**Boris Yakovlevic Bukreev**'s father was a graduate of Kharkov University and had become a school teacher in Lgov. The city is in western Russia in the province of Kursk, about 65 km west of the provincial capital Kursk. The family had a tradition of school teaching, with Boris Yakovlevic's paternal grandfather also being a teacher. His early education was at home, after which he attended the classical Kursk Gymnasium. From the second grade onwards he received a scholarship of 16 rubles per month. He showed great mathematical ability when at the Gymnasium as well as a great love of mathematics. In the summer of 1878 he graduated from the Gymnasium having been awarded the silver medal in his final year.

In the autumn of 1878 Bukreev entered the Faculty of Physics and Mathematics of Kiev University. The university had been founded in 1834 and had, from its foundation, a strong school of mathematics. In particular Vasilii Petrovich Ermakov (1845-1922), who was especially interested in the pedagogy of mathematics, taught Bukreev. Ermakov was the first to show that some nonlinear differential equations of second order are simply related with linear differential equations of second order. His method became known as the 'Method of Ermakov'. Among all Bukreev's lecturers, Ermakov was the one who exerted the greatest influence on him as an undergraduate. But other outstanding lecturers in the Department of Mathematics also taught Bukreev including Mikhail Egorovich Vashchenko-Zakharchenko and P E Romer. In addition to mathematics, he studied physics and among his teachers on this topic special mention should be made of M P Avenarius and N Schiller, the latter having worked with Helmholtz in Berlin before coming to Kiev. Other topics which Bukreev studied include: mechanics where be was taught by Ivan Ivanovich Rakhmaninov (1826-1897) from the Department of Mechanics; astronomy taught by Mitrofan Fedorovich Handrikov (1837-1915), the Professor of astronomy; and chemistry taught by A Bazarov.

In 1880 Bukreev showed he was one of the best students in the Faculty of Physics and Mathematics when he was awarded the Faculty's gold medal. He completed his first degree in 1882 and remained at the University of Kiev to train to become a university professor. His interests at this time were focussed on Weierstrass's theory of elliptic functions and this was the topic which he chose as the subject of his Master's Thesis (equivalent to an American/British Ph.D.). This thesis was entitled *On the expansion of transcendental functions in partial fractions* and as well as containing an impressive collection of new results, it also contained an historical introduction describing developments from Gauss to Mittag-Leffler. The authors of [2] write:-

Publication of this thesis represented a significant contribution to Russian mathematical literature.

Before his thesis was published, Bukreev had published several articles such as *On some applications of a theorem of Mittag-Leffler* and, around the time his thesis was published, another two articles appeared. Both were on evolutes, one being simply titled *A note on evolutes*.

After completing his Master's thesis, Bukreev went on a research visit abroad. He visited Berlin where he heard lectures on the theory of hyperelliptic functions by Karl Weierstrass, lectures on the theory of Abelian functions and linear differential equations by Lazarus Fuchs, and lectures on the theory of numbers from Leopold Kronecker. Bukreev undertook research on Fuchsian functions and, guided by Fuchs, he had completed the work by the end of 1888. The results of this research were submitted as his doctoral thesis *On the Fuchsian functions of rank zero*. He defended his thesis in May 1889. Fuchsian functions had been studied earlier by Poincaré but his approach had been geometrical, while Bukreev's approach was entirely analytical. Bukreev determined the conditions for continuity of Fuchsian groups and constructed differential equations corresponding to each of the discontinuous groups.

He became an extraordinary professor of mathematics at the University of Kiev from 1889, later being made an ordinary professor. Vashchenko-Zakharchenko was teaching at Kiev when Bukreev was appointed and he had a considerable influence on the direction of Bukreev's research. During the 1890s Bukreev produced a series of high quality papers including: *On the theory of gamma functions*; *On some formulas in the theory of elliptic functions of Weierstrass*; *On the distribution of the roots of a class of entire transcendental functions*; and *Theorems for elliptic functions of Weierstrass*. By the end of the 1890s Bukreev's research interests had moved somewhat and he began to undertake research into differential geometry; in 1900 he published *A Course on Applications of Differential and Integral Calculus to Geometry*. Although at this time his main position was in the University of Kiev, he also taught at the Women's College from 1896 and at the Kiev Polytechnic Institute from 1898. He continued to teach at the Polytechnic Institute until 1926.

Bukreev's work was broad and in addition to the areas of complex functions, differential equations, the theory and application of Fuchsian functions of rank zero, and geometry, he published papers on algebra such as *On the composition of groups* (1900). After 1900 he became interested in the theory of series, publishing papers such as *Notes on the theory of series* and he also worked on the Calculus of Variations. His vigorous research activity did not prevent him from devoting time to teaching of the highest quality [2]:-

His scientific and pedagogical activities are inextricably linked together harmoniously and complement each other.

Bukreev incorporated into his lectures new mathematical ideas that he had developed himself and also the latest research which he had read about in current journals. He taught courses on analysis, differential and integral calculus and their applications to geometry, the theory of integration of differential equations, the theory of series, algebra, and other topics. He was the first to introduce a course entirely devoted to the theory of surfaces, beginning in 1897. The way that he taught was greatly influenced by the style of teaching he had encountered from Weierstrass, Kronecker and Dedekind [2]:-

Due to his teaching of the foundations of mathematical analysis in the1890s and1900s, the level at the University of Kiev was raised to new heights.

The basic principles Bukreev adopted as a teacher included ensuring completeness of coverage of the topic under discussion, accuracy and clarity of presentation, and presenting his material in simple language.

The Russian tsar was deposed in 1917 and the Bolsheviks, under the leadership of Lenin, came to power in the October Revolution of 1917. The Communist government introduced major changes to the education system which greatly affected all university staff. This was a period in the Ukraine when new organizational forms and new methods of teaching at secondary and tertiary levels were introduced. Kiev National University was transformed first into the Higher Institute of Education, then into the Institute of Education, then into the Physics, Chemistry and Mathematics Institute. Bukreev worked in all these different versions of the university. In 1933 the University of Kiev was restored and at this time a Department of Geometry was created in the Mechanics and Mathematics Faculty; the first head of this Department was Bukreev.

Bukreev published a number of books which proved influential. For example:* Introduction to the theory of series*; *Elements of the theory of determinants*; *Course on definite integrals* (1903); and *Elements of algebraic analysis* (1912). The authors of [2] write:-

These books are written at a high scientific level, yet in simple and clear language. In many places, he provides historical information as well as literary references for those wishing to deepen their knowledge in a particular matter.

In 1934, he published *An Introduction to the Calculus of Variations*. In the Introduction he explains how important he sees the practical applications of his mathematics:-

The Calculus of Variations is central to the physical and mathematical sciences. All the rays coming from a number of subjects: geometry, mechanics, physics, engineering ... converges in that topic. One who is studying the Calculus of Variations not only repeats and learns infinitesimal analysis, but also understands that this analysis is a powerful tool to address the many issues that are of a purely practical nature.

He was interested in both projective and non-Euclidean geometry, publishing about fifteen articles on this latter topic including *Equidistant lines of constant geodesic curvature in the planimetry of Lobachevsky* (1955) and *Lobachevskian geometry *(1957). Note that, remarkably, Bukreev was ninety-eight years old when this last mentioned paper was published. His most important book on non-Euclidean geometry was *Non-Euclidean Planimetry in Analytic Terms* which he published in 1951. The book contains nine chapters: I. The geodesics as solutions of the Euler equation, Gauss curvature, geodesic curvature; II. Formula for the parallel angle; III. Equidistant curves and limit circles; IV. Motions; V. Geodesic triangles, concurrence of altitudes, medians, etc.; VI. Hyperbolic trigonometry; VII. Lambert quadrilaterals; VIII. Area of triangles and circles, length of circles; IX. Realisation as geometry on the pseudosphere. H Busemann, in a review of the work, is not entirely complimentary:-

The book develops hyperbolic geometry from the point of view of differential geometry. It is very elementary and reliable, but without zest and elegance.

Bukreev also made many contributions to the history of mathematics writing biographies on the life and work of a number of mathematicians including Vasilii Petrovich Ermakov, Mikhail Egorovich Vashchenko-Zakharchenko, and Gaspard Monge. Having seen many events in the history of Russian mathematics and the history of the University of Kiev, Bukreev was able to recount many interesting episodes from his personal experience. In 1956 Bukreev spoke at the special ceremonial meeting of the Academic Council of the Faculty of Mechanics and Mathematics of Kiev University on the topic *A hundred years from the death of N I Lobachevsky*. We note that Lobachevsky died only three years before Bukreev was born, yet he was able to give the address at the celebrations for the centenary of his birth.

Bukreev also contributed to high school education showing a keen interest in the teaching of elementary mathematics. From 1907 to 1917, he gave the annual reviews of the written examinations in mathematics in high schools in the Kiev district. He also played a major role in university affairs serving as Chairman of the Finance Committee and Vice Dean of the Faculty of Natural Sciences. In 1922 he was elected a member of the Central Committee of the USSR Union of Educators, and was elected as chairman of the Qualification Commission. In 1925, Bukreev travelled to Leningrad, as a delegate from the Kiev Polytechnic Institute, to deliver his address on 200^{th} anniversary of the founding of the Russian Academy of Sciences.

The authors of [4] write:-

For60years, Boris Yakovlevic was the Head of Department at the University of Kiev, first in the chair of general mathematics, and then in the chair geometry. He directed the preparation of scientific workers and graduate students, helping young teachers just coming from school to master the skill of teaching. Due to his personal charm, Boris Yakovlevic was to his students not only a teacher but an older friend. His paternal attitude to students and young teachers who he cared about, and his faithful performance of duty as a teacher and educator, served as an example to many of his young colleagues.

In 1939, when World War II broke out, Bukreev was eighty years old. This proved an extremely difficult time for him. The German invasion in 1941 brought severe suffering and destruction to Kiev. After a fierce 80-day battle, German forces entered it on the 19^{th} September 1941. In 1943, after bitter fighting, Soviet troops liberated Kiev on 6 November. The city itself had suffered great destruction but Bukreev was one of the first to return to the university. He found that the mathematical library and the mathematical models he had made had been destroyed by the occupying troops. Despite his advanced age, he began to rebuild the department, lecturing in half unheated lecture theatres during severe frosts in January 1944. Bukreev had joined the Moscow Mathematical Society in 1893 and the fiftieth anniversary of his joining fell in the middle of the war. However, in 1946 the Moscow Mathematical Society honoured his achievement, making him an honorary member.

When he reached his 100^{th} birthday he was greatly honoured. The authors of [4] write this summary of his contributions:-

The work of Boris Yakovlevic Bukreev shows the closely intertwined aspirations of the teacher and scholar. Boris Yakovlevic worked on the theory of functions of a complex variable, on mathematical analysis, on algebra, on the calculus of variations, and on differential geometry. Recently, he has focused on the geometry of Lobachevsky, promoting and developing the eternal ideas of the great Russian geometer. Boris Yakovlevic educated generations of mathematicians. Many of his students later became outstanding scientists. His lectures inspired generations of listeners. He aimed to kindle in them a spark of independent creative thought and carefully fostered the development of their scientific interests. In many parts of the Soviet Union, even in its most remote corners, there are people who know Boris Yakovlevic and remember him with love. There were many telegrams and letters for him on the day of his anniversary. Boris Yakovlevic did not always personally know the authors of these letters, but they all considered themselves his disciples by reading his books from which they learned to love mathematics.

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

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