and also an unescorted married woman, as a result the University declined her a position. Eventually Mittag-Leffler was forced to leave the University as well, he became resented as he was a Swede among Fins. He returned to Sweden to become a founder of the more liberal institution, which was to become the University of Stockholm.
In the autumn of 1881 Sonya moved to Paris. The following year she sent Fufa and her nanny back to Russia, whilst she carried on her work with Weierstrass’ problem. In the spring of 1883 events took a turn for the worse, on 27th April Vladimir faced prosecution over the stock scandal and he drank a whole bottle of chloroform which killed him. He left a note for Sonya asking for her forgiveness concerning the emotional and financial mess that he had made of their lives, and protesting his innocence. Sonya was devastated by this news, she felt awful for not remaining in Russia with him. Her grief was tremendous and she shut herself in her room refusing to eat, drink or receive medical help. After four days she lost consciousness and her doctor force-fed her liquid food. On the sixth day she woke, sat up in bed, and without saying a word began to trace symbols on the blanket. She then asked for pencil and paper, and furiously covered the page with mathematical formulae. Sonya never fully recovered from this, her grief seemed to age her.  
In September 1883 Sonya attended the scientific congress in Odessa. Here she gave the results of her research on the refraction of light in a crystalline medium. The paper was received well although many years later the Italian mathematician Vito Volterra discovered an error that had not been noted by either Sonya or Weierstrass. Weierstrass had the excuse of being ‘brain-weary’, with all of his official duties and responsibilities at seventy years old he was getting tired. Sonya’s mishap though reflected her sustained leave from mathematics.
Although Vladimir’s death was a dreadful event, it was a breakthrough for Sonya’s career. Callous as it was, the widow status gave Sonya a greater independence than if she were married or single. In Europe in the nineteenth century widowhood was very respectable, and opportunities that were before closed to her began to open up.
With Mittag-Leffler’s assistance Sonya managed to secure a teaching position at the liberal University of Stockholm where Mittag-Leffler had become a professor of mathematics. He was eager that the new institution be the first to attract a woman lecturer. Initially, Mittag-Leffler advertised the lectures as a one-off chance to hear the words of a distinguished woman scholar, just in case the lectures were not successful and compromised further women’s careers. Fortunately though her lectures on partial differential equations were received well and she continued to lecture after this initial one-off period.
Sonya lectured in German, although she could not be as expressive as she wished to be she was still very popular with her students. This was demonstrated at the end of her first semester when her pupils delivered a warm speech expressing their praise for her efforts, along with a framed photograph of the class. She was definitely a success as a teacher. She boasted course titles that included subjects such as, the theory of derived partial equations, the theory of abelian functions according to Weierstrass, curves defined by differential equations according to Poincaré, the theory of elliptic functions, application of analysis to the theory of whole numbers and so on. Therefore her capabilities as a mathematician were vast!
Sonya also took on the task of liasing with the mathematicians of Paris and Berlin, and took part in the organisation of international conferences.
Sonya re-established her literary talents by collaborating with Mittag-Leffler as an editor of ‘Acta Mathematica’, a new journal. She also became great friends with his sister Anna Carlotta who was a writer, and collaborated with her on many plays and poetry. Even in this field she received ill treatment from males in the literary society. The famous playwright and misogynist August Strindberg was most annoyed at her presence, and wrote an article portraying this, Sonya’s response to this was –
“ I have received an article by Strindberg in which he proves as clearly as two times two equals four how monstrous a phenomenon is a woman professor of mathematics, how pernicious, useless, and unpleasant. I think he is essentially correct; the only thing I object to is his assertion that there are many male mathematicians in Sweden who are better than I and that I have been invited here only out of a sense of chivalry.” 
During 1886 Sonya’s sister, Anya, became gravely ill. Sonya went to Russia to be with her and once again the difficulties of penetrating a male-only society presented themselves. Mittag-Leffler wrote to her denying leave of absence for personal reasons, in his letter he stated –
“A man may not request, and would never be granted, leave to care for a sick wife, child or other relative.” 
He went on to say that if she took leave to look after her sister it would cause a riot re-igniting the issues of employing women. Male professors therefore appeared to be able to depend on female relatives to look after family sicknesses, an option Sonya did not have.
Another disadvantage Sonya had to deal with was salary. It is believed that she was paid far less than her male colleagues in the same position. It was only after much arguing from Sonya and effort from Mittag-Leffler that her wages were as generous as they were. In 1888 Mittag-Leffler had to volunteer 1000 kroner out of his own pocket to substitute her wages to 6000 kroner for the year.
It took five years of lecturing for the University of Stockholm to realise her importance and consequently she finally secured a tenured appointment.
In1886 Sonya made a mathematical discovery, which she planned to submit to a competition at the Paris Academy in 1888. The topic for the competition was significant contributions to the problem of the study of a rigid body. She discussed her project with several French mathematicians, one being Charles Hermite who arranged competitions to obtain recognition for his protégés. Members of the committee who chose the topic for the competition also discussed the project with Sonya and expressed that she had a good chance of winning.
The rules of the competition dictated that each entry had to be submitted anonymously, the author’s name was sealed into an envelope bearing the same motto as that inscribed on the paper, and the envelope would not be opened until the winning paper was declared. This meant that any prejudices against women would be eradicated, the works were sexless.
On Christmas Eve in 1888, Sonya’s paper ‘On the Problem of the Rotation of a Solid Body about a Fixed Point’ won the competition, and Sonya was presented with the famous ‘Prix Bordin’ from the French Academy of Sciences. This was the high point of her career and Weierstrass was overjoyed that finally her pure genius had been recognised.
Sonya’s entry was regarded as so exceptional that the value of the prize money was increased from 3000 francs to 5000 francs. The motto that she chose for her paper was ‘Say what you know, do what you must, come what may.’
It was believed Sonya’s paper rendered an extraordinary service to mathematical physics. Prior to this there had only been solutions to the motion of a rigid body about a fixed point for the two cases where the body is symmetric. Whereas Sonya’s paper developed the theory for an unsymmetrical body where the centre of its mass is not on an axis in the body. It has been stated that it was not even understood at the time why her methods worked, this was because she used complex analysis – a branch of mathematics that was still in its infancy. An impressive feat!
The competition was the certification necessary for the University of Stockholm to offer Sonya the coveted professorship in June 1889, and she became the first woman since the physicist Laura Bassi and Maria Gaetana Agnesi to hold a chair at a European university.
The Swedish Academy of Sciences awarded Sonya with an extra 1500 kroner for two more works based on her original prize-winning paper.
Finally in December 1889 Sonya got the recognition that she had been striving for. Although the Tsaris and government had repeatedly refused her a university position in Russia, the rules of the Imperial Academy were changed specifically for Sonya to allow the election of women. Sonya therefore became the first woman Corresponding Member of the Russian Academy of Sciences. Although still no teaching position in Russia was offered.
It is amazing that during this period she managed to achieve so much, as well as worrying about the ill-health of her sister, childcare, and her teaching position, she also had another distraction. She had fallen in love. Maksim Kovalevsky was visiting Stockholm in 1885 to deliver a series of lectures and it is believed that their mail got mixed up as they both had the same surname. He was a radical sociologist and legal scholar, but more importantly to Sonya he was understanding, kind, caring, and compromising on Sonya’s behalf. They both seemed to be devoted to one another, but practicalities of time and location meant it was hard to see one another. Sonya would not give up her work for the competition as in her mind it proved to academia that women were incompetent, but as a result her personal life suffered. In addition to this Sonya kept demanding unreasonable shows of affection and love from Maksim (as mentioned earlier in this chapter a trait of her personality that stemmed from her childhood insecurities), eventually he could not fulfil these demands, which caused a strain on their relationship. Maksim’s work relocated him from Stockholm to France, and he wanted Sonya to go with him. Sonya could not give up her hard-earned positions and flatly rejected his offer. Sonya fell into one of her depressions and turned to her writing for comfort, and it was whilst she was visiting him in France for the summer of 1889 that she completed her novel ‘The Rayevsky Sisters’ – her recollections of childhood.
Once published the novel received generous praise from literary critics who declared that it “equalled the best writers of Russian literature in style and content” . It was translated into many languages. Sonya also published novels entitled ‘Vera Vorontzoff’ which reflected on life in Russia, and ‘Nihilist Girl’.
Sonya and Maksim parted company in 1890 after a holiday in Genoa. On the return journey to Stockholm Sonya took a different route to avoid Denmark which had an epidemic of smallpox. As a result of worry and upset she had not thought through her alternative journey and consequently she found herself caught in the middle of the night at a cold, deserted station where she had no Danish money to tip the porter to carry her luggage. Struggling to carry her own luggage, freezing and wet, her already run down body was susceptible to influenza, which was also epidemic at the time. When she finally reached Stockholm she was feverish and ill, and on 10 February 1891 she died.  
Some sources state that she was not returning from a holiday with Maksim but was returning from one of her many trips to Moscow to visit her dying sister, Anya, and Fufa who was also still in Russia . This contradicts further sources that record Anya’s death in 1887 .
Sonya’s death was a dreadful shock to her family and friends, especially to Weierstrass who was devastated – he burnt all of their correspondence. Sonya’s papers were left in disarray, an utter confusion. Therefore considerable scientific correspondence was lost. Her life and career was cut short at just forty-one years of age. She had talked with friends about a move to France (maybe to follow Maksim) and new projects, so much potential was lost. The mathematical world also mourned the loss.
Sonya was buried in Stockholm at the Norra Begravningsplats.
During Sonya’s career she published ten papers in mathematics and mathematical physics, and several literary works. Many of these papers contained ground-breaking theories or were reference for future discoveries. Her last published work was a short article in which she gave a new, simpler proof of Bruns’ theorem on a property of the potential function of a homogenous body.
Her prize-winning paper on partial differential equations produced results that are still of importance today for finding solutions to differential equations with initial conditions. This is known as the Cauchy problem. Therefore Sonya must also be given recognition for the resulting Cauchy-Kovalevskaya theorem. This concerns second order linear partial differential equations in one dependent and ‘n’ independent variables. Sonya’s theorem helped to establish rigorously the first existence theorem associated with the problem. The Cauchy-Kovalevskaya theorem is the foundation of most graduate courses in partial differential equations.
Even after her death Sonya was subject to experiments to demonstrate if the female intelligence was inferior to the males. Her brain was preserved, and four years later it was weighed and compared to the weight of Hermann von Helmholtz’s brain. It was concluded that, after considering body weight, Sonya’s brain tissue was greater than Helmholtz’s i.e. a woman’s intelligence/brain tissue was greater than a man’s. Once again scientists failed to show that a male brain was superior to a females.
In 1951 and 1996 Russia issued commemorative stamps to honour Sonya. Russia has been one of the most generous countries for representing mathematicians on postage stamps, so it is even more of an accomplishment that Sonya is the only woman in her field to have such an honour.
Sonya was not just content to utilise her abilities to translate the works of others – she had an extraordinary talent for mathematical research and she derived numerous new discoveries that altered the mathematical world. She had a fantastic ability to change the dauntingly complex into clearly simple. The President of the Academy of Sciences, which awarded Sonya the Prix Bordin, once said about her –
“Our co-members have found that her work bears witness not only to profound and
broad knowledge, but to a mind of great inventiveness.”