**Aleksandr Kotelnikov**'s father P I Kotelnikov was a colleague of Lobachevsky and, in fact, the only one of his colleagues to publicly praise his great geometrical achievements during his lifetime. This connection between Kotelnikov's father and Lobachevsky is important since the Lobachevsky connection was to play a large role in Kotelnikov's work throughout his life.

Kotelnikov was educated at the University of Kazan, graduating in 1884. After this he taught at a gymnasium in Kazan before entering the Department of Mechanics at Kazan to work for his university teachers qualification. He began teaching at Kazan University in 1893, then in 1896 he received his Master's Degree.

The thesis he presented for the Master's Degree was *The Cross-Product Calculus and Certain of its Applications in Geometry and Mechanics.* This thesis applied vector methods in theoretical mechanics, and he was to teach this vector approach to mechanics throughout his life.

Kotelnikov obtained his doctorate in 1899 for a thesis *The Projective Theory of Vectors* which generalised the vector calculus to the non-euclidean spaces of Lobachevsky and Riemann. He also applied this to mechanics in non-euclidean spaces. Much of his career is spent working on physics and non-euclidean geometry.

In 1899, the year he received his doctorate, he was appointed professor and Head of the Department of Pure Mathematics at the University of Kiev. Perhaps it is worth pointing out that in many countries today a doctorate is a lower degree, essentially a degree of a level to start university teaching. This was equivalent to the Master's Degree referred to here, where the doctorate was really the qualification necessary to become a professor.

In 1904 Kotelnikov left Kiev and returned to Kazan to become professor and Head of the Department of Pure Mathematics at the University. After 10 years he went back to Kiev again, this time o become Head of Theoretical Mechanics at the Polytechnical Institute. After another 10 years he moved, in 1924, to Moscow, where he worked at the Bauman Technical College until his death in 1944.

In 1927 he published one of his most important works, *The Principle of Relativity and Lobachevsky's Geometry.* He also worked on quaternions and applied them to mechanics and geometry.

Among his other major pieces of work was to edit the Complete Works of two mathematicians, Lobachevsky and Zhukovsky. He received many honours for his work, being named Honoured Scientist in 1934, then one year before he died he was awarded the State Prize of the USSR.

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

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