Felix Klein and Sophus Lie: Evolution of the Idea of Symmetry in the Nineteenth Century

Description

cited in the references of the historical introduction of Cantwell's Introduction to Symmetry Analysis (pp. xl-xli, PDF pp. 42-43)

mentions Weber (and his relationship with Gauss, Riemann, Felix Klein, et al.) passim

This is a lucid exposition of the interrelationship of geometry and the emerging group theory in the 19th century. It starts with a short exposition of Galois' work in modern terminology, connecting both the mathematical introduction of finite groups and historical information on their origin in early Galois theory and the investigation of permutation groups. The middle part of the book deals with three aspects of the development of geometry in the 19th century, the rise of projective geometry, of non-Euclidean geometries, and of higher-dimensional vector spaces (and hypercomplex number systems).
That sets the stage for an exposition of the central ideas of Lie's theory of continuous transformation groups and Felix Klein's "Erlanger Programm''. Here again the author's style is that of a lucid blend of exposition of mathematical ideas in modern notation and terminology with historical and biographical information. In Lie's case he includes a short discussion of the latter's intended use of the new concept in analysis (differential equations) and geometry (contact transformations).
The author attains a very lucid narrative of his story directed to a mathematically literate, broad audience. This is partly due to the export of the bulk of the more detailed information, be it historical or mathematical, into an extensive appendix of notes, which in total is of the same length as the core text. These notes fill in many of the historical details and contain interesting side remarks referring to later mathematical developments and recent mathematical textbooks. The cited literature is of a remarkable international range and is thus as useful for the Western reader without any restriction.
Thus the translation of this book into English is a useful undertaking. It can be warmly recommended to any reader interested in the subject matter who is not too keen on details of the latest historiographical research.

Reviewed by Erhard Scholz