The present volume is a translation of the first of the two chapters that make up Part III of Geometric Pansformations by I.M. Yaglom. Parts I and II are available in English as NML volumes 8 and 21. The second, not yet translated, chapter of Part III deals with inversions, that is, transformations of the plane that take circles (including lines which may be viewed as circles of infinite radius) into circles, but need not carry lines into lines; it also deals with transformations of the plane that take circles (including points which may be viewed as circles of zero radius) into circles, and lines into lines, but need not carry points into points. Like the first chapter, so too the second chapter concludes with a Supplement on hyperbolic geometry. For technical reasons, it was not possible to include both chapters of the Russian original Part III in a single NML volume; it is hoped that the remaining untranslated chapter will, a t some future time, be published in English as a separate volume.

This NML volume deals with affine and projective transformations of the plane. As in NML 8 and 21, the 112 problems interspersed throughout the text are an essential part also of this book; their detailed solutions are included.

The translator wishes to thank Professor Yaglom for his valuable assistance in preparing this American edition. He examined the draft of the translation, made several additions and corrections, and prepared a number of new problems for the translation, not present in the Russian version.