Felix Klein and Sophus Lie: Evolution of the Idea of Symmetry in the Nineteenth Century
By I. M. Yaglom
* Publisher: Birkhauser
* Number Of Pages: 237
* Publication Date: 1988-04
* ISBN-10 / ASIN: 0817633162
* ISBN-13 / EAN: 9780817633165
* Binding: Hardcover
Summary: A great history of geometry and groups in the 19th century
Rating: 5
Starting with Galois and his contribution to the evolving subject of group theory Yaglom gives a beautiful account of the lives and works of the major players in the development of the subject in the nineteenth century: Jordan, who was a teacher of Lie and Klein in Paris and their adventures during the Franco-Prussian War. Monge and Poncelet developing projective geometry as well as Bolyai, Gauss and Lobachevsky and their discovery of hyperbolic geometry. Riemann's contributions and the development of modern linear Algebra by Grassmann, Cayley and Hamilton are described in detail. The last two chapters are devoted to Lie's development of Lie Algebras and his construction of the geometry from a continuous group and Klein's Erlanger Programm unifying the different approaches to geometry by emphasizing automorphism groups. These last pages are definitely the climax of the book.
All of this only occupies the first half of the book, the second half consists of extensive notes to the former, intended to help the reader deepen her understanding of the subject in a second reading.
I think this book is very readable and should be accessible to any interested reader who has a slight understanding of the terms used so far in this review. In particular I enjoyed the details of the lives of the mathematicians involved and how the whole story seemed to culminate in the works of Lie and Klein.
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