Zur Theorie der partiellen Differentialgleichungen.

Berlin: Georg Reimer, 1874.

First edition, rare, of the doctoral thesis of “the greatest woman mathematician prior to the twentieth century” (DSB) and the first woman in Europe to obtain a doctorate in mathematics. The thesis contains what is now called the ‘Cauchy-Kowalevsky’ theorem on the existence of solutions of partial differential equations. It generalises a result obtained by the French mathematician Augustin-Louis Cauchy in 1842, although “it goes without saying that Kovalevskaya’s work was completely independent of the work of Cauchy, even though there is some duplication. The important concept of normal form, which brings order to the whole topic, is due to Kovalevskaya, though probably inspired by [Carl Gustav Jacob] Jacobi” (Cooke, p. 35). Henri Poincaré said that “Kovalevsky significantly simplified the proof and gave the theorem its definitive form,” and the German mathematician Karl Weierstrass, under whose supervision the thesis was written, wrote: “So you see, dear Sonya, that your observation (which seemed so simple to you) on the distinctive property of partial differential equations … was for me the starting point for interesting and very elucidating researches.” In addition to being a brilliant mathematician, Kovalevskaya was also a political activist and a public advocate of feminism.

Provenance: Signed and inscribed on title page by Mariia Vasil’evna Pavlova (1854-1938) to Alexandra Mirozkina in 1911. Pavlova, a palaeontologist well known for her research on fossil hoofed-mammals, studied at the Sorbonne from 1880 to 1884, and from 1885 worked in Moscow University, initially as an unpaid volunteer in the geological museum. Shortly after the Russian Revolution she became a professor of palaeontology at Moscow University, later acting as the first female head of its department of palaeontology. She was an honorary member of the USSR Academy of Sciences, and academician of the Ukrainian Academy of Sciences. In 1926 the French Geological Society jointly awarded its prestigious Albert Gaudry Medal to her and her husband Aleksei Pavlov.

The simplest form of the Cauchy-Kowalevsky theorem states that any equation of the form ∂z/∂x = f(x, y, z, ∂z/∂y), where the function f is analytic (has a convergent power series expansion) near a point (x0, y0, z0, w0) has an analytic solution z(x, y) which is analytic near (x0, y0) and which is equal to a given analytic function g(y) when x = x0. Eighteenth-century mathematicians had developed a successful approach to solving such partial differential equations, which was called the ‘method of undetermined coefficients.’ This involved writing down a power series for the unknown solution f in which the coefficients of the powers of the variables were treated as unknown coefficients. Substitution in the equation to be solved then led to an infinite set of algebraic equations for the unknown coefficients. By solving these algebraic equations a solution f of the partial differential equation was found.

Eighteenth-century mathematicians used the method of undetermined coefficients in a formal way, without worrying about the convergence of the power series they obtained as a solution. The first person to worry in print about such convergence was Cauchy (1789-1857). In ‘Mémoire sur l'emploi du calcul des limites dans l'intégration des équations aux dérivées partielles’ (Comptes rendus, 1842), Cauchy used a technique which he called the ‘calcul des limites’, now called the ‘method of majorants’ in English, to establish that the power series obtained by the method of undetermined coefficients always converge. This was the first version of the ‘Cauchy-Kowalevsky’ theorem to be proved.

Weierstrass (1815-97) was working on problems similar to those which Cauchy studied at about the same time, apparently in ignorance of Cauchy’s work. Weierstrass’s work was not published until 1894, and was communicated only to his students at Berlin after 1857, one of whom was Kovalevskaya. Weierstrass pursued these questions because he wanted to use differential equations in order to define analytic functions, and so he naturally had to consider the question of the existence of solutions. “Apparently [Weierstrass] thought that a general theorem could be proved to the effect that a power series obtained formally from a partial differential equation in which only analytic functions occur would necessarily converge. He says in a letter of 25 September 1874 to [Paul] du Bois-Reymond that he had made such a conjecture. It is this conjecture which Kovalevskaya was evidently supposed to prove, if possible” (Cooke, p. 30). She succeeded completely. “In the general theorem, the simple case illustrated [above] is generalized to functions of more than two independent variables, to derivatives of order higher than the first, and to systems of equations” (DSB).

“Kovalevskaya worked the material into its final form in July 1874 and submitted it to the University of Gottingen as one of the dissertations for the doctoral degree. The following month she submitted it to the Journal für die reine und angewandte Mathematik, where it appeared in 1875 [Bd. 80, pp. 1-32]. While the article was in press, Kovalevskaya returned to Russia. The final episode in the story is told in a letter from Weierstrass to Kovalevskaya dated 21 April 1875. Weierstrass says that he received the early 1875 issues of the Comptes rendus rather late, as a consequence of a delay in renewing his subscription. He was amazed to find in them two articles by [Jean Gaston] Darboux ‘On the existence of an integral in differential equations containing an arbitrary number of functions and independent variables.’ He continues, “So you see, my dear, that this question is one which is awaiting an answer, and I am very glad that my student was able to anticipate her rivals” … Weierstrass sent a copy of Kovalevskaya’s dissertation to Darboux’s mentor [Charles] Hermite in Paris and advised Kovalevskaya to ask [Carl] Borchardt, the editor of the Journal für die reine und angewandte Mathematik, to state in the journal that Kovalevskaya’s article had been received in August 1874. His fears of a priority dispute were exaggerated, however. Hermite and Darboux became Kovalevskaya’s closest friends and admirers in Paris and were later instrumental in promoting her participation in the Bordin Competition for 1888, in which she won international fame and 5,000 francs …

“Taken in the context of the time, Kovalevskaya’s paper can be considered significant for at least three reasons. First, it gave systematic conditions which the method of undetermined coefficients must work. Second, it charted the terrain, so to speak, for the application of analytic function theory in differential equations, showing under what conditions a differential equation was likely to have an analytic solution. Third, it showed that a differential equation could be used as the definition of an analytic function, when taken together with certain initial conditions” (Cooke, pp. 34-6).

Kovalevskaya (1850-91) “was the daughter of Vasily Korvin-Krukovsky, an artillery general, and Yelizaveta Shubert, both well-educated members of the Russian nobility … In Recollections of Childhood (and the fictionalized version, The Sisters Rajevsky), Sonya Kovalevsky vividly described her early life: her education by a governess of English extraction; the life at Palabino (the Krukovsky country estate); the subsequent move to St. Petersburg; the family social circle, which included Dostoevsky; and the general’s dissatisfaction with the “new” ideas of his daughters. The story ends with her fourteenth year. At that time the temporary wallpaper in one of the children’s rooms at Palabino consisted of the pages of a text from her father’s schooldays, namely, Ostrogradsky’s lithographed lecture notes on differential and integral calculus. Study of that novel wall-covering provided Sonya with her introduction to the calculus. In 1867 she took a more rigorous course under the tutelage of Aleksandr N. Strannolyubsky, mathematics professor at the naval academy in St. Petersburg, who immediately recognized her great potential as a mathematician.

“Sonya and her sister Anyuta were part of a young people’s movement to promote the emancipation of women in Russia. A favorite method of escaping from bondage was to arrange a marriage of convenience which would make it possible to study at a foreign university. Thus, at age eighteen, Sonya contracted such a nominal marriage with Vladimir Kovalevsky, a young paleontologist, whose brother Aleksandr was already a renowned zoologist at the University of Odessa. In 1869 the couple went to Heidelberg, where Vladimir studied geology and Sonya took courses with Kirchhoff, Helmholtz, Koenigsberger, and du Bois-Reymond. In 1871 she left for Berlin, where she studied with Weierstrass, and Vladimir went to Jena to obtain his doctorate. As a woman, she could not be admitted to university lectures; consequently Weierstrass tutored her privately during the next four years. By 1874 she had completed three research papers on partial differential equations, Abelian integrals, and Saturn’s rings. The first of these was a remarkable contribution, and all three qualified her for the doctorate in absentia from the University of Göttingen.

“In spite of Kovalevsky’s doctorate and strong letters of recommendation from Weierstrass, she was unable to obtain an academic position anywhere in Europe. Hence she returned to Russia where she was reunited with her husband. The couple’s only child, a daughter, “Foufie,” was born in 1878. When Vladimir’s lectureship at Moscow University failed to materialize, he and Sonya worked at odd jobs, then engaged in business and real estate ventures. An unscrupulous company involved Vladimir in shady speculations that led to his disgrace and suicide in 1883. His widow turned to Weierstrass for assistance and, through the efforts of the Swedish analyst Gösta Mittag-Leffler, one of Weierstrass’ most distinguished disciples, Sonya Kovalevsky vas appointed to a lectureship in mathematics at the University of Stockholm. In 1889 Mittag-Leffler secured a life professorship for her.

“During Kovalevsky’s years at Stockholm she carried on her most important research and taught courses (in the spirit of Weierstrass) on the newest and most advanced topics in analysis. She completed research already begun on the subject of the propagation of light in a crystalline medium. Her memoir, On the Rotation of a Solid Body About a Fixed Point (1888), won the Prix Bordin of the French Academy of Sciences. The judges considered the paper so exceptional that they raised the prize from 3,000 to 5,000 francs. Her subsequent research on the same subject won the prize from the Swedish Academy of Sciences in 1889. At the end of that year she was elected to membership in the Russian Academy of Sciences. Less than two years later, at the height of her career, she died of influenza complicated by pneumonia …

“An unusual aspect of Sonya Kovalevsky’s life was that, along with her scientific work, she attempted a simultaneous career in literature. The titles of some of her novels arc indicative of their subject matter: The University Lecturer, The Nihilist (unfinished), The Woman Nihilist, and, finally, A Story of the Riviera. In 1887 she collaborated with her good friend and biographer, Mittag-Leffier’s sister, Anne Charlotte Leffler-Edgren (later Duchess of Cajanello), in writing a drama, The Struggle for Happiness, which was favorably received when it was produced at the Korsh Theater in Moscow. She also wrote a critical commentary on George Eliot, whom she and her husband had visited on a holiday trip to England in 1869” (DSB).

Cooke, The Mathematics of Sonya Kovalevskaya, Springer, 1984.

4to (254 x 198 mm), pp [4], 32, contemporary half calf over blind-tooled cloth boards, front inner hinge week. A fine and fresh copy. Very rare.

Item #4203

Price: $12,500.00