Mathematics – Geometry

Marcel Berger978-3-540-11658-5, 978-3-540-93815-6

This book атом essentially from two sources: a course called ”Maihe-maliques Element aires Approfondies”, taught in 1972-1973 and 1973-1974 at the University of Paris VII; and the author’s fifteen years of experience in preparing the geometry part of the orals in the Agrigation de Mathemaliquee. a competition lo select the best high school teachers in France.The main objectives pursued in this book, more precisely, in the framework of elementary geometry, are the following:— to emphasise the vbual, or ”artistic”, aspect of geometry, by using figures in abundance;— to accompany each nrw notion with as interesting a result as possible, preferably one with a simple statement but a non-obvious proof;— finally, to show ibal this simple-looking mathematics does not belong in a museum, that it is an everyday tool in advanced mathematical research, and that occasionally one encounters unsolved problems at even the most elementary level.Here are some particularities of this book that derive from the general principles above. Main definitions are, whenever possible, followed by non-trivial (and sometimes new) examples. Figures abound: at the beginning, especially, each geometric reasoning is illustrated by a diagram. (Such was the general practice fifty years ago, but pictures have all but disappeared from modern geometry books. One reason seems to be that authors think that readers keep pencil and paper next to them and draw figures as they go along, or else draw mental pictures. But the author’s experience from university examinations shows that students aren’t likely to draw pictures, either on paper or in their heads. Thus one of the aims of this book is to teach the reader to make systematic use of figures as he reads.)Notes are also common, referring to both the historical development and (he current, often very advanced, applications of the ideas introduced. This is meant to convey to the reader a feeling that the elementary mathematics that he is studying is an integral part of the living, continuing corpus of mathematical knowledge. The notes are backed by an extensive bibliography.