History of Science and Engineering Annotation << Back On the History and Methodology of Progress of the Problem of Rotation of a Rigid body Around a Fixed Point in the 18–20th Centuries Yulina A.O. The article considers the history of the analytical solution of the classical problem of spherical motion of a rigid body. The methodology of the presented problem is shown in close connection with the history of the development of mathematical methods for solving problems in mechanics. Based on the obtained historical and mathematical research, a new chronology of the main cases of solving the problem of rotation of a rigid body about a fixed point is obtained. The work recreates a complete picture of the study of issues of rigid body dynamics in the 17th 19th centuries. The main methods of the 18th century for solving the problem of rotation of a rigid body about a fixed point are classifi ed. Based on the study of the works of mathematicians and mechanics of the 17th-18th centuries, the relationship of this problem with the development of mathematical methods for solving it in mechanics is shown. As a result, a new classification of various cases of integrability of the rotation problem is presented. The description of L. Euler's approach to solving the classical problem, the practical and analytical advancement of the problem by J.L. Lagrange, and, fi nally, the fi nal and complete solutions performed by K. Jacobi and O.I. Somov using hyperelliptic functions are clarifi ed and supplemented. The result of the analysis was a problem-chronological systematization of the complete analytical solution to the problem of rotation of a rigid body around a fi xed point. The work contains a historical and scientifi c analysis of analytical solutions in the works of L. Euler, J.L. Lagrange, L. Poinsot; an analysis of the development of the apparatus of the theory of elliptic functions in the works of N. Abel, O.I. Somov, K. Jacobi, K. Weierstrass and B. Riemann. The result of this analysis was the definition of two main theoretical approaches applicable to the problems of rigid body dynamics: geometric in the works of Riemann, and analytical based on the theory of elliptic functions. Keywords: Rotation problem, impact, elliptic functions, L. Euler, J.L. Lagrange, O.I. Somov, K. Jacobi, K. Weirstrass. DOI: 10.25791/intstg.2.2025.1526 Pp. 03-12.