John Stillwell's latest book, "Elements of Mathematics," is a fine ~400-page overview of the field of mathematics. The chapter titles lend a sense of the range of material in the volume:
Arithmetic
Computation
Algebra
Geometry
Calculus
Combinatorics
Probability
Logic
...meanwhile, the first chapter touches on "Elementary," and the final chapter on "Advanced" topics in math
Obviously, entire books have been written on each of these topics, so the 30-40 pages Stillwell delegates to each subject can only hit upon various historical, philosophical, and mathematical high points. Yet the volume does definitely go beyond elementary concepts. The writing is clear, succinct, organized, and the diagrams/illustrations excellent. It is a rich, but somewhat dry read (as most mathematics is), and not directed to a lay audience, so much as the college-level math student or professional. While some of the discussion is introductory or elementary, it always leads to deeper, more challenging ideas. The entire last chapter brings up still more advanced tidbits related to each of the topics already covered. I especially like the way the volume weaves between historical, philosophical, elementary, and more modern content, as well as math problems/examples, without lingering too long in any one spot.
I think this will make a fine basic addition to most mathematicians' bookshelves, unless you are so specialized that general math overviews don't interest you. The only other question is whether you may already have enough of this content on your shelf already, and even then I suspect you will find several new, interesting ideas in Stillwell's presentation, including perhaps the discussion of "reverse mathematics" within modern-day logic.
This isn't one of the most scintillating reads in popular math so far this year, but for the pure mathematician it is one of the more instructive.
Stillwell is the author of several other math volumes as well:
http://www.amazon.com/John-Stillwell/e/B001IQWNS2
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