This textbook comprises lectures read by the author to the first-year students of mathematics at Moscow State University. The book is divided into two parts containing the texts of lectures read in the first and second semesters, respectively. Part One contains 29 lectures and read in the first semester.The subject matter is presented on the basis of vector axiomaticsof geometry with special emphasis on logical sequence in introduction of the basic geometrical concepts. Systematic exposition and application of bivectors and trivectors enables the author to successfully combine the above course of lectures with the lectures of the following semesters. The book is intended for university undergraduates majoring in mathematics.