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The philosopher Immanuel Kant writes in the popular introduction to his philosophy: There is no single book about metaphysics like we have in mathematics. If you want to know what mathematics is, just look at Euclid’s Elements. (Prolegomena Paragraph 4) Even if the material covered by Euclid may be considered elementary for the most part, the way in which he presents essential features of mathematics in a much more general sense, has set the standards for more than 2000 years. He displays the axiomatic foundation of a mathematical theory and its conscious development towards the solution of a specific problem. We see how abstraction works and how it enforces the strictly deductive presentation of a theory. We learn what creative definitions are and how the conceptual grasp leads to the classification of the relevant objects. For each of Euclid’s thirteen Books, the author has given a general description of the contents and structure of the Book, plus one or two sample proofs.In an appendix, the reader will find items of general interest for mathematics, such as the question of parallels, squaring the circle, problem and theory, what rigour is, the history of the platonic polyhedra, irrationals, the process of generalization, and more. This is a book for all lovers of mathematics with a solid background in high school geometry, from teachers and students to university professors. It is an attempt to understand the nature of mathematics from its most important early source.
