For the last thirty or forty years, most Americans have somehow lost interest in geometry. The present book constitutes an attempt to revitalize this sadly neglected subject. The four parts correspond roughly to the four years of college work. However, most of Part II can be read before Part I, and most of Part IV before Part III. The first el.even chapters (that is, Parts I and II) will provide a course for students who have some knowledge of Euclid and elementaw analytic geometry but have not yet made up their minds to speialize in mathematics, or for enterprising high school teachers who wish to
see what is happening just beyond their usual curriculum. Part Ill deals with the foundations of geometry, including projective geometry and hyperolic non-Euclidean geometry. Part IV introduces differential geometry, combinatorial topology, and four-dimensional Euclidean geometry. In spite of the large number of cross references, each of the twenty-two chapters is reasonably self-contained; many of them can be omitted on first reading without spoiling one's enjoyment of the rest. For instance, Chapters
1, 3, 6, 8, 13, and 17 would make a good short course. There are relevant exercises at the end of almost every section; the hardest of them are proided with hints for their solution. (Answers to some of the exercises are given at the end of the book. Answers to many of the remaining exercises are provided in a separate booklet, available from the publisher upon request.) The unifying thread that runs through the whole work is the idea
of a group of transformations or, in a single word, symmetry.

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