| Summary: | The present book differs widely in content, methods, and point of view from traditional presentations of the subject. Herein more space is devoted to the discussion of the basic concepts of distance, motion, area and perpendicularity. In fact, the non-Euclidean geometries are reached via general metric spaces and the Hilbert problem of finding those geometries in which straight lines are the shortest connections. Of course, the general problem is only formulated here; but this leads naturally to the consideration of geometries other than the Euclidean and two non-Euclidean ones, and thus to the modern view in which the three classical geometries are very special, and closely related, cases of general geometric structures. The overall aim is to counteract the impression of geometry as an isolated and static subject, and to present its methods and essential content as part of modern mathematics
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