Analysis by Its History E. Hairer & G. Wanner Published by Springer, as part of their Undergraduate Texts in Mathematics series. ISBN 0-387-94551-2 http://www.springer-ny.com Introduction: For many first year Mathematics undergraduates an introduction to Analysis is often accompanied by a sensation verging close to shell shock. One moment they are at one with the world, enjoying watching their familiarity with the Calculus and Linear Algebra, first established at high school, bloom and grow. The next moment they find themselves trapped on foreign soil, menaced by predicates, quantifiers and a professor hurling propositions and connectives at them with seemingly endless ferocity. This may be somewhat of an exaggeration, but nevertheless it is undoubtably the case that one of the major hurdles that faces the student in understanding and gaining confidence with Analysis is the apparent 'disconnected' and almost artificial nature of the subject. This is where 'Analysis by Its History' by E. Hairer and G. Wanner comes in to lend a helping hand. Audience: Mainly of interest to those studying undergraduate level Analysis, although possibly also having some appeal for those studying the natural sciences and/or engineering subjects who are curious about the development and definitions of the mathematics they regularly use. Pros: Places analysis within its historical context, illustrating the connections between definitions and concepts used in analysis with the practical problems that led to their formulation. Covers topics in Analysis chronologically (i.e. beginning with the infinite series as they first arose in antiquite, progressing to the established mathematical rigour of the 19th century), thus offering a different method of presentation to the conventional 'back-to-front' method. Clearly written with many illustrative examples and a vast number of historical quotes begging to be appropriated for use on personal websites. Cons: Occassionally topics are covered slightly too quickly, with scant mention to or explanation of set theory and logic. Cursed with the unspeakably vile yellow cover of Springer publications (and don't even begin to pretend that one of the concerns when buying textbooks isn't how cool it will look on your bookshelf). Conclusion: A valuable addition to the library of anyone wanting to study pure mathematics but is left feeling uncertain and lacking confidence in their understanding of analysis by lectures and other textbooks. Having said this, however, 'Analysis by Its History' cannot be recommended as a 'main' textbook for the subject. It proves most useful in setting topics covered in more orthodox books, such as 'A First Course in Mathematical Analysis' by J.C. Burkill, or even online lecture notes (I recommend those of Vitali Liskevich of the University of Bristol for a good coverage of set theory, logic, and analysis. But, you'll understand, I am exceptionally biased in this matter), in some kind of wider context.