**Boris Nikolaevich Delone**'s name is sometimes written Delaunay for, although the Russian transliterates into Delone, it is pronounced 'Delaunay'. The name sounds French, and indeed it is for it comes from an ancestor, a French Army officer named 'de Launay' who was a nephew of the Marquis de Launay the governor of the Bastille. The army officer served in Napoleon's army that invaded Russia in 1812 and, after being captured, married an aristocratic Russian lady and remained in Russia. Boris Nikolaevich's mother was Nadezhda Alexandrovna and his father was Nikolai Borisovich Delone, a professor of mechanics and physics at St Petersburg University. As one might expect of the son of a university professor, Boris Nikolaevich had an excellent education that developed his many skills and talents. As a child he took music very seriously, playing all the Beethoven sonatas and writing his own compositions that made his talent clear to his teachers. His music teacher insisted that the gifted boy should study music composition at the conservatory after graduating from his high school, while his drawing teacher strongly advised the Delone family that their talented son should continue his education at the Art Academy. Meanwhile, Boris Nikolaevich was also falling in love with science turning the room in his home into a physics laboratory and, at the age of fifteen, building his own telescope with a bronze mirror which he polished himself. Around this time he was also conducting his own mathematical researches and discovered for himself a proof of Gauss's reciprocity law. In addition to this remarkable range of talents, he became an exceptional mountain climber, learning to climb in the Swiss Alps where his family spent most summers when he was a child. He made his first ascent in 1903 in the Eastern Dolomites.

In the middle of his years at high school the Delone family had moved from St Petersburg to Kiev where Nikolai Borisovich completed his secondary education. He entered Kiev University in 1908 but by this time he had become involved with Russia's first aeronautic circle which he helped organise with his father who had been influenced by Nikolai Egorovich Zhukovsky. They built five gliders, each one an improvement on the one before as they learnt from their experiences. In 1908-1909 the eighteen-year-old Delone flew the gliders he was constructing, so becoming one of the first Russian glider pilots. In fact his first publication was *How to build and fly a cheap and light glider* in 1910. At Kiev University he studied under Vasilii Petrovich Ermakov and Dmitrii Aleksandrovich Grave. At this stage in his career, Grave was undertaking research in algebra and number theory, particularly Galois theory and the theory of ideals, and it was these topics that formed Delone's areas of research. As an undergraduate he wrote the prize-winning essay *The connection between the theory of ideals and Galois theory* which, despite its brilliance, was never published. He graduated from Kiev University in 1913 and after the award of his Master's Degree (equivalent to a doctorate) in 1916, he began teaching at the University as a Privatdozent. He became a member of the Mathematical Society which had among its members Ch T Bialobzeski, P V Voronets, N B Delone (the father of the subject of this biography), D A Grave, A A Friedmann, A P Kotelnikov, V P Linnik (I V Linnik's father) and O Yu Schmidt. We should also mention Delone's influence on Nikolai Grigorievich Chebotaryov, who began studying under Grave in 1912. The two young men were both very active participants in the algebra and number theory seminar at Kiev. Delone continued his passion for mountaineering and by the time he graduated in 1913, he had become one of the three leading mountaineers in Russia.

Following the Revolution of 1917 there was a change in policy towards education which, certainly in the Ukraine, had to become more technology based and more practical. Algebra certainly did not fit into this new educational philosophy and Grave's algebra seminar was forced to close. Some mathematicians, such as Grave himself, changed to study applied mathematical topics. Delone, however, chose to continue to study algebra and so was forced, in the 1920s, to leave the Ukraine. Delone moved to Petrograd in 1922. Petrograd was the name that St Petersburg had been given in 1914 and, two years after Delone began working there, in 1924, it was again renamed, this time to Leningrad. Delone worked at Leningrad University from 1922 until 1935. In August 1924 he attended the International Congress of Mathematicians in Toronto, Canada, where he delivered the invited address *Sur la sphére vide* . Delone married and on 22 May 1926 his son Nikolai Borisovich Delone was born. Nikolai Borisovich became a professor of physics with a high international reputation in nuclear physics. Nikolai Borisovich's son Vadim Delone (grandson of the subject of this biography) was a famous poet and human-rights activist.

The St Petersburg Mathematical Society had been founded in 1890 but disbanded at the time of the 1917 Revolution. However the Society was reformed in 1921 as the Petrograd Physical and Mathematical Society and Delone joined in the following year after his move to Petrograd. He played an active role in the Society along with other outstanding mathematicians such as Ya V Uspenskii, V I Smirnov, V A Steklov, A A Friedmann, V A Fok, A S Besicovitch, Sergei Bernstein, Ya D Tamarkin, R O Kuzmin and B G Galerkin. There he continued his interest in number theory, particularly the geometry of numbers, but he also worked on polyhedra, and on crystallography with important papers such as *On the question of the uniqueness of the determinations of the foundations of a parallelopiped crystal structure by the method of Debye* (1926). Regarding his work on polyhedra at this time, Igor Shafarevich (who was a doctoral student of Delone) writes [14]:-

In number theory, Delone studied cubic fields, in particular investigating the correspondence between binary cubic forms and rings in cubic fields. He studied Tschirnhaus's inverse problem, producing methods to determine whether or not two given cubic equations determine the same field. He invented a geometric method to tabulate all cubic rings whose discriminant does not exceed a given absolute value. He gave a geometric interpretation of binary cubics and their covariants which he then used in a reduction method.The work of Boris Nikolaevich in the theory of polyhedra was in the theory of the regular partition of spaces, the elements of which were initiated by the crystallographer Pedorov and continued by Voronoi. In this field Boris Nikolaevich originated two deep methods - "The method of the empty sphere" and "The method of the foliated construction". These two methods decided the extraordinarily difficult problem of the determination of all regular partitions of ternary and quaternary spaces.

The 1917 Revolution had made it impossible for Delone to go mountaineering for a number of years but he was able to start climbing again in the West Caucasus in 1923. His favourite area for climbing, however, was the Russian portion of the Altai Mountains. The highest mountain in the range is the twin-peaked Mt Belukha reaching 4500 m in height. It is close to the point where Russia, Kazakhstan, China and Mongolia (almost) meet. The third highest mountain in the Altai range, the 4070 m Delone Peak, close to Mt Belukha, has been named after Delone. In addition to Delone Peak, there is also Delone Col and Delone Pass (3400 m) leading to the Mensu glacier. In 1930 Delone was named "Master of Soviet mountaineering" and around the same time he organised mountaineering camps, being the first person to do so. He published the mountaineering guidebook *The peaks of the Western Caucasus* in 1938.

The Institute of Physics and Mathematics had been established by** **Vladimir Andreevich Steklov in Petrograd (as it was called at the time) in 1921. In 1932 the Institute of Physics and Mathematics was divided into two independent Departments, the Steklov Mathematical Institute headed by Ivan Matveevich Vinogradov and the Lebedev Physics Institute headed by Sergei Ivanovich Vavilov. Vinogradov invited some outstanding mathematicians to join the new Mathematical Institute including Delone. In this new Mathematics Institute, Delone became a colleague of Sergei Bernstein, N N Luzin, V I Smirnov, R O Kuzmin, N S Koshlyakov, N Y Kochin, S L Sobolev and D K Faddeev. From 1932 to 1960 Delone was Head of the Department of Algebra in Steklov Mathematical Institute and from 1960 to 1980 he was Head of the Department of Geometry. However, the Steklov Mathematical Institute moved to Moscow and, in 1935 Delone also moved to Moscow. He was professor of mathematics at Moscow State University from 1935 to 1943, being Head of the Department of Higher Geometry in the Faculty of Mechanics and Mathematics. In 1943, remaining at Moscow State University, he was made Professor of Higher Geometry and Topology, holding this position until 1960.

Delone published a number of important monographs. In 1940, in collaboration with D K Faddeev, he published *Theory of Irrationalities of Third Degree* (Russian). J V Uspensky begins a review as follows:-

The importance of the book is clearly illustrated by the fact that in 1964 the American Mathematical Society published an English translation of the treatise.The purpose of this outstanding monograph is to present all that is known at the present time about cubic irrationalities and such problems in number theory as are intimately connected with them. The book for the most part consists of the original investigations of its authors, and even that which has been contributed by other investigators is presented from a new and original point of view. Two features are very characteristic of the mode of presentation: on the one hand the extensive use of geometrical considerations as a background for the true understanding of complicated situations which otherwise would remain obscure, and on the other hand, the care shown by the authors in inventing effective methods of solution, illustrated by actual application to numerical examples and to the construction of valuable tables.

In 1947 Delone published *The St Petersburg school in the theory of numbers* (Russian) which described the works of the early number theorists who worked at St Petersburg University, namely P L Chebyshev, A N Korkin, E I Zolotarev, A A Markov, G F Voronoy and I M Vinogradov. An English translation, published by the American Mathematical Society, appeared in 2005. In 1948 he published the sixteen-page pamphlet *Mathematics and its development in Russia* (Russian) and in the following years he published some further historical work with papers such as *The work of Gauss in number theory *(1956), written to mark the 100^{th} anniversary of the death of Gauss, and *Euler as a geometer* (1958), written to mark the 250^{th} anniversary of the birth of Euler. In the 1960s he published *X-rays and crystals, 50 years since Max Laue's discovery* (1962), *An outline of the history of the development of mathematics in the Academy of Sciences of the USSR during the Soviet period (1917-1960)* (1963), and *The great French mathematician Josef Lagrange and his analytic mechanics* (1963).

Delone was also involved with secondary school mathematics, and in 1934, together with his pupil V A Tartakovskii, initiated and organized the first School Mathematics Olympiad. He also published a number of texts aimed at school pupils including (with O K Zhitomirski) *Problems with solutions for a revision course in elementary mathematics* (1928), (with O K Zhitomirski) *Problems in geometry* (1935), *Analytical geometry I* (1948), and (with D A Raikov) *Analytical geometry II* (1949). His book *A short exposition of the noncontradictoriness of the planimetry of Lobachevsky* (1953) is aimed at those with no more than a secondary school education. N A Court praises the book, saying that it:-

At Moscow State University, Delone taught a course on computers and in 1952 published... is written with consummate skill and great care. The author takes only for granted the rudiments of the mathematical equipment of those for whom the book is intended. Figures are abundant(about a hundred in all). Nothing is dismissed cryptically as being "perfectly obvious". And when the phrase occurs, it is followed by a suitable explanation.

*A short course in mathematical machines. Part 1. Small computing machines and mathematical instruments*which discussed calculators, the polar planimeter and other integrating instruments, and the construction and operation of a mechanical differential analyzer.

As a final comment let us note that in 1947 Delone published *On a duplicator linkage of Prof N B Delone* containing a four-line proof of a result on linkages due to his father Nikolai Borisovich Delone.

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