Preface

The reader is invited to immerse himself in a “love story” which has been unfolding for 35 centuries: the love story between mathematicians and geometry. In addition to accompanying the reader up to the present state of the art, the purpose of this Trilogy is precisely to tell this story. The Geometric Trilogy will introduce the reader to the multiple complementary aspects of geometry, first paying tribute to the historical work on which it is based and then switching to a more contemporary treatment, making full use of modern logic, algebra and analysis. In this Trilogy, Geometry is definitely viewed as an autonomous discipline, never as a sub-product of algebra or analysis. The three volumes of the Trilogy have been written as three independent but complementary books, focusing respectively on the axiomatic, algebraic and differential approaches to geometry. They contain all the useful material for a wide range of possibly very different undergraduate geometry courses, depending on the choices made by the professor. They also provide the necessary geometrical background for researchers in other disciplines who need to master the geometric techniques. The present book leads the reader on a walk through 35 centuries of geometry: from the papyrus of the Egyptian scribe Ahmes, 16 centuries before Christ, to Hilbert’s famous axiomatization of geometry, 19 centuries after Christ. We discover step by step how all the ingredients of contemporary geometry have slowly acquired their final form. It is a matter of fact: for three millennia, geometry has essentially been studied via “synthetic” methods, that is, from a given system of axioms. It was only during the 17th century that algebraic and differential methods were considered seriously, even though they had always been present, in a disguised form, since antiquity. After rapidly reviewing some results that had been known empirically by the Egyptians and the Babylonians, we show how Greek geometers of antiquity, slowly, sometimes encountering great difficulties, arrived at a coherent and powerful deductive theory allowing the rigorous proof of all of these empirical results, and many others. Famous problems—such as “squaring the circle”—induced the development of sophisticated methods. In particular, during the fourth century BC, Eudoxus overcame the puzzling difficulty of “incommensurable quantities” by a method which is