Berlin and Weierstrass
Throughout life she preferred to meet situations head-on rather than to wait in doubt or trepidation for their slow development.
It was this attitude which saw her appealing personally to Weierstrass despite the fact that she did not think she would be accepted to study at the University of Berlin.
Weierstrass was opposed to the admission of women to university in general, but was inclined to go against his better judgement by a number of factors. Not only did Sofia have excellent letters of recommendation from Königsberger and Du Bois-Reymond, but the war at that time had greatly reduced the number of students that he was working with. He was also greatly impressed by her solutions to a number of test questions which he had set her on hyperelliptic functions in order to test her ability. He very quickly became convinced that she had,
the gift of intuitive genius to a degree he had seldom found among even his older and more developed students.
The authorities at Berlin University could not be persuaded to allow Sofia to attend lectures but,
Ironically this actually helped her since over the next four years Weierstrass tutored her privately.
He was a kind man who did not wish to see Sofia's talent go to waste, but when he made his offer of private tuition,
he had no idea that Kovalevskaya would become the closest of his disciples and remain so until her death.
Weierstrass was the most noted mathematician in the world when Sofia first met him. Students were attracted to working with the man who, fathered modern mathematical analysis, by his reputation for lecturing on the latest mathematical theories. He was the leader of what was known as, the German tendency in mathematics, and believed in carrying out mathematical analysis for analysis' sake. This was in contrast to the more concrete approach of Russian mathematicians for example, who preferred to work with a grounding in more practical problems and thus were reluctant to admit to the merits of Weierstrass's work. The man himself said that when it came to a scientific work he demanded,
unity of method, the sequential following of a definite plan, and the appropriate working out of details, and that it should bear the stamp of independent investigation.
This demand goes a long way towards explaining the attention to detail and efforts to obtain explicit formulas which Sofia shows throughout her work. By his,
uncompromising development of a systematic foundation and his
pursuit of a fixed plan after appropriate preparation of detail,
he continues to influence the world of mathematics greatly to this day.
Weierstrass soon developed into something of a surrogate father for Sofia. Roger Cooke has stated that,
It is impossible to doubt that Weierstrass was in love with Kovalevskaya, in a way that only a middle-aged bachelor can be, 19
but it is also true that there is almost no doubt at all that their relationship ever developed into anything more than a deep friendship between a professor and his student. The incredible impact which her private lessons with Weierstrass had on Sofia can best be highlighted by her own words. She said of them that,
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.
To begin with he taught Sofia the same topics that he happened to be lecturing on at the time. These included elliptic and Abelian functions as well as synthetic geometry and the calculus of variations. This working relationship soon became mutually beneficial when Weierstrass began discussing his latest research with Sofia. In a letter which he wrote to her, Weierstrass stated that,
never have I found anyone who could bring me such understanding of the highest aims of science and such joyful accord with my intentions and basic principles as you!
Sofia clearly shared this view of their bond, writing of their relationship in a letter to Vladimir in 1873 that,
there is much that is poetic, ideal, open-hearted, and it gives me a terrible amount of happiness and delight.
Weierstrass was firmly of the opinion that married women did not need either university degrees or careers of any kind. In October, 1872 however, Sofia was so miserable as far as her relationship with Vladimir was concerned that she make a full explanation to Weierstrass about their unconventional marriage. From this point on he was able to be more outspoken regarding his opinion that Sofia and Vladimir were completely unsuited, saying of Vladimir that he, has no true appreciation of her intellectual merits. 23 He disapproved of her marital arrangement, but felt a great deal of compassion towards Sofia regarding this matter. Having had her position explained to him, Weierstrass came to believe that Sofia my in fact require a career after all. As a result he felt that she should work towards the attainment of a doctoral degree.
The evidence which more than anything else supports the assertion that Weierstrass was an inspirational teacher and advisor is the fact that, somehow or another he made creative mathematicians out of a disproportionately large fraction of his students. The students who had the pleasure of working under Weierstrass included such mathematical greats as Königsberger, Hensel and Schwartz, yet Weierstrass, often said that Sofia was his most talented and best pupil. As a result of working together so closely, Sofia came to understand his work better than anyone excluding the man himself. Due to this fact she was often called upon to explain his work to others. Weierstrass would be upset when students would use his work without giving him the credit for it, but in Sofia he found someone who,
accorded him the affection of a daughter, propagandised his methods in her work, and always credited him as the source of her ideas.