Divine Proportions - Rational Trigonometry to Universal Geometry

This book deals with the Euclidean geometry of plane and space, not only over the field of real numbers but over a general field, in particular over the field Fp, p 2 and prime. The reader will become acquainted with many theorems of elementary geometry such as Menelaus’ theorem, Ceva’s theorem, Ptolemy’s theorem and Stewart’s theorem. Feurbach’s nine point circle and the Euler line are presented. Conics are given by equations in linear coordinates and then parabolas, quadrolas and grammolas are defined. In three-dimensional space, Platonic solids are investigated. Besides linear coordinates the book also contains an explanation of polar and spherical coordinates, but not in the usual way (angles are not used). There is an interesting study of circles and other conics and their tangents in geometries over finite fields.

The book differs from current textbooks; it gives a new foundation for Euclidean geometry and trigonometry. The author suggests working with quadrance, i.e. using the square of distance and “spread” and “gross” instead of sine and cosine, which means the square of sine and the square of cosine in the case of real numbers. The author calls this technique “Rational Trigonometry” and adopts a purely algebraic approach, which is in his opinion a conceptually simpler framework for students. The reviewer does not share his optimistic point of view. For example, it is not clear in his modification of the cosine law which of two angles he is dealing with (acute or obtuse). The method used by the author has a lot of negative effects, nevertheless the book presents a very interesting exposition of many facts and theorems of elementary geometry.

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