The present volume is a reprint, under one cover, of four short books written by outstanding German mathematicians. The first book, The Continuum by Hermann Weyl, is the study of the continuum, particularly its role in the foundations of analysis. The second piece is Mathematical Analysis of Space Problems, also by Weyl. The two main topics are infinitesimal geometry and the use of group theory in the analysis of space problems. These are followed by a selection of twelve additional topics, such as Mobius transformations, computing Riemannian curvature, and elements of Lie theory. Next in the collection is Edmund Landau's short book Statement and Proof of Some New Results in Function Theory. In eight short chapters, Landau considers a variety of topics, such as Tauberian theorems, Hadamard's three circle theorem, and Koebe's distortion theorem for schlicht functions. In his preface, Landau writes that he has chosen the topics for their great elegance. The final piece of this volume is Bernard Riemann's ground-breaking lecture On the Hypotheses Which Lie at the Foundations of Geometry.
In this one remarkable lecture, Riemann laid out the foundations of what became Riemannian geometry, particularly the notion of curvature for arbitrary manifolds. The story is told that, of the audience, only Gauss fully appreciated the depth of what Riemann was saying.
Publisher: American Mathematical Society