Aleksandr was the son of P.I. Kotelnikov, a colleague of Nikolai Lobachevsky. The subject of hyperbolic geometry was non-Euclidean geometry, a departure from tradition. The early exposure to Lobachevsky's work eventually led to Aleksandr undertaking the job of editing Lobachevsky's works.
Kotelnikov studied at Kazan University, graduating in 1884. He began teaching at a gymnasium. Having an interest in mechanics, he did graduate study. His thesis was The Cross-Product Calculus and Certain of its Applications in Geometry and Mechanics. His work contributed to the development of screw theory and kinematics, as noted by Wilhelm Blaschke (Kinematics and Quaternions (1960)). Kotelnikov began instructing at the university in 1893. His habilitation thesis was The Projective Theory of Vectors (1899).
In Kiev, Kotelnikov was professor and head of the department of pure mathematics until 1904. Returning to Kazan, he headed the mathematics department until 1914. He was at the Kiev Polytechnic Institute directing the department of Theoretical Mechanics until 1924, when he moved to Moscow and took up teaching at Bauman Technical University.
One reviewer put Kotelnikov at the head of a chain of investigations of Spaces over Algebras. Successive researchers included D.N. Zeiliger, A.P. Norden, and B. A. Rosenfel’d.
- 1925: Introduction to Theoretical Mechanics, Moscow-Leningrad
- 1927: The Principle of Relativity and Lobachevsky's Geometry, Kazan
- 1950: The Theory of Vectors and Complex Numbers, Moscow-Leningrad
- A.P. Shirokov (2002) "Spaces over algebras and their applications", Journal of Mathematical Sciences 108(2): 232–48
- A.T. Grigorian (1976) "Aleksandr Petrovich Kotelnikov", Dictionary of Scientific Biography.
- B.L. Laptev & B.A. Rozenfel'd (1996) Mathematics of the 19th Century: Geometry, page 87, Birkhäuser Verlag ISBN 3-7643-5048-2 .
- O'Connor, John J.; Robertson, Edmund F., "Aleksandr Petrovich Kotelnikov", MacTutor History of Mathematics archive, University of St Andrews.