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This book is based on lectures delivered by the authors at the Physics Department of Moscow University. In addition to the traditional material usually presented in the course of analytic geometry, this textbook includes some problems which find application in physics and theoretical mechanics (such as the notion of barycentric coordinates, the role of Euler's angles in problems related to the transformation of coordinates, and the presentation of arbitrary transformations as translations and rotations in space, optical properties of second-order curves, etc.). Also of interest is the Appendix containing Hilbert's system of axioms, the substantiation of the method of coordinates, and the fundamentals of non-Euclidean geometry. This textbook was awarded the 1980 State Prize of the USSR