**Dmitrii Matveevich Sintsov** was born in Viatka (sometimes written as Vyatka) which was a large city in western Russia and the administrative centre of the Kirov Province. The change of name of this city to Kirov did not happen until 1934 when it was renamed after the Soviet official Sergey M Kirov. He attended the Third Kazan High School, graduating with the Gold Medal in 1886. Later in the year in which he graduated from the High School, he began his studies at Kazan University, graduating in 1890. This University, the result of one of the many reforms of the Emperor Alexander I, was founded in 1805, and was famed in mathematics by having Lobachevsky as its rector from 1827 to 1846. By the time Sintsov began his university studies he was already convinced that mathematics was the topic for him to concentrate on, and he became a member of the mathematics section of the Physics and Mathematics Faculty of the university. His lecturers in mathematics were A V Vasil'ev, F M Suvorov, V V Preobrazhenskii and P S Nazimov. He also took courses in astronomy with D I Dubyago.

Sintsov's first research was on Bernoulli functions of fractional order and he carried this out while taking his fourth year undergraduate courses. His paper on the topic was published in the *Notices of the Kazan Physics and Mathematics Society* in 1890. This was a remarkable piece of work for a student at this stage in his undergraduate studies and it earned him a Gold Medal. Although Sintsov's interests moved away from the areas of his first scientific investigations, nevertheless he did undertake further research into Bernoulli functions and published further papers on this topic near the beginning of his career. Having made such an excellent start to his research, his "esteemed teacher" Aleksandr Vasil'evich Vasil'ev (1853-1929) recommended that he continue his studies at the University of Kazan with the aim of qualifying as a High School teacher. He spent three years, from the beginning of February 1891 to the beginning of February 1894, taking the necessary courses to obtain his teaching qualification. During this period he was being advised on research topics by Vasil'ev and, following his advice, he wrote his Master's Thesis *The Theory of Connexes in Space in Connection with the Theory of First Order Partial Differential Equations*. I A Naumov explains in [7]:-

The German mathematician A Clebsch was the first to investigate the theory of connexes in the period1870-1872. He considered plane connexes i.e., plane geometrical objects, where the point-straight line combination was chosen as the basic element of the plane. Such connexes are termed ternary. Clebsch constructed the geometry of a ternary connex and applied it to the theory of ordinary differential equations.

Sintsov was appointed to the staff of Kazan University and taught there from 1894 to 1899. After leaving Kazan, Sintsov taught at the Odessa Higher Mining School, then, in 1903, he was appointed to Kharkov University where he taught until his death in 1946. He took a leading role in the development of mathematics at Kharkov University and, for many years, he was President of the Kharkov Mathematical Society. This Society is one of the early mathematics societies, being founded in 1879. Following Vladimir Andreevich Steklov's presidency from 1902 to 1906, Sintsov took over as President, and held the position until his death forty years later [8]:-

Through Sintsov's initiative, the Kharkov Mathematical Society was deeply involved in the improvement of mathematical education in the schools of the Kharkov region. Sintsov also put considerable effort into maintaining the Kharkov Mathematical Society mathematical library which is still one of the most complete mathematical libraries in the Ukraine.

Sintsov had an outstanding research record, and published 267 works during his long and productive scientific and teaching career. Of course through his many years of research his interests varied but the main areas on which he worked were the theory of conics and applications of this geometrical theory to the solution of differential equations and, perhaps most important of all, the theory of nonholonomic differential geometry. I A Naumov writes [7]:-

His classical work on the theory of connexes, of which he was one of the founders, and on nonholonomic differential geometry are well known far beyond the frontiers of our country.

The book in which the articles [2] (written by Ja P Blank who was a student of Sintsov) and [5] appear, contains a selection of the Sintsov's major works on nonholonomic geometry. These were first published during the years 1927-1940 and include: *A generalization of the Enneper-Beltrami formula to systems of integral curves of the Pfaffian equation Pdx + Qdy + Rdz = *0 (1927); *Properties of a system of integral curves of Pfaff's equation, Extension of Gauss's theorem to the system of integral curves of the Pfaffian equation Pdx + Qdy + Rdz = *0 (1927); *Gaussian curvature, and lines of curvature of the second kind* (1928); *The geometry of Mongian equations* (1929); *Curvature of the asymptotic lines (curves with principal tangents) for surfaces that are systems of integral curves of Pfaffian and Mongian equations and complexes* (1929); *On a property of the geodesic lines of the system of integral curves of Pfaff's equation* (1936); *Studies in the theory of Pfaffian manifolds (special manifolds of the first and second kind)* (1940) and *Studies in the theory of Pfaffian manifolds* (1940).

At Kharkov University, Sintsov created a school of geometry which became the leading school in this field in the Ukraine and has continued to flourish through the years still today being a leading centre. There he studied the geometry of Monge equations and he introduced the important ideas of asymptotic line curvature of the first and second kind. In 1903 he published two papers on the functional equation *f* (*x*, *y*) + *f* (*y*, *z*) = *f* (*x*, *z*), now called the 'Sintsov equation,' which are discussed by Detlef Gronau in [4]. He writes:-

Sintsov gave in1903an elegant proof of its general real solution, which has the formf(x,y) =q(x) -q(y), where q is an arbitrary function in one variable. ...[Sintsov]was the first who gave(in two papers ... in1903)elementary simple proofs of its general real solutions. But before, it was Moritz Cantor who proposed these equations(there are two equations). In his journal 'Zeitschrift fur Mathematik und Physik,' ... he published[a note on them]in1896. Cantor quotes these equations as examples of equations in three variables which can be solved by the method of differential calculus due to Niels Henrik Abel. ... The proof of Sintsov is much simpler and elegant.

Sintsov also took an interest in the history of mathematics and one of the major projects which he undertook in this area was the detailed study of the work of previous mathematicians at Kharkov University. This work provides a fascinating account of the development of mathematics there from the founding of the university in 1805.

The Ukrainian Academy of Sciences honoured Sintsov by electing him to membership on 22 February 1939.

**Article by:** *J J O'Connor* and *E F Robertson*

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