Pangeometry

Nikolai I. Lobachevsky, Athanase Papadopoulos

(Heritage of European Mathematics)

European Mathematical Society | 2010 | 332 páginas | PDF | 12,5 Mb

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Lobachevsky wrote his Pangeometry in 1855, the year before his death. This memoir is a résumé of his work on non-Euclidean geometry and its applications, and it can be considered as his clearest account on the subject. It is also the conclusion of his lifework, and the last attempt he made to acquire recognition. The treatise contains basic ideas of hyperbolic geometry, including the trigonometric formulae, the techniques of computation of arc length, of area and of volume, with concrete examples. It also deals with the applications of hyperbolic geometry to the computation of new definite integrals. The techniques are different from those found in most modern books on hyperbolic geometry since they do not use models.

Besides its historical importance, Lobachevsky’s Pangeometry is a beautiful work, written in a simple and condensed style. The material that it contains is still very alive, and reading this book will be most useful for researchers and for students in geometry and in the history of science. It can be used as a textbook, as a source book and as a repository of inspiration.

The present edition provides the first complete English translation of thePangeometry that appears in print. It contains facsimiles of both the Russian and the French original versions. The translation is accompanied by notes, followed by a biography of Lobachevky and an extensive commentary.

Contents

Foreword v

On the present edition xi

I. Pangeometry 1

English translation 3

French original from 1856 79

Russian original from 1855 145

II. Lobachevsky’s biography 203

Preface to Lobachevsky’s 1886 biography 205

Lobachevsky’s biography (1886) 217

III. A commentary on Lobachevsky’s Pangeometry 227

Introduction 229

1. On the content of Lobachevsky’s Pangeometry 235

2. On hyperbolic geometry and its reception 260

3. On models, and on model-free hyperbolic geometry 280

4. A short list of references 285

5. Some milestones for Lobachevsky’s works on geometry 290

Bibliography 299

Review in Zentralblatt MATH 1208.51013

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