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Appendix is an outstanding document in the history of mathematics and human culture. It presents the solution of a more than two thousand year old problem in connection with parallel lines and the description of János Bolyai's discovery of non-Euclidean and the more general absolute geometries. Non-Euclidean geometry changed our thinking not only about geometry, but mathematics, in general, and paved the way for the creation of the modern physical theories of the twentieth century.