This part of Geometric Transformations is devoted to the study of transformations of the plane which carry lines into lines. These are known as affine and projective transformations, or simply as collineations, and studied a t the university in the USSR. Yet, the present work is aimed primarily at readers concerned with high school mathematics: high school students and teachers as well as prospective high school teachers and their teachers. Thus, the main objective of the present work is to demonstrate the close connection between affine and projective transformations (affinities and projectivities) and elementary geometry.

Considerations of space ruled out almost entirely a discussion of more advanced theories connected with geometric transformations. The one significant excursion into “higher geometry” is the Supplement devoted to hyperbolic geometry. But even here no effort was spared to keep the exposition elementary in the hope of making the Supplement accessible to the more persistent high school students.

The problems form an essential part of the book; their solutions appear in the second half of the book. While the basic text is entirely independent of the problems, the author believes that the reader's attempts to solve at least some of them is bound to deepen his understanding of the text. All the problems pertain to elementary geometry, except those in the Supplement which are intended to acquaint the reader with concrete theorems of hyperbolic geometry. (To keep the exposition at once concise and elementary the author refrained from introducing the concept of “conic section”.