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Rating: 
Summary: Makes the others Look bad
Review: I got this book as a gift from a long time friend. He had trouble with reading it. It is only for that reason I give it only 4 stars. These authors make others that I have read on this range of subjects look bad: Fields Medalists included! A lot of it is that they just bother to give you the real mathematics with examples. I think the initial miss definition of the Riemann surface gives a false impression, because the explanations of ramified covers and toral elliptic lattices is just wonderful. Reading this book makes Dr. Singerman's papers look so much better! I was disappointed in the treatment of triangle groups, but the treatment of modular functions and gamma1 and gamma2 makes up for that. It is a masterful work... the best I have seen by a modern author. It reminds me of books by Ulam or Russell. Sawyer's little book is not as good!
Rating: 
Summary: The unity of math!
Review: The popular press leaves us with the impression that math is
intimidating. This wasn't always the case. In my time, the approach to how we teach math, and write books about it, went through a number of cycles, or trends; some of them now discredited;--or not!? Here is a sample: (1) I grew up with the boot-camp approach with its endless drills, (2) then came "The New-Math approach", followed by (3) "The back-to-basics" trend. (4)Following Eric Temple Bell, it became popular for a time to mix into the teaching of math a lot of history/ or dramatic stories about the heros in the subject. Finally, more recently:(5) "The Make-it-Seem-Easy-and Fun approach" and the motivational speakers; imitating popular TV shows.---Seriously, what I like about this lovely book is that it treats mathmatics as one unified subject, and that the authors masterfully highlight a number of unexpected connections between what otherwise are thought of as isolated specialties within math: The exciting new problems are at the same time also the old and classic problems in math: The elliptic integrals of Abel and Gauss, Jacobi's theta functions, modular functions, quadratic fields, elliptic curves, and Mordell-Weil. It is all beautifully presented. The book is selfcontained, and it is a pleasure to read. The clear and concise presentation is what makes the subject seem easy, or more importantly interesting and useful. I hope it will be a model for other math books to follow.
Rating:
Summary: Highly recommended
Review: This book avoids the traps which would make this subject so inaccessible. Rather than frightening the reader with group theory and the sort of very advanced material that would fit it into a post graduate slot, the book starts with very little beyond geometry and complex number theory. The book carefully progresses to discussions on the projective line, and Riemann surfaces (never too much at once) to the inevitable subjects of the Icosohedral group, and invariant theory. It manages to do this almost without you noticing the depth of maths that is being covered - quite a feat!From here on, elliptic integrals are discussed, and the work of Jacobi, Gauss, Legendre and Abel discussed freely, with many examples and clear pictures. The text is interspersed with exercises (some of which you can do with a few moments thought, others more difficult). I enjoyed this section (and the remainder of the book) for several very interesting short accounts of subjects slightly tangential to the main material. [One of my favorites was the account of a letter with a amazingly strange but elegant identity with a continued fraction sent by Ramanujan to Hardy, and Hardy's subsequent absolute amazement... You MUST NOT miss reading that, even if it isn't what you picked the book up for!] Then the book goes into the area I bought the book for - modular groups, and the solution of the Quintic. This subject draws mostly on work by Hermite, and later, Klein, but is presented carefully and slowly. I was very glad to find this book. It doesn't race through the subject at breakneck speed, which is what some books on Galois Theory or Algebraic Curves do, and has illuminated quite a few additional topics for me. I guess that now I will be able to recognize the origins of so much hard maths now (and all those entries in the tables of integrals I never understood) After all, this subject is now very important. Elliptic curves occur in many subjects - Cryptography, Information Theory, and of course, the proof of Fermats last theorem.
Rating:
Summary: long on content, short on abstract nonsense
Review: This is a great book because it presents some of the neatest topics in mathematics, without the usual discouraging layers of abstraction and notation. It attacks the topics historically so you get some idea of the motivation and steps followed, instead of a compendium of the most general results and their most elegant proofs. Also, as a previous reviewer mentioned, the book derives the bizarre and amazing continued fraction formula from Ramanujan's letter to Hardy. I had always wanted to see this, ever since reading "The Man Who Knew Infinity." It is satisfying to see this demystified, even if you don't fully master the argument. If you literally have not seen most of these topics before, as I had not, you won't find this an easy read, but it's well worth while. I spent a long time on it, and couldn't absorb it all, but I plan to read it again one day.
Rating:
Summary: long on content, short on abstract nonsense
Review: This is a great book because it presents some of the neatest topics in mathematics, without the usual discouraging layers of abstraction and notation. It attacks the topics historically so you get some idea of the motivation and steps followed, instead of a compendium of the most general results and their most elegant proofs. Also, as a previous reviewer mentioned, the book derives the bizarre and amazing continued fraction formula from Ramanujan's letter to Hardy. I had always wanted to see this, ever since reading "The Man Who Knew Infinity." It is satisfying to see this demystified, even if you don't fully master the argument. If you literally have not seen most of these topics before, as I had not, you won't find this an easy read, but it's well worth while. I spent a long time on it, and couldn't absorb it all, but I plan to read it again one day.
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