On the hypotheses which lie at the bases of geometry. Edited, historically and mathematically commented by Jürgen Jost
The series ``Klassische Texte der Wissenschaft'' aims at providing reprints of important publications in mathematics and the sciences. The present book deals with Riemann's famous habilitation lecture, ``On the hypotheses upon which geometry is based'' [Gött. Abhandl. XIII. 1868. 1--20 (1868; JFM 01.0022.02)], edited by Jürgen Jost. The first chapter deals with the ``prehistory'' of Riemann's lecture: the problem of space in science from Aristotle until Newton, the discovery of non-Euclidean geometry, and the development of differential geometry. Chapter 2 is a reprint of Riemann's lecture from his collected works [Zbl 0703.01020] (see also [Zbl 0957.01031]) along with Hermann Weyl's comments from the same source. The next 20 pages (Chapter 4) are devoted to an analysis of Riemann's ideas by the editor, and this is followed (Chapter 5) by a study of the influence of Riemann's lecture on subsequent developments (key figures are Helmholtz, Lie and Weyl). Chapter 6 is a short essay on the relevance of Riemann's idea in modern research, and Chapter 7 consists of a commented bibliography. This is completed by an index, a glossary, and a time table. It is to be hoped that this excellent edition of Riemann's habilitation lecture gets a wide audience.
scientific article; zbMATH DE number 6153727