"The present second volume deals with many more advanced topics from Euclidean geometry and contains a long treatise about hyperbolic geometry."

This book is a continuation of Rothe's geometry magnum opus. The first volume presented "axioms from Hilbert's Foundations of Geometry" and progressed to "projective, neutral and basic Euclidean geometry." In contrast, the second volume presents advanced topics from Euclidean geometry, as well as in-depth studies of hyperbolic geometry. Familiar tools are mentioned, such as the straightedge and compass, which are well-known to students of Euclidean methods. The preface briefly talks about the theories of mathematicians Hilbert, Poincaré, and Gaus. Their work is subsequently explored, as are a plethora of many other theorems, models, and conclusions.

This work is geared toward the serious intermediate or advanced student and is well-structured and researched. In its 500 pages, the author expounds on practically every aspect of the field. Examples range from "Standard Euclidian Triangle Geometry" to the "Lunes of Hippocrates." The reader wades in, aware that the pool is familiar and soon will become deeper and more complex. Before the end of the book, the mathematician and geometry aspirant alike are stimulated and enlightened by topics such as Rothe's presentations on "About Gauss' differential geometry," the "Riemann metric of the Poincaré disk," the "Riemann metric of Klein's model," and "A second proof of Gauss' remarkable theorem." The author comes across throughout the book as knowledgeable of his applications as well as respectful of and grateful to his colleagues of the past and present. Rothe's book is an enjoyable read, particularly for those familiar with the work of the researchers to whom the author refers frequently.

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