Читать книгу A course of plane geometry - Carlos Alberto Cadavid Moreno - Страница 7

Preface

In 2010 I was asked to teach the course Geometry in Context, a first semester course for students of the programs of Mathematical Engineering and Physical Engineering, at Universidad EAFIT, in Medellín, Colombia. The goal of the course was that students learned euclidean plane geometry in the classical way, and also that they applied this theoretical knowledge in a real situation, by requiring them to build a mechanism of their choice, which applied some geometrical principle. In preparing the course I consulted several books presenting euclidean plane geometry in the classical way. In studying these books I started to feel uneasy, because the discussion did not adhere to the modern standard of rigor, making proofs kind of difficult to follow, at least to me. I was aware that the great mathematician David Hilbert had proposed a rigorous presentation of the euclidean geometry of space in his book Grundlagen der Geometrie (Foundations of Geometry)[4], published in 1899. I went to our university’s library, looking for a copy of Hilbert’s book. Old books are usually hard to read, and Hilbert’s book was in that category. I kept looking for a modern book which explained Hilbert’s work more clearly, and I stumbled upon the book Geometry: Euclid and Beyond [1], by R. Hartshorne. I browsed the book and I knew I had found the reference I needed. The first chapter of the book presents classical euclidean geometry pointing out its shortcomings, and the second chapter presents Hilbert’s axiomatization of (plane) geometry and discusses how to interpret and prove the propositions in Book I of the Elements, in the light of Hilbert’s approach. When I began reading chapter two more carefully I was amused when I found out that Hartshorne’s presentation made it possible to introduce, in the most natural way, non-euclidean geometries from the very beginning! A subject that is usually taught in a Differential Geometry course. But I soon realized that Hartshorne’s book is rather dense, going too fast for a first semester student. My task in teaching the course was to explain slowly and in complete detail, most of chapter two of Hartshorne’s book. This book is an account of this effort. I follow Hartshorne’s approach throughout, and I even paraphrase some of the interesting problems he proposes.