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Its lively, fast-paced style makes for enjoyable reading, and the mathematical exposition is generally very lucid with thoughtful, well-chosen examples. For example, the connection between geometrical symmetry and the Galois group of a quintic equation is nicely illustrated in Chapter 1, which discusses the achievements of Lagrange, Ruffini, Abel, and Galois.

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

Yaglom also succeeds in conveying the essential ingredients which distinguish a Lie group from its associated Lie algebra and in explaining how these structures are related to work of Cayley, Hamilton, and Grassmann on n-dimensional spaces and hypercomplex number systems. His most penetrating remarks, however, are generally reserved for the supplementary notes, which fill nearly as much space as the text itself of the pages.

More often than not, the author relies on obsolete or dated sources e. Indeed, his often idiosyncratic reflections on the major figures discussed in this book would appear to be based on a combination of folklore, conjecture, and a superficial reading of sometimes notoriously unreliable secondary works. Errors of fact or interpretation abound on nearly every page, and the author himself even feels obliged to straighten the record out from time to time by correcting his own oversimplified statements through clarifications that appear in the notes.

The problem is that by no stretch of the imagination can Lie and Klein properly be called students of Jordan.

The Discovery of Lie Groups and Algebras and Their Properties

Whether or not they actually studied it carefully is another matter, and its actual influence on them is certainly debatable. And although both certainly hoped to widen their horizons while in Paris, they had no formal affiliation with any institution or personduring their brief stay there. The decision whether to offset the post to the candidate was made after the lecture was discussed. The twenty-three-year-old Klein chose as his subject a Comparative review of recent research in geometry iust as, in a similar situation, eighteen years before, Kiemann had spoken On the hypotheses that lie at the foundations of geometry.

The lecture soon became known as The Erlangen Program, a title which underscores both the broad vistas opened by Klein for further progress in geometry and his clear standpoint.

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In this passage,however, Yaglom finds severalways to embellish on this standarderror. Only after the candidatewas formally appointed would he be asked to deliver such an address. At times, however, he exhibits a tendency to engage in gross oversimplification. Lie himself acknowledged the fact that he was a horribly muddled writer, and anyone who has struggled with his collected works will surely concur with this opinion.

These weaknesses reflect many of the standard problems that arise when mathematicians undertake historical studies of their discipline. In the present case, the author clearly has a solid grasp of the mathematics under discussion and considerable insight into the modern developments that have grown up out of them.

What is lacking here, however, is historical sensibility, and without that the history of mathematics can never be more than a playground for anecdotes, tall tales, and a fundamentally ahistorical interest in mathematics as a collection of disembodied ideas.

Lie and Klein step on stage briefly in chapter 2 as the "pupils" of Camille Jordan the leadiang heir to Galois's legacy during their brief visit to Paris in Yaglom then goes back to recount the emergence of projective geometry during the preceding 50 years, highlighting the work of Poncelet, Mobius, Steiner, Plucker, and Chasles.

He then gives a reasonably complete narrative of contemporaneous developments in the field of non-Euclidean geometry, rightly chiding Gauss for his failure to promote the work of Bolyai and Lobachevsky, despite the fact that it was fully in accord with his own unpublished ideas.

Felix Klein's Erlangen Program

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